Edinburgh Hodge Institute

Abstract: Speaker: Simon Goodwin (Birmingham) Title: Representations of finite W-algebras Abstract: To each nilpotent element e in a complex semisimple Lie algebra \g, one can associate a finite W-algebra denoted by U(\g,e). This algebra can be viewed as the enveloping of the Slodowy slice through the adjoint orbit of e, and has many connections to other areas of Lie theory. After presenting some history and motivation we will present an approach, due to Brundan, Kleshchev and the author, to highest weight representation theory of finite W-algebras. There is not a natural comultiplication on finite W-algebras; however, it is possible to give the tensor product of a U(\g,e)-module with a finite dimensional U(\g)-module the structure of a U(\g,e)-module. We will discuss properties of these tensor products, which are expected to be of importance in understanding the representation theory of U(\g,e).

Abstract: Title: The grammar of the word problem Speaker: Sarah Rees, University of Newcastle. Abstract: I shall discuss the word problem for groups, introduced a century ago by Dehn, and solved by him for surface groups. I am interested in how the structure of the set of words representing the identity element of a group (this set is called the `word problem' for the group) is related to the structure of the whole group. In the late 1980's Muller and Schupp proved that the word problem for a group can be recognised using a pushdown automaton (a machine using a simple stack memory) precisely when the group is virtually free. In this case it is Dehn's algorithm that can be programmed on the pushdown automaton. A set of words that can be recognised using a pushdown automaton can be constructed using a context-free grammar, and conversely. The proof of Muller and Schupp's result is based on the realisation that the underlying context-free grammar of the word problem puts a restriction on the geometry of the group. There's a correspondence between the types of machines that recognise sets of strings and the grammars that build them, but this result suggests that the grammar constructing the word problem of a group is more clearly related to the structure of the group than is the machine that recognises it. I shall report on my recent joint work with Holt and Shapiro. This examines the grammar associated with word problems that can be solved using a generalisation of Dehn's algorithms developed by Goodman and Shapiro; we see that in this case the grammar is always `growing context-sensitive'. We extend Goodman and Shapiro's work, and find a host of examples of groups with word problems that are context-sensitive but not growing context-sensitive. Hence we can answer questions of Kambites and Otto, who found the first example of a word problem in that category.

Abstract: Speaker: Jasper Stokman Title: Double affine Hecke algebras and bispectrality Abstract: One of the Macdonald conjectures is the duality -by now- theorem. The duality theorem points out the bispectral nature of the Macdonald polynomials. It was proven by Cherednik using the double affine Hecke algebra. In this talk I will establish the interplay between the double affine Hecke algebra and bispectrality on a more fundamental level. It leads to a bispectral version of the quantum Knizhnik-Zamolodchikov equations and to an integrable bispectral problem associated to the Macdonald operators. This is joint work with Michel van Meer.

Abstract: Speaker: Jaimal Thind (Stony Brook) Title: Coxeter Elements and Periodic Auslander--Reiten Quiver Abstract: Traditionally, to study a root system $R$ one starts by choosing a set of simple roots $\Pi\subset R$ (or equivalently, polarization of the root system into positive and negative parts) which is then used in all constructions and proofs. We discuss a different approach, starting not with a set of simple roots but with a choice of a Coxeter element $C$ in the Weyl group. We show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of $R$ with a certain subset $\Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $\Ihat$ has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram $I$: roots are vertices of $\Ihat$, and the root lattice and the inner product admit an explicit description in terms of $\Ihat$. Time permitting we will discuss how this picture can be used to obtain a description of the corresponding Lie algebra. (This is joint work with A. Kirillov Jr)

Abstract: Speaker: Chris Smyth (Maxwell Institute) Title: Integer symmetric matrices and Coxeter graphs

Abstract: Speaker: Mathieu Carette (Brussels) Title: The automorphism group of accessible groups

Abstract: Speaker: Will Turner (University of Aberdeen) Title: Representation theory and four dimensional topology

Abstract: Speaker: Andrew Duncan (Newcastle) Title: From pregroups to groups: decision problems and universal theory

Abstract: Speaker: Paul Turner (Heriot-Watt) Title: Homology of algebras with coefficients in a graph Abstract: Starting with a directed graph, I will describe the construction of a homology theory for algebras, related to the Khovanov homology of graphs. When the graph is the n-gon, this homology agrees with Hochschild homology up to degree n.

Abstract: Speaker: Paul Martin (University of Leeds) Title: The complex representation theory of the Brauer algebra

Abstract: Speaker: Nicolas Guay (Maxwell Institute) Title: Representations of double affine Lie algebras

Abstract: Speaker: Rudolf Tange (University of York) Title: The centre of the universal enveloping algebra in characteristic p. Abstract: Let U be the universal enveloping algebra of the Lie algebra g of a reductive group G over an algebraically closed field of characteristic p and let Z be the centre of U. The algebraic variety corresponding to Z is called the Zassenhaus variety of g. Unlike in characteristic 0, Z is not a polynomial ring, in fact the Zassenhaus variety is not smooth. I will show (under certain mild assumptions) that Z is a unique factorisation domain and that its field of fractions is purely transcendental over k (i.e. the Zassenhaus variety is rational). If time allows I will indicate the relevance of the Zassenhaus variety for the representation theory of g and a relation with the Gelfand-Kirillov conjecture.

Abstract: Speaker: Saul Schleimer (University of Warwick) Title: Polynomial-time word problems Abstract: We will sketch a proof that Aut(G) has polynomial-time word problem when G is a word hyperbolic group. The heart of the argument is the idea from computer science; straight-line programs are widely studied in the field of data compression. As it so happens, they are also well suited for analyzing group automorphisms.

Abstract: Speaker: Pepe Burillo (Barcelona) Title: Higher-dimensional Thompson's groups Abstract: Higher-dimensional analogs of Thompson's group V have been introduced recently by Brin. We will recall their definition and find the analog of the standard interpretation of Thompson's groups by tree pair diagrams. We will use this interpretation to give presentations for them (both finite and infinite), and to find estimates for the word metric of these groups in terms of the number of carets in the tree pair diagram. Finally, we will show that the inclusion of F, T and V in the higher-dimensional groups is exponentially distorted

Abstract: Speaker: Arun Ram (University of Melbourne) Title: Two boundary Braid groups, Hecke algebras and tantalizer algebras Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will a medley of vignettes around double affine type C braid groups and quotient algebras.

Abstract: Speaker: Pramod Achar (Louisiana State University) Title: Introduction to staggered sheaves Abstract: Perverse sheaves, introduced around 1980, have many remarkable properties, involving such notions as Poincare-Verdier duality, weight filtrations and "purity," and the celebrated Decomposition Theorem. These properties have made perverse sheaves into an incredibly powerful tool, especially for applications in representation theory. "Staggered sheaves" are a new attempt to duplicate some of these properties in the setting of vector bundles and coherent sheaves. I will discuss the ingredients that go into defining staggered sheaves, state the main results that are known so far, and perhaps speculate on potential applications. This will be an introductory talk: I will not assume any familiarity with perverse or staggered sheaves, and I will try to focus on examples on A^1 or P^1.Some of the results on staggered sheaves are joint work with D. Sage and D. Treumann.

Abstract: Speaker: Rick Thomas (University of Leicester) Title: FA-presentable structures Abstract: We are interested in the notion of computing in structures. One approach would be to take a general model of computation such as a Turing machine. A structure would then be said to be computable if its domain can represented by a set which is accepted by a Turing machine and if there are decision-making Turing machines for each of its relations. However, there have been various ideas put forward to restrict the model of computation used; whilst the range of possible structures decreases, certain properties of the structures may become decidable. One interesting approach was introduced by Khoussainov and Nerode who considered structures whose domain and relations can be checked by finite automata; such a structure is said to be FA-presentable. This was inspired, in part, by the theory of automatic groups; however, the definitions are somewhat different. We will survey some of what is known about FA-presentable structures, contrasting it with the theory of automatic groups and posing some open questions. The talk is intended to be self-contained, in that no prior knowledge of these topics is assumed. Reply

Abstract: Speaker: Max Neunhoeffer (University of St Andrews) Title: Finding normal subgroups Abstract: In the context of the Matrix Group Recognition Project the following is an important task: Given G=< g_1, ..., g_k >, find a non-trivial element x in G that is contained in a proper normal subgroup or fail if G is simple. In this talk I explain why this problem is important and present some ideas how to tackle it.

Abstract: Speaker: Susan Cooper (University of Nebraska - Lincoln) Title: In Search of Exactness Abstract: Certain data about a finite set of distinct, reduced points in projective space can be obtained from its Hilbert function. It is well known what these Hilbert functions look like, and it is natural to try to generalize this characterization to non-reduced schemes. In particular, we consider a fat point scheme determined by a set of distinct points (called the support) and non-negative integers (called the multiplicities). In general, it is not yet known what the Hilbert functions are for fat points with fixed multiplicities as the support points vary. However, if the points are in projective 2-space and the number of support points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities (due to Guardo-Harbourne and Geramita-Harbourne-Migliore). In this talk we focus on what can be said, in projective 2-space, given information about what collinearities occur among the support points. Using this information we measure how far related sequences can be from being exact on global sections. Doing so, we obtain upper and lower bounds for the Hilbert function of the fat point scheme. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case. This is joint work with B. Harbourne and Z. Teitler.

Abstract: Speaker: Charudatta Hajarnavis (University of Warwick) Title: Polynomial Identity Rings of Finite Global Dimension Abstract: A commutative Noetherian ring of finite global dimension is a direct sum of integral domains (including fields). In the dimension 1 case (i.e. hereditary rings) these are Dedekind domains. In the non-commutative case there is an extensive theory of hereditary rings showing a much more complex situation. In this survey talk we look at the situation for dimension 2 and higher and also mention some recent work.

Abstract: Speaker: Emmanuel Letellier (Université de Caen) Title: Topology of character varieties and Macdonald polynomials Abstract: We conjecture a formula for the mixed Hodge polynomials of representations varieties of the fundamental group of punctured Riemann surfaces in terms of Macdonald polynomials. In this talk we will bring evidences for this conjecture and see some applications in the representation theory of quivers.

Abstract: Speaker: Vladimir Dotsenko (University of Dublin Trinity College) Title: Parking functions and vertex operators Abstract: The goal of this talk is to discuss several series of graded vectors spaces whose series of dimensions include the series of Catalan numbers (and their generalisations), and the sequence (n+1)^{n-1} of "parking functions numbers". First of all, we show how these vector spaces arise from representation-theoretical constructions for some associative algebras. Another way to construct vector spaces with same dimensions and graded characters is to consider spaces of global sections of certain vector bundles on (zero fibres of) Hilbert schemes (for the latter the dimension and character formulae were obtained by Haiman in his works on diagonal harmonics and the "n! conjecture"). I shall formulate a conjecture relating these two constructions and try to explain some reasons for this conjecture to be true.

Abstract: Speaker: Nick Gilbert Title: Diagram groups and rewriting for words and trees.

Abstract: Speaker: Natalia Iyudu (University of Belfast) Title: Quadratic algebras: the Anick conjecture, representation spaces and Novikov structures

Abstract: Speaker: Toby Stafford (University of Manchester) Title: Equidimensionality for Cherednik algebras

Abstract: Speaker: Kenny Brown (University of Glasgow) Title: Connections between generic q and roots of unity: q-modular systems

Abstract: Speaker: Sinead Lyle (University of East Anglia) Title: Jucys-Murphy elements and homomorphisms between Specht modules. Abstract: In the representation theory of the symmetric group, an important open problem is to determine the structure of certain objects known as Specht modules. I will talk about a method of constructing homomorphisms between pairs of Specht modules using the Jucys-Murphy elements. This is joint work with Andrew Mathas.

Abstract: Speaker: Sue Sierra (University of Washington) Title: Primitivity of twisted homogeneous coordinate rings Abstract: Let B = B(X, L, f) be the twisted homogeneous coordinate ring associated to a complex projective variety X, an automorphism f of X, and an appropriately ample invertible sheaf L. We study the primitive spectrum of B, and show that there is an intriguing relationship between primitivity of B and the dynamics of the automorphism f. In many cases Dixmier and Moeglin's characterization of primitive ideals in enveloping algebras generalizes to B; in particular, this holds if X is a surface. This is joint work with J. Bell and D. Rogalski.

Abstract: Speaker: Jochen Heinloth (University of Amsterdam) Title: Twisted groups on curves and some related moduli spaces Abstract: To study the spaces of bundles on a Riemann surface (or algebraic curves), the so called uniformization theorem has been a very useful tool. This result says that these spaces can be viewed as a quotient of the space of all maps of the circle into a Lie group. A similar result has been conjectured by Pappas and Rapoport for spaces of bundles equipped with different types of extra structure. I would like to explain, what these are, how they are related to twisted loop groups and why the general setup allows to give a short proof of the conjecture. At the end of the talk I will try to indicate an application of this to point counting arguments for these moduli spaces.

Abstract: Speaker: Vic Reiner (University of Minnesota). Title: Extending the coinvariant theorems of Chevalley, Shephard-Todd and Springer. Abstract: (This is joint work with B. Broer, L. Smith, and P. Webb.) The theorems in the title are classical results in the invariant theory of finite subgroups of GL_n(C) generated by reflections. After reviewing these results, we show how to extend them in several directions, removing many of their hypotheses. In particular, our results work over an arbitrary field k rather than the complex numbers. Also our version of the Chevalley-Shephard-Todd theorem applies to any finite subgroup of GL_n(k), not just reflection groups. If time permits, we will mention the combinatorial applications in characteristic p that motivated us.

Abstract: Speaker: Brendan Owens (Glasgow) Ttile: Knot surgeries bounding definite 4-manifolds

Abstract: Speaker: Tara Brendle (Glasgow) Title: The symmetric Torelli group

Abstract: Rouquier families for Hecke algebras

Abstract: Speaker: Gerald Williams (Essex) Ttile: Finiteness of some cyclically presented and Fibonacci-like groups

Abstract: Multiplicity-Free Actions of Classical Groups

Abstract: Mirror symmetry, Langlands duality and the Hitchin system

Abstract: Speaker: Alina Vdovina Title: Cayley graph expanders, pro-p-groups and buildings

Abstract: http://www.maths.abdn.ac.uk/artin/meeting.php?id=21

Abstract: http://www.maths.abdn.ac.uk/artin/meeting.php?id=21

Abstract: TBA

Abstract: Speaker: Owen Cotton-Barratt (Oxford) Title: When good groups go bad Abstract: Much of group theory is concerned with whether one property entails another. When such a question is answered in the negative it is often via a pathological example. The Rips construction is an important tool for producing such pathologies. We will consider the construction and a recent refinement which makes the output group conjugacy separable. The motivation for this was an application in profinite group theory; the context for this theorem will be described.

Abstract: Speaker: Tom Lenagan (UoE) Ttile: From totally nonnegative matrices to quantum matrices and back, via Poisson geometry

Abstract: Speaker: Mark Lawson (HWU) Title: A non-commutative generalization of Stone duality

Abstract: Speaker: David Jones (HWU) Title: Strong representations of the polycyclic monoids: cycles and atoms.

Abstract: TBA

Abstract: Speaker: Michael Heusener (Clermont-Ferrand) Title: Infinitesimal inflexibility under Dehn filling (joint work with Joan Porti) Abstract: A closed hyperbolic 3-manifold inherits a natural projective structure. Though the hyperbolic structure is rigid (Weil,Mostow), the projective one may be rigid or not. Johnson and Millson provide nonrigid examples, by means of bending along totally geodesic surfaces. Rigid examples are constructed by Cooper, Long and Thistlethwaite and are called \emph{inflexible}. The aim of this talk is to present a method to construct infinite families of closed inflexible manifolds. In particular we shall show that for $n$ sufficienly large, the homology sphere obtained by $1/n$-Dehn filling on the figure eigth knot is infinitesimally inflexible.

Abstract: In general, local models are schemes defined over DVRs. In all examples known to me, one of the most important techniques to study a given local model is to embed its special fiber in an appropriate affine flag variety; in this way, the special fiber becomes stratified into Schubert cells. In this talk I will discuss some of the combinatorial and algebro-geometric problems in the affine flag variety that arise from these considerations. As in my other talk, the emphasis will be placed heavily on understanding concrete examples.

Abstract: Speaker: Martin Bridson (Oxford) Ttile: Curvature, dimension, and representations of mapping class groups Abstract: In this talk I'll discuss constraints on the way in which mapping class groups of surfaces can act on spaces of non-positive curvature and explain how these constraints lead to conclusions about homomorphisms between mapping class groups, and how they inform us about the (linear) representation theory of such groups.

Abstract: Speaker: Cornelius Reinfeldt (HWU) Title: The structure of homomorphisms into hyperbolic groups

Abstract: Speaker: Gwyn Bellamy Title: Cuspidal representations for rational Cherednik algebras

Abstract: In the theory of quantum groups, Lie subalgebras of semisimple Lie algebras should be realised as coideal subalgebras of quantised enveloping algebras. While many classes of such coideal subalgebras of are known, there is so far no general classification. In this talk, a classification of coideal subalgebras of the positive Borel part of a quantised enveloping algebra is presented. The result is expressed in terms of characters of quantisations of nilpotent Lie subalgebras, which were introduced by de Concini, Kac, and Procesi for any element in the Weyl group. The study of such characters naturally leads to fun Weyl group combinatorics. The talk is based on joint work with I.~Heckenberger.

Abstract: Schubert calculus answers enumerative questions such as, how many lines meet four given (sufficiently general) lines in 3-dimensional space? The answer to such a question is a non-negative integer, and can be arrived at by a calculation in the cohomology ring of the variety of lines in 3-dimensional projective space. Analogous calculations can be carried out for more general cohomology theories and more general homogeneous spaces. In these more general settings it may not be clear what the answers are enumerating, that they should be positive, or even what "positive" should mean. We'll explain what type of positivity holds for equivariant K-theory of compact homogeneous spaces, and how to prove it using souped-up versions of Kleiman transversality and Kodaira vanishing.

Abstract: Speaker: Tom Leinster (Glasgow) Title: Rethinking set theory Abstract: At the heart of mathematical culture is a niggling worry. We use basic set-theoretic language all the time, and we are informed that ZFC is the "foundation of mathematics". Yet most of us sail through life neither knowing nor much caring what the axioms of ZFC are; and if we do stop to look at the axioms, they seem curiously remote from what we actually do as mathematicians. I will present a solution to this problem, due to Lawvere. It is a radical reshaping of set theory. The axioms boil down to 10 totally mundane properties of sets, used every day by ordinary mathematicians. In this way, I hope to persuade you that set theory is not to be sniffed at.

Abstract: Speaker: Christopher Phan (Glasgow) Title: Generalised Koszul properties for noncommutative graded algebras Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.

Abstract: I will introduce some of the important models of computation, give some examples of important techniques, highlight some central results and address some open questions.

Abstract: Joint with Geometry

Abstract: Cluster Algebras come equipped with a natural Poisson bracket , or potentially a family of Poisson brackets which are "related" via the actions of an algebraic torus. In this talk I will show how to classify the torus invariant Poisson prime ideals of a cluster algebra using the combinatorial data obtained from the exchange matrix. Finally I will apply this method to the case of the cluster algebra of functions on matrices. Of course, I will also introduce all the relevant notions.

Abstract: Jorgensen is speaking on "A Caldero-Chapoton map for infinite clusters" Minasyan is speaking on ""Fixed subgroups of automorphisms of non-positively curved surfaces."

Abstract: Some aspects of the modular representation theory of a finite group can be described by a tree. Such trees have been determined for almost all finite simple groups, but some cases remain unknown. Starting from the example of the group SL2(q) I will explain how geometric methods can be used to solve this problem for finite reductive groups.

Abstract: Title: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators''

Abstract: Title: "Quadratic algebras: the Anick conjecture on Hilbert series, Koszulity, NCCI and RCI" Abstract: We present several results on the Anick conjecture which asserts that the lower bound for the Hilbert series, known as the Golod-Shafarevich estimate is attained on generic quadratic algebra. The technique (due to Anick), allowing to write down precisely the formula for the Hilbert series will be demonstrated. We will discuss also related questions of Koszulity and being noncommutative complete intersection (NCCI). Connections to the latter property on the level of finite dimensional representations, namely, introduced by Ginsburg and Etingof notion of representational complete intersection (RCI) will be considered and some examples given.

Abstract: Title: Braid group actions on quantum symmetric pair coideal subalgebras Abstract: It was noted recently by Molev and Ragoucy, and idependently by Chekhov, that the nonstandard quantum enveloping algebra of so(N) allows an action of the Artin braid group. We interpret and generalize this action within the theory of quantum symmetric spaces.

Abstract: Title: Additive polylogarithms Abstract: In this talk we will define additive polylogarithms and describe how they are related to motivic cohomology over the dual numbers of a field of characteristic zero. In the characteristic p case, and in weight 2, we will also describe how the additive dilogarithm is related to Kontsevich's logarithm.

Abstract: Title: Mutation of rigid objects and partial triangulations. Abstract: By several results (due to Amiot, Fomin--Shapiro--Thurston, Labardini-Fragoso and Keller--Yang) a cluster category can be associated with any compact Riemann surface with boundaries and marked points. The triangulations of the marked Riemann surface correspond to the so-called cluster-tilting objects of the cluster category. These objects are of particular interest since they categorify the clusters of Fomin--Zelevinsky's cluster algebras. In particular, they have a nice theory of mutation. This mutation turns out to be the categorical analogue of the flip of triangulations. Brustle--Zhang proved that some more general objects, the rigid objects, categorify the partial triangulations of the surface. In this talk, based on a joint paper with Robert Marsh, I will explain how both flips and mutations can be generalised to this situation. Our main tool is a result showing that Iyama--Yoshino reduction for cluster categories correspond to cutting along an arc the associated Riemann surface. All statements and results will be illustrated with some (small) geometric examples.

Abstract: Title: Canonical birationally commutative factors of noetherian graded algebras Abstract: It is known that if a graded k- algebra R is strongly noetherian (that is, it remains noetherian upon commutative base-change), then there is a canonical map from R to a twisted homogeneous coordinate ring on some projective scheme. We show this can be generalized to algebras that are merely noetherian, and the resulting factor satisfies a universal property. Further, we show that under suitable conditions on the geometry of the Hilbert schemes of point modules over R, this canonical factor is a naive blowup algebra, in the sense of Keeler-Rogalski-Stafford.

Abstract: Title: Some quantum analogues of properties of Grassmannians Abstract: The classical coordinate ring of the Grassmannian has many nice structural properties and one expects these to carry over to its quantum analogue. We will discuss two properties for which this does indeed happen, namely a cluster algebra structure (recent work with Launois, quantizing work of Scott) and an action of the dihedral group (work with Allman, extending a recent construction of Launois and Lenagan). We will also mention an extension in a different direction, namely to infinite Grassmannians (work with Gratz).

Abstract: Title: "Slow but not too Slow: Nil Algebras and Growth"

Abstract: Title: "Parahoric torsors and Parabolic bundles on compact Riemann surfaces and representations of Fuchsian groups." Abstract: Let X be an irreducible smooth projective algebraic curve of genus g ≥ 2 over the ground field of complex numbers and let G be an arbitrary semisimple simply connected algebraic group. The aim of the talk is to introduce the notion of a semistable and stable parahoric torsor under certain Bruhat-Tits group schemes and construct the moduli space of semistable parahoric G –torsors and identify the underlying topological space of this moduli space with spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles. The talk is on a joint work with C.S. Seshadri.

Abstract: Title: Fourier--Mukai partners of elliptic surfaces. Abstract: For smooth projective varieties X and Y, when their derived categories are equivalent we say that X is a Fourier--Mukai partners of Y. We study the set of isomorphism classes of Fourier--Mukai partners of elliptic surfaces with negative Kodaira dimensions.

Abstract: Title: Flops and mutations of polyhedral singularities Abstract: Let G be a finite subgroup of SO(3) and consider the so called polyhedral singularity C^3/G. It is well known that the G-Hilb is a distinguished crepant resolution which plays a central role in the so called McKay correspondence. I will explain in the talk how every crepant resolution of C^3/G is a moduli space of quiver representations showing that there exists a 1-to-1 correspondence between between flops of G-Hilb and mutations of the McKay quiver. This is a joint work with Y. Sekiya.

Abstract: Title: Totally nonnegative matrices Abstract: A real matrix is totally nonnegative if each of its minors is nonnegative (and is totally positive if each minor is positive). The talk will survey elementary properties of these matrices and present new results which have surprising links with the theory of quantum matrices.

Abstract: Title: "J. Sally's question and a conjecture by Y. Shimoda" Abstract: In 2007, Y. Shimoda, in connection with an open question of J. Sally from 1978, conjectured that a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. This talk surveys the results that have been obtained to date concerning this conjecture. First we indicate that we can reduce to the case of dimension three, and that the conjecture has a positive answer if the ring is either regular, or is complete with infinite residue field and multiplicity at most three. Finally we consider the case of the appropriate analogue of the conjecture for standard graded rings, and indicate how a mix of algebraic and geometrical methods yields a definite answer in this setting. (Joint work with S. Goto and F. Planas-Vilanova)

Abstract: Title: Finite Subgroups of the Classical Groups Abstract: A theorem of Jordan (1878) states that there is a function f on the natural numbers such that if G is a finite subgroup of GL(n,C), then G has an abelian normal subgroup of index at most f(n). Several years ago, I determined the optimal value for f(n), and I will talk about this and recent work that extends the result to the finite subgroups of all classical groups, both real and complex.

Abstract: Title: Affine W-algebras Abstract: Affine W-algebras may be considered as a generalization of infinite dimensional Lie algebras such as Kac-Moody algebras and the Virasoro algebra. They may be also regarded as an affinization of finite W-algebras, but affine W-algebras are introduced earlier than finite W-algebras in physics literature. Affine W-algebras are related with conformal field theories, integrable systems, quantum groups, the geometric Langlands program, and 4 dimensional gauge theories. In my talk I will discuss about their structure and describe their representation theory by focusing on type A cases.

Abstract: Title: Rational Cherednik algebras in positive characteristic. Abstract: In this talk I will describe some of the basic features of rational Cherednik algebras in positive characteristic. There is a close relationship between the representation theory of these algebras and the geometry of their centres. I will show how their representation theory can be used to determine when the centre of the algebra is a regular ring. This is based on joint work with M. Martino.

Abstract: Title: Modular representation theory of the Ariki-Koike algebra in characteristic 0. Abstract: The Ariki-Koike algebra is a natural generalisation of the Iwahori-Hecke algebras of types A and B. Much of its representation theory is controlled by the Schur elements, which are Laurent polynomials attached to its irreducible representations. We will give a new, pretty formula for these elements, and study the applications of our result to the representation theory of the Ariki-Koike algebra in characteristic 0

Abstract: Title: Higgs algebra of weighted projective lines and loop crystals. Abstract: In this talk we contruct enveloping algebras of loop Lie algebras via geometry, considering constructible functions on the space of Higgs bundles on a weighted projective line. The geometry of this space then leads to nice elements in the algebra, which forms a basis called the semicanonical basis. Another interested feature coming from geometry is the construction of a loop crystal, which is an analog of a crystal in the loop case.

Abstract: The abstract notion of recognition: algebra, logic and topology (Joint work with M. Gehrke and S. Grigorieff) We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topology is the key ingredient. We prove the existence of a minimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a compact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular languages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered.

Abstract: Title: On 1-dimensional representations of finite W-algebras. Abstract: 1-dimensional representations of finite W-algebras enable one to construct completely prime primitive ideals with a prescribed associated variety and quantise coadjoint nilpotent orbits. A few years ago I conjectured that all finite W-algebras admits such representations. In my talk I am going to discuss the current status of this conjecture.

Abstract: Title: Symmetric Auslander and Bass categories Abstract: We define the symmetric Auslander category $\sA^{\s}(R)$ to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains $\sA(R)$, the ordinary Auslander category. It is well known that $\sA(R)$ is intimately related to Gorenstein projective modules, and our main result is that $\sA^{\s}(R)$ is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories \[ \underline{\GMor}(R) \stackrel{\simeq}{\rightarrow} \sA^{\s}(R) / \sK^{\bounded}(\Prj\,R) \] where $\underline{\GMor}(R)$ is the stable category of Gorenstein projective objects in the abelian category $\Morph(R)$ of homomorphisms of $R$-modules. This result is set in the wider context of a theory for $\sA^{\s}(R)$ and $\sB^{\s}(R)$, the symmetric Bass category which is defined dually. This is joint work with Peter Jorgensen.

Abstract: Title: "Harish-Chandra bimodules of rational Cherednik algebras" Abstract: "Harish-Chandra bimodules are a class of bimodules defined for rational Cherednik algebras that have attracted much recent research interest. In this talk, I will attempt to explain some of the motivation behind this interest and then move on to present some results regarding Harish-Chandra bimodules of rational Cherednik algebras, with particular emphasis on the case of cyclic groups."

Abstract: Title: "Conjugacy of algebraic numbers with rational parameters" Abstract: "We consider algebraic numbers having either rational real part, rational imaginary part or rational modulus, and discuss the question of whether such numbers can share their minimal polynomial. To answer this question, we apply some Galois theory and group theory." This work is joint with Karl Dilcher and Rob Noble.

Abstract: Title: "Automorphisms of generalized R. Thompson groups via dynamics"

Abstract: Title: Height zero characters of covering groups. Abstract: The characters of height 0 of finite groups are the object of numerous theorems and conjectures. If G is a finite group, and p a prime, we write Irr_0(G) for the set of characters of p-height 0 of G. The Alperin-Mckay Conjecture states that, if B is a p-block of G with defect group D, and with Brauer correspondent b in N_G(D), then |Irr_0(B)|=|Irr_0(b)|. In 2002, Isaacs and Navarro formulated a refinement of this conjecture. For any integer 0 < k < p, we denote by M_k(B) the set of height 0 characters of B whose degree has a p'-part congruent to ± k modulo p. The Isaacs-Navarro Conjecture then states that |M_{ck}(B)|=|M_k(b)|, where c is the p'-part of the index of N_G(D) in G. In this talk, I want to present (an idea of) the proof of this result in the Schur extensions of the symmetric and alternating groups. As in the symmetric groups, it is in this case possible to exhibit an explicit bijection, by using the combinatorics that describes the characters and blocks. I also show how these groups fit within the frame of a recent conjecture by Malle and Navarro on nilpotent blocks. Finally, I want to conclude with some related results about the combinatorics we use, in particular about hooks in partitions and bars in bar-partitions.

Abstract: Title: "Geometry of random groups"

Abstract: Title: A family of 4-dimensional algebras Abstract: We construct an interesting family of algebras of global dimension and GK-dimension 4, and show that the general member of this family is noetherian and birational to P2 (in the appropriate sense). Such algebras were conjectured not to exist by Rogalski and Stafford. We show also that these algebras have counterintuitive homological properties: in particular, the Auslander-Buchsbaum formula fails for them. This is joint work with Rogalski.

Abstract: Title: On the categorification of Reid's recipe Abstract: For a finite abeilan subgroup G of SL(3,C), Reid's recipe is a combinatorial cookery that describes very simply the relations between tautological line bundles on the G-Hilbert scheme. Building on results of Cautis-Logvinenko, I'll describe joint work that reveals the importance of this cookery for the derived category of the G-Hilbert scheme.

Abstract: Title: Generation of Finite Groups

Abstract: Title: "Classifying Noncommutative surfaces: Subalgebras of the Sklyanin algebra" Abstract: Noncommutative projective algebraic geometry aims to use the techniques and intuition of (commutative) algebraic geometry to study noncommutative algebras and related categories. A very useful intuition here is that (the category of coherent sheaves over) a noncommutative projective scheme is simply the category of finitely generated graded modules modulo those of finite length over a graded algebra R. One of the major open problems here is to classify the noncommutative projective irreducible surfaces aka noncommutative graded domains of Gelfand-Kirillov dimension three. After surveying some of the known result on this question I will describe some very recent work of Rogalksi, Sierra and myself describing the subalgebras of the Sklyanin algebra.

Abstract: Title: "Spectral Curves ans Number Theory" Abstract: The modern approach to integrable systems typically proceeds via a curve, the parameters of the curve encoding the actions and its Jacobian (or possibly some related Prym) encoding the angles. Physically relevant families of curves are often described by fixed relations amongst differentials on the curve. We shall look at number theoretic properties of these curves. For many integrable systems the curves are transcendental. I shall review W\"ustholz's Analytic subgroup theorem giving simple examples before applying this in the spectral curve context.

Abstract: Speaker: Christian Korff Title: Quantum cohomology via vicious and osculating walkers Abstract: We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder. These lattice paths can be described in terms of the combinatorial R-matrix of Kirillov-Reshetikhin crystal graphs. Crystal graphs are special directed, coloured graphs which are combinatorial objects encoding the representation theory of quantum algebras. Using the path description we identify the quantum Kostka numbers of Bertram, Ciocan-Fontanine and Fulton with the cardinality of a special subset of vertices in these graphs. Speaker: James Mitchell Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set Abstract: In this talk I will review some recent results relating to the lattice of subgroups of the symmetric group and its semigroup theoretic counterpart the lattice of subsemigroups of the full transformation semigroup on an infinite set. As might be expected, these lattices are extremely complicated. I will discuss several results that make this comment more precise, and shed light on the maximal proper sub(semi)groups in the lattice. I will also discuss a natural related partial order, introduced by Bergman and Shelah, which is obtained by restricting the type of sub(semi)groups and considering classes of, rather than individual, (semi)groups. In the case of the symmetric group, this order is very simple but in the case of the full transformation semigroup it is again very complex.

Abstract: Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set

Abstract: Title: n-representation infinite algebras Abstract: We introduce a distinguished class of finite dimensional algebras of global dimension n which we call n-representation infinite. For the case n=1, they are path algebras of non-Dynkin quivers. Taking (n+1)-preprojective algebras, they correspond bijectively with (n+1)-Calabi-Yau algebras of Gorenstein parameter 1. I will discuss 3 important classes of modules, preprojective, preinjective and regular as an analogue of the classical case n=1. This is a joint work with Martin Herschend and Steffen Oppermann.

Abstract: Title : Quadratic differentials as stability conditions Abstract : This talk is about how spaces of quadratic differentials on Riemann surfaces arise as stability conditions on certain CY3 categories. These categories are defined by quivers with potential but can also be viewed (heuristically?) as Fukaya categories of symplectic manifolds. I will try to explain what all this means, and give the main idea of the construction. This is joint work with Ivan Smith, inspired by a paper of physicists Gaiotto, Moore and Neitzke.

Abstract: Title: 'The mixed Littlewood Conjecture'.

Abstract: Title: $\tau$-tilting theory Abstract: Mutation is a basic operation in tilting theory, which covers reflection for quiver representations, APR tilting modules and Okuyama-Rickard construction of tilting complexes. In this talk we introduce the notion of (support) $\tau$-tilting modules, which `completes' tilting modules from viewpoint of mutation in the sense that any indecomposable summand of a support $\tau$-tilting module can be replaced in a unique way to get a new support $\tau$-tilting module. Moreover, for any finite dimensional algebra we show that there exist bijections between (1) support $\tau$-tilting modules, (2) functorially finite torsion classes, and (3) two-term silting complexes. Moreover if the algebra comes from a 2-Calabi-Yau triangulated category, (4) cluster-tilting objects also correspond bijectively. This is a joint work with Takahide Adachi and Idun Reiten.

Abstract: We will tell some of the story around a classical elementary congruence due to Wolstenholme (1862) that deal with prime numbers. Like for the classical Fermat and Wilson congruences various generalizations were soon discovered. Surprisingly, yet another generalization was discovered only very recently. It involves Lucas sequences, which are a generalization of the Fibonacci numbers.

Abstract: Title: Rational Cherednik algebras and Schubert cells Abstract: I will recall the connection between rational Cherednik algebras, the Calogero-Moser space and the adelic Grassmannian. Then I will try to explain how one can interoperate Schubert cells, and conjecturally their intersection, in the Grassmannian in terms of the representation theory of the rational Cherednik algebra.

Abstract: Lie superhomomorphisms of superalgebras. The relationship between the associative and Lie structure of an associative algebra was studied by Herstein and some of his students in the 1950's and 1960's. After some initial partial results the complete classification of Lie isomorphisms was obtained by Matej Bre\vsar in 1993. It now seems natural to continue the investigation of Lie homomorphisms in the setting of superalgebras. Let $A=A_0\oplus A_1$ be an associative superalgebra over a field $F$ of characteristic not $2$. By replacing the product in $A$ by the superbracket $[\cdot,\cdot]_s$, $A$ becomes a Lie superalgebra. Recall that $[\cdot,\cdot]_s$ is defined for homogeneous elements $a,b\in A$ as $[a,b]_s=ab-(-1)^{|a||b|}ba$. A bijective linear map $\phi:A\to A$ is a Lie superautomorphism of $A$ if $\phi(A_i)=A_i$, $i\in \mathbb{Z}_2$, and $\phi([a,b]_s)=[\phi(a),\phi(b)]_s$ for all $a,b\in A$. We will present a characterization of Lie superautomorphisms of simple associative superalgebras, obtained in a joint work with Yuri Bahturin and Matej Bre\vsar.

Abstract: Title: Chevalley restriction theorem for vector-valued functions on quantum groups Abstract: For a simple finite dimensional Lie algebra g, its Cartan subalgebra h and its Weyl group W, the classical Chevalley theorem states that, by restricting ad-invariant polynomials on g to its Cartan subalgebra, one obtains all W-invariant polynomials on h, and the resulting restriction map is an isomorphism. I will explain how to generalize this statement to the case when a Lie algebra is replaced by a quantum group, and the target space of the polynomial maps is replaced by a finite dimensional representation of this quantum group. I will describe all prerequisites for stating the theorem and sketch the idea of the proof, most notably the notion of dynamical Weyl group introduced by Etingof and Varchenko.

Abstract: An endomorphism T of an object can be viewed as a discrete-time dynamical system: perform one iteration of T with every tick of the clock. This dynamical viewpoint suggests questions about the long-term destiny of the points of our object. (For example, does every point eventually settle into a periodic cycle?) A fundamental concept here is the "eventual image". Under suitable hypotheses, it can be defined as the intersection of the images of all the iterates T^n of T. I will explain its behaviour in three settings: one set-theoretic, one algebraic, and one geometric. I will then present a unifying categorical framework, using it to explain how the concept of eventual image is a cousin of the concepts of spectrum and trace.

Abstract: Path algebras with relations constructed from a superpotential were studied by Bocklandt, Schedler and Wemyss in 'Superpotentials and Higher Order Derivations'. I consider when deformations of these algebras have relations given by an inhomogenous superpotential. This encompasses many interesting examples, such as deformed preprojective algebras and symplectic reflection algebras.

Abstract: Title: Characterizations of left orders in left Artinian rings. Abstract: Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In my talk, three more new criteria are given.

Abstract: Stephane Launois (Kent) Efficient recognition of totally nonnegative cells. In this talk, I will explain how one can use tools develop to study the prime spectrum of quantum matrices in order to study totally nonnegative matrices.

Abstract: Adrien Brochier (Edinburgh) On finite type invariants for knots in the solid torus. Finite type knot invariants are those invariants vanishing on the nth piece of some natural filtration on the space of knots. This notion was introduced by Vassiliev and it turns out that most of known numerical invariants are of finite type. Kontsevich proved the existence of a "universal" invariant, taking its values in some combinatorial space, of which every finite type invariant is a specialization. This result involves some complicated integrals, but can be made combinatorial using the theory of Drinfeld associators. We will review this construction and explain why the naive generalization of this theory for knot in thickened surfaces fails. We will suggest a general way of overcoming this obstruction, and prove an analog of Kontsevich theorem in this framework for the case M=C^*, i.e. for knots in a solid torus. Time permitting, we will give an explicit construction of specializations of our invariant using quantum groups.

Abstract: Oleg Chalykh (Leeds) Calogero-Moser spaces for algebraic curves. I will discuss two existing definitions of Calogero-Moser spaces for curves: one in terms of Cherednik algebras, another - in terms of deformed preprojective algebras, the link between them, and explain how one can compute geometric invariants of these spaces, such as the Euler characteristic and Deligne-Hodge polynomial.

Abstract: Michele D'Adderio (University Libre de Bruxelles) A geometric theory of algebras. I will introduce some classical notions of geometric group theory (like growth and amenability) in the setting of associative algebras, and I will show how they interact with other classical invariants (like the Gelfand-Kirillov dimension and the lower transcendence degree).

Abstract: Wendy Lowen (Antwerp) On compact generation of deformed schemes. We discuss a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's Result on compact generation of quasi-compact separated schemes. We give an application of our theorem to non-commutative deformations of such schemes.

Abstract: Dirac Operators on Quantised Hermitian Symmetric Spaces In this joint work with Matthew Tucker-Simmons (U Berkeley) the \bar\partial-complex of the quantised compact Hermitian symmetric spaces is identified with the Koszul complexes of the quantised symmetric algebras of Berenstein and Zwicknagl. This leads for example to an explicit construction of the relevant quantised Clifford algebras. The talk will be fairly self-contained and begin with three micro courses covering the necessary classical background (one on Dirac operators, one on symmetric spaces, one on Koszul algebras), and then I'll explain how noncommutative geometry and quantum group theory lead to the problems that we are dealing with in this project.

Abstract: Title: Beilinson-Drinfeld factorization algebras and QFT. Abstract: I will explain what are factorization algebras and how they can be defined in very general settings of geometries with a good notion of D-modules. I will then talk about applications of this theory. In particular I will discuss the differentiable case and its relations to quantum field theory.

Abstract: Title: Morita equivalences from Higgsing toric superpotential algebras Abstract: Let A and A' be superpotential algebras of brane tiling quivers, with A' cancellative and A non-cancellative, and suppose A' is obtained from A by contracting, or 'Higgsing', a set of arrows to vertices while preserving a certain associated commutative ring. A' is then a Calabi-Yau algebra and a noncommutative crepant resolution of its prime noetherian center, whereas A is not a finitely generated module over its center, often not even PI, and its center is not noetherian and often not prime. I will present certain Morita equivalences that relate the representation theory of A with that of A'. I will also describe the Azumaya locus of A, and relate it to the Azumaya locus of A'. Along the way, I will introduce the notion of a non-local algebraic variety, and show how this notion is intimately related to these algebras.

Abstract: I will explain how some nice Lie structures appear when one is trying to compute Ext and Tor of a closed subvariety. I will use it as an excuse to introduce some nice concepts of derived geometry. I'll end the talk with a striking analogy between two great results: one in Lie theory and the other one in algebraic geometry.

Abstract: The multiple CSP is the following: Given two conjugate lists of elements A=[a_1,a_2,...,a_n] and B=[b_1,b_2,...,b_n], can we find a conjugator x such that x^{-1}a_{i}x=b_i ? We show that one can put a linear upper bound on the geodesic length of a shortest conjugator.

Abstract: I will talk about symmetries of derived categories of symmetric algebras, known as spherical twists, and explain when they satisfy braid relations. Using a more general collection of symmetries, known as periodic twists, I will explain how lifts of longest elements of symmetric groups to braid groups act on the derived category. This was first described in a special case by Rouquier and Zimmermann and, if time permits, I hope to present a new proof of their result.

Abstract: I will discuss properties of resolutions of powers of ideals. This will be done in the framework of Betti diagrams and their polyhedral structure. At the end a conjecture regarding monomial ideals will be stated.

Abstract: We explain how one can relate the Hochschild (co)homologies of Hopf algebras having equivalent tensor categories of comodules, in case the trivial module over one of the Hopf algebras admits a resolution by free Yetter-Drinfeld modules. This general procedure is applied to the quantum group of a bilinear form, for which generalizations of results by Collins, Hartel and Thom in the orthogonal case are obtained. It also will be shown that the Gerstenhaber-Schack cohomology of a cosemisimple Hopf algebra completely determines its Hochschild cohomology. Basic facts and definitions about Hopf algebras will be recalled first.

Abstract: We describe the recent research started by Nekrasov Shatashvilly Braverman Maulik Okounkov on correspondence between the quantum cohomology of the quiver varieties and the quantum integrable systems. Our main example will be the cotangent spaces to partial flag varieties.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: For title and abstract, click here

Abstract: Click here for title and abstract.

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: For title and abstract, click here

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: See the maximals webpage for detalis.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract:

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://www.ma.hw.ac.uk/~ndg/nbsan.html

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: Tom Leinster recently introduced an invariant of graphs called the magnitude. In this talk I will define a homology theory for graphs that categorifies the magnitude, in the sense that the magnitude of a graph can be recovered from its homology by taking the Euler characteristic. (Thus this is categorification in the same sense that Khovanov homology categorifies the Jones polynomial.) Important properties of the magnitude can then be seen as shadows of properties of homology. For example, magnitude satisfies an inclusion exclusion formula that can be recovered from a Mayer-Vietoris theorem in homology. The talk will (hopefully) be accessible for anybody who knows what graphs and chain complexes are, and I will try to illustrate it with lots of pictures and examples.

Abstract: Abstract: A uniform categorical description for both the Drinfeld center and a Heisenberg analogue called the Hopf center of a monoidal category (relative to a braided monoidal category) is presented using morphism categories of bimodules. From this categorical definition, one obtains a categorical action as well as a definition of braided Drinfeld and Heisenberg doubles via braided reconstruction theory. In examples, this categorical picture can be used to obtain a categorical action of modules over quantum enveloping algebras on modules over quantum Weyl algebras. Moreover, certain braided Drinfeld doubles give such an action on modules over rational Cherednik algebras using embeddings of Bazlov and Berenstein of these algebras into certain braided Heisenberg doubles which can be thought of as versions of the Dunkl embeddings. We argue that the corresponding braided Drinfeld doubles can serve an quantum group analogues in the setting of complex reflection groups. Finally, the categorical description can be extended naturally to give TQFTs with defects using recent work of Fuchs-Schaumann-Schweigert.

Abstract: We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Poisson Gelfand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as a Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients. For instance this class includes Poisson determinantals varieties.

Abstract: Abstract: The study of separating invariants is a relatively recent trend in invariant theory. For a finite group acting linearly ona vector space, a separating set is a set of invariants whose elements separate the orbits of the action. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the leastpossible cardinality of a separating set for a given action. Our main result is a lower bound which generalizes the classical result of Serre that if the ring of invariants is polynomial, then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples. This is based on joint work with Emilie Dufresne.

Abstract: I will explain how (quasi-)Hamiltonian reduction fits into the framework of derived symplectic geometry. (Quasi-) Hamiltonian spaces are interpreted as Lagrangians in shifted symplectic stacks and the reduction corresponds to Lagrangian intersection. This gives a new perspective on deformation quantization of Hamiltonian spaces. This could also be used to make sense of deformation quantization of quasi-Hamiltonian spaces.

Abstract: Starting from solutions of the Yang-Baxter equation we construct a noncommutative bi-algebra which can be described in purely combinatorial terms using non-intersecting lattice paths. Inside this noncommutative algebra we identify a commutative subalgebra, called the Bethe algebra, which we identify with the direct sum of the equivariant quantum cohomology rings of the Grassmannian. We relate our construction to results of Peterson which describe the quantum cohomology rings in terms of Kostant and Kumar's nil Hecke ring and the homology of the affine Grassmannian. This is joint work with Vassily Gorbounov, Aberdeen.

Abstract: Abstract: I am going to discuss two subalgebras in the rational Cherednik algebra associated with a Coxeter group. These subalgebras satisfy Poincare-Birkhoff-Witt property and they are given by quadratic relations. They deform semidirect product of quotients of universal enveloping algebras of so(n) and gl(n) with the Coxeter group algebra, and they are related to quantisation of functions on the Grassmanian of two-planes and on the space of matrices of rank at most 1 respectively. The centres of these subalgebras give quantum Hamiltonians related to Calogero-Moser integrable systems which I plan to discuss as well. This is based on joint work with T. Hakobyan.

Abstract: I will discuss work (in progress) with Ben Webster on homological mirror symmetry for hypertoric varieties. Hypertoric varieties are a family of noncompact algebraic symplectic spaces associated to hyperplane arrangements; we show how the quantization of such spaces in finite characteristic has a natural description on the mirror side.

Abstract: Abstract: The idea behind this talk is to describe two-sided ideals of U(g) for an infinite dimensional Lie algebra g using known classification of prime two-sided ideals of U(g') for finite dimensional Lie subalgebras g' of g. In particular, if g is semisimple, by analogy with a finite dimensional case, one may expect that all primitive two-sided ideals are annihilators of highest weight modules. To start with we focus on infinite-dimensional Lie algebra sl(\infty). We will see that the annihilators in U(sl(\infty)) of most highest weight sl(\infty)-modules equal (0) and explicitly describe all highest weights for which this annihilator is not (0). We also prove that in the latter case the annihilator is an integrable ideal and provide a classification of such ideals. The proof will use the classification of two-sided ideals of U(sl(n)) (and thus a little bit of Young diagrams and Robinson–Schensted algorithm). Title/Abstract for the second part: Title: On ideals in the enveloping algebra of a locally simple Lie algebra Abstract: Let g be a Lie algebra with universal enveloping algebra U(g). To a two-sided ideal I of U(g) one can canonically assign a Poisson ideal gr I in S(g). It turns out that very frequently S(g) has no non-trivial Poisson ideals (and I hope I give some idea why it is so). As a consequence very frequently U(g) has no non-trivial two-sided ideals. As a final result I will provide some quite explicit description of countable dimensional locally simple Lie algebras g such that U(g) affords a non-trivial two-sided ideal.

Abstract: Abstract: I will discuss a cohomological field theory associated to a quasihomogeneous isolated singularity W with a group G of its diagonal symmetries (a Landau-Ginzburg A-model, in physical parlance). The state space of this theory is the equivariant Milnor ring of W and the corresponding invariants can be viewed as analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). In the case of simple singularities of type A they control the intersection theory on the moduli space of higher spin curves. The construction is based on derived categories of (equivariant) matrix factorizations of singularities with the role of the virtual fundamental class from the Gromov-Witten theory played by a "fundamental matrix factorization" over a certain moduli space.

Abstract: Abstract: A rack is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode two of the three Reidemeister moves. We will begin by generalizing Whitehead's notion of a crossed module of groups to that of a crossed module of racks. Motivated by the relationship between crossed modules of groups and strict 2-groups, we then will investigate connections between our rack crossed modules and categorified structures including strict 2-racks and trunk-like objects in the category of racks. We will conclude by considering topological applications, such as fundamental racks. This is joint work with Friedrich Wagemann.

Abstract: Abstract: A nonnoetherian singularity may be viewed geometrically as an algebraic variety with positive dimensional `smeared-out' points. I will describe how isolated nonnoetherian singularities admit noncommutative blowups which are smooth, in a suitable geometric sense. Furthermore, I will describe how a class of isolated nonnoetherian noncommutative singularities, namely non-cancellative dimer algebras, also admit noncommutative blowups which are smooth.

Abstract: Abstract: In search of L-infinity mappings between homotopy Poisson algebras of functions, we discovered a construction of a certain nonlinear analog of pullbacks. Underlying such "nonlinear pullbacks" there are formal categories that are "formal thickening" of the usual category of smooth maps of manifolds (or supermanifolds). Morphisms in these categories are called ''thick morphisms'' (or microformal morphisms). They are defined by formal canonical relations between the (anti)cotangent bundles of the manifolds. Thick morphisms have beautiful properties. For example, we can define the adjoint of a nonlinear morphism of vector bundles, as a thick morphism of the dual bundles, which reduces to the ordinary adjoint in the linear case. In the talk, I will explain the construction of microformal morphisms and nonlinear pullbacks, and their applications to homotopy Poisson structures, vector bundles and L-infinity algebroids. (See preprints: http://arxiv.org/abs/1409.6475 and http://arxiv.org/abs/1411.6720.)

Abstract: The 45th ARTIN meeting will take place at the University of Edinburgh on the 11th and 12th of September 2015. All talks will be in the James Clerk Maxwell Building, room 6206. The theme of the meeting is noncommutative ring theory, with an emphasis on noncommutative algebraic geometry. See http://hodge.maths.ed.ac.uk/tiki/ARTIN-45 for more details

Abstract: The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. One can enlarge these algebras by a commutative subalgebra $X$ to anaffine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right $X$-module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of $X$-semisimple, or calibrated, representations, tha t don't necessarily factor through the affine Hecke algebra. I will discuss Walker's TQFT-motivated 2-handle construction of the $X$-semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description. This is joint work with Kevin Walker.

Abstract: Abstract: In this talk I will present basic results of cluster-tilting theory developed in [Iyama Yoshino 2008: Mutation in triangulated categories and Rigid CM modules] and [Buan-Iyama-Reiten-Scott 2009: Cluster structures for 2-Calabi-Yau categories]. I will explain how these results were a motivation for generalising the construction of cluster categories. I will first recall the motivation and definition of the acyclic cluster category due to Buan Marsh Reineke Reiten Todorov in 2006, and then focus to the construction of the generalised cluster category associated with algebras of global dimension 2 [Amiot 09]. Then I will explain how cluster-tilting theory can apply in classical tilting theory via graded mutation in a joint work with Oppermann.

Abstract: In this talk I will explain a joint work with Y. Grimeland. Surface algebras are algebras of global dimension 2 associated to an unpunctured surface $S$ with an admissible cut. It is possible to associate to each such algebra an invariant in an affine space of $H^1(S,\mathbb Z)$ up to an action of the mapping class group which determines the derived equivalence class of the algebra. The proof uses strongly the graded mutation introduced in a joint work with S. Oppermann. This invariant is closely linked with the Avella-Alaminos-Geiss invariant for gentle algebras, and gives some information on the AR structure of the corresponding derived category.

Abstract: I will discuss what a graded version of a category is and why graded versions of categories of representations are interesting to study. I want to discuss how recent advances in the theory of motives help to better understand these graded versions.

Abstract: Open Richardson varieties are the intersections of opposite Schubert cells in full flag varieties. They play a key role in Schubert calculus, total positivity and cluster algebras. We will show how to realize the quantized coordinate ring of each open Richardson variety as a normal localization of a prime factor of a quantum Schubert cell algebra. Using a combination of ring theoretic and representation theoretic methods, we will produce large families of toric frames for all quantum Richardson varieties. This has applications to cluster algebras and to the construction of a Dixmier type map from the symplectic foliation of each Schubert cell to the primitive spectrum of the corresponding quantum Schubert cell algebra. This is a joint work with Tom Lenagan (University of Edinburgh).

Abstract: We review the definition of Hattori-Stallings traces of projective modules and their relation to Morita equivalence. As an application, we will discuss Berest-Etingof-Ginzburg's work on Morita equivalence of rational Cherednik algebras of type A.

Abstract: The mapping class group Mod(S) of a surface S appears in a variety of contexts, for example, as a natural analogue both of arithmetic groups and of automorphism groups of free groups, and as the (orbifold) fundamental group of the moduli space of Riemann surfaces. However, its subgroup structure is not at all well understood. In this talk we will discuss a certain rigidity displayed by a wide class of subgroups of Mod(S): any normal subgroup of Mod(S) that contains a "small" element has Mod(S) as its group of automorphisms. This result is proved using a resolution of a metaconjecture posed by N. Ivanov stating that every stating that every "sufficiently rich" complex associated to S has Mod(S) as its group of automorphisms. (This is joint work with Dan Margalit.)

Abstract: The natural problem of determining characters of simple objects in suitable representation categories can be rephrased in terms of multiplicities of irreducible modules in standard objects. In this talk, I will focus on the case in which these multiplicities are governed (or expected to be governed) by certain combinatorial families of polynomials, and explain how moment graph techniques can be used to approach such a problem.

Abstract: The Birch—Swinnerton-Dyer conjecture is one of the most mysterious open problems in number theory, predicting a relation between arithmetic objects, such as the points on an elliptic curve, and certain complex-analytic functions. A powerful approach to the conjecture is via a tool called an ‘Euler system’. I will explain the idea behing this approach, and some recent new results in this direction.

Abstract: In this talk I will discuss how categorification methods can be used to obtain a graded version of the Brauer algebra, especially in non semi-simple cases. This will involve the category O for certain orthogonal Lie algebras and their parabolic and graded analogues. The categorifications involved in this process are a generalised version of the ones introduced by Rouquier and Khovanov-Lauda in the sense that the categorified object is not a Kac-Moody algebra or quantum group, but a so-called quantum symmetric pair. This is joint work with Catharina Stroppel.

Abstract: Abstract: Flag varieties are fertile soil where it grows geometry, algebra and combinatorics. Motivated by the PBW filtration of Lie algebras, E. Feigin defined the degenerate flag varieties, which are flat degenerations of the corresponding flag varieties. The purpose of this talk is to introduce a new family of (flat) degenerations of flag varieties of type A, called linear degenerate flag varieties, by classifying flat degenerations of a particular quiver Grassmannian. The geometry of these degenerations will also be presented. This talk is based on a joint work (in progress) with G. Cerulli Irelli (Rome), E. Feigin (Moscow), G. Fourier (Glasgow) and M. Reineke (Wuppertal).

Abstract: If X is a determinantal variety then there are a number of objects that may be regarded as "resolutions of singularities" of X: (1) the classical Springer resolution by a vector bundle over a Grassmannian, (2) a suitable quotient stack, (3) various non-commutative resolutions. In the talk we will discuss how these different resolutions are related. For ordinary determinal varieties this is joint work with Buchweitz and Leuschke. For determinantal varieties of symmetric and skew-symmetric matrices new phenomena occur due to the fact that the Springer resolution is no longer crepant. This is joint work with Špela Špenko.

Abstract: In pioneering work, Khovanov introduced the so-called arc algebra. His arc algebra is a certain algebra built from cobordisms turning up in TQFT’s and is known to be related to Khovanov homologies, categorification of tensor products of sl(2) and to a certain 2-block parabolic of category O for gl(n) - as follows from work of many researchers in the last 15 years. Sadly, in Khovanov’s original construction, the functoriality of Khovanov homologies cannot be encoded directly nor is it clear how to generalize his construction to obtain relations to e.g. tensor products of sl(N) or N-block parabolics of category O for gl(n). For this purpose, one needs to modify his arc algebra by using singular TQFT’s instead of “usual” TQFT’s. In this talk I will explain Khovanov’s topologically and elementary, yet powerful, construction in details as well as its relations to categorification and category O. I will then sketch how to use singular TQFT’s to generalize the construction.

Abstract: We describe the finite dimensional representation category of gl(m|n) and of its quantized enveloping algebra using variations of Howe duality, and we review the Reshetikhin-Turaev construction of the corresponding link invariants of type A. We discuss then some results (and some open questions) on their categorification, in particular using the BGG category O.

Abstract: Spherical subgroups of finite type of a Kac-Moody group have been recently introduced, in the framework of a research project aimed at bringing the classical theory of spherical varieties to an infinite-dimensional setting. In the talk we discuss their definition and some of their properties. We introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, which satisfies the same axioms as in the finite-dimensional case.

Abstract: Roughly speaking, an element s in a commutative ring A is said to be weakly proregular if every module over A can be reconstructed from its localisation at s considered along with its local cohomology at the ideal generated by s. This notion extends naturally to finite sequences of elements: a precise definition will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. In particular, every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A. In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.

Abstract: Let G be a complex reductive group and B a Borel subgroup of G, a subgroup H of G is called spherical if it acts with finitely many orbits on the flag variety G/B. For example, if H coincides with B, then the orbits are parametrized by the Weyl group of G and the orbits are the Schubert cells. Even though spherical subgroups are classified combinatorially, the corresponding orbit decompositions of G/B are not yet understood in general. In this seminar I will consider two special cases, namely that of a solvable spherical subgroup and that of a symmetric subgroup of G corresponding to an involution of Hermitian type. In these cases, I will explain how to attach a root system to every H-orbit in G/B, and how these root systems allow to parametrize the H-orbits in G/B. These parametrizations are compatible with a general action of the Weyl group of G that Knop defined on the set of H-orbits in G/B, and I will explain how to recover the Weyl group action from the parametrization of the orbits. The talk is based on two joint works, respectively with Andrea Maffei and with Guido Pezzini.

Abstract: In the late 1970s and early 1980s, Dixmier and Moeglin gave algebraic and topological conditions for recognising the primitive ideals (namely the kernels of the irreducible representations) of the enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, such algebras are said to satisfy the "Dixmier-Moeglin equivalence". Many interesting families of algebras, including many families of quantum algebras, have since been shown to satisfy this equivalence. I will outline work of Jason Bell, Stephane Launois, and myself, showing that in several families of quantum algebras, an arbitrary prime ideal is equally close (in a manner which I will make precise) to being primitive, rational, and locally closed. The family on which I shall focus is that of the quantum Schubert cells U_q [w]. For a simple complex Lie algebra g, a scalar q which is not a root of unity, and an element w of the Weyl group of g, U_q [w] is a subalgebra of U_q^+(g) constructed by De Concini, Kac, and Procesi; familiar examples include the algebras of quantum matrices.

Abstract: I will begin presenting (part of) the history of the Krull-Schmidt-Remak Theorem. Then I will show a number of results concerning uniqueness of direct-sum decompositions of right modules over a ring R and uniqueness of direct-product decompositions of a group G. I will conclude giving some results about the category of G-groups, which is a category rather similar to the category Mod-R of right R-modules. Here a G-group is a group H on which G acts as a group of automorphisms.

Abstract: Factorisation algebras and equivalently chiral algebras are geometric versions of vertex algebras, introduced by Beilinson and Drinfeld. There is also a non-linear analogue of a factorisation algebra, called a factorisation space. I will define these objects, and furthermore show how we can use the Hilbert scheme of points of a smooth d dimensional variety X to construct examples.

Abstract: The prime spectrum of a quantum algebra has a finite stratification in terms of a set of distinguished primes called H-primes, and we can study these strata by passing to certain nice localizations of the algebra. H-primes are now starting to show up in some surprising new areas, including combinatorics (totally nonnegative matrices) and physics, and we can borrow techniques from these areas to answer questions about quantum algebras and their localizations. In particular, we can use Grassmann necklaces -- a purely combinatorial construction -- to study the topological structure of the prime spectrum of quantum matrices.

Abstract: Motivic Donaldson-Thomas invariants of quivers were defined by M. Kontsevich and Y. Soibelman as a mathematical definition of string-theoretic BPS state counts. We discuss several results relating these invariants to the geometry of moduli spaces of quiver representations.

Abstract: By a result of J. Scott, the homogeneous coordinate ring of the Grassmannian Gr(k,n) can be realised as a cluster algebra. The Pluecker coordinates of the Grassmannian are all cluster variables. I will talk about joint work with J. Scott. We introduce a twist map on the Grassmannian and show that it is related to a twist of Berenstein-Fomin-Zelevinsky and can be implemented by a maximal green sequence, up to frozen variables. We give Laurent expansions for twists of Pluecker coordinates as scaled dimer partition functions (matching polynomials) on weighted versions the plabic (planar bicoloured) graphs arising in the cluster structure.

Abstract: Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, or all of $M_n(K)$. We prove the conjecture for $K=\mathbb{R}$ or for quadratically closed field and $n=2$, and give a partial solution for an arbitrary field $K$. We also consider homogeneous and Lie polynomials and provide the classifications for the image sets in these cases.

Abstract: Set-theoretic solutions of the Yang--Baxter equation form a meeting ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. Associated to each set-theoretic solution are several algebraic constructions: the monoid $S(X, r)$, the group $G(X, r)$, the semigroup algebra $kS(X, r)$ over a field k, generated by X and with quadratic relations $xy = .r(x, y)$, a special permutation group $\mathcal{G}$ and a left brace $(G, +,.)$. In this talk I shall discuss some of the remarkable algebraic properties of these object.

Abstract: A hovel is degraded building. It can be used to associate to a Kac-Moody group over a local field a bunch of Hecke algebras. First, I will explain the definition of the hovel which generalises the construction of the Bruhat-Tits building associated to a reductive group. Then I will present three kind of algebras that one can associate to the hovel: the spherical Hecke algebra, the Iwahori-Hecke algebra and the Bernstein-Lusztig-Hecke algebra.

Abstract: By quantum matrix algebras I mean algebras related to quantum groups and close in a sense to that Mat(m). These algebras have numerous applications. In particular, by using them (more precisely, the so-called reflection equation algebras) we succeeded in defining partial derivatives on the enveloping algebras U(gl(m)). This enabled us to develop a new approach to Noncommutative Geometry: all objects of this type geometry are deformations of their classical counterparts. Also, with the help of the reflection equation algebras we introduced the notion of braided Yangians, which are natural generalizations of the usual ones and have many beautiful properties.

Abstract: The noncommutative and nonassociative algebra which arises in the description of the target space of non-geometric string theory fits naturally as a commutative and associative algebra object in a certain closed braided monoidal category, the representation category of a triangular quasi-Hopf algebra. In this talk I will show how exploring the syntax of category theory enables one to express notions of geometry on an algebra object in terms of universal constructions internal to the representation category of any triangular quasi-Hopf algebra.

Abstract: We study the representation theory of finite-dimensional graded algebras which admit a triangular decomposition similar to universal enveloping algebras of Lie algebras. Such a decomposition implies a rich combinatorial structure in the representation theory (this was discovered in this generality by Holmes and Nakano) and there are many examples like restricted quantized enveloping algebras at roots of unity, restricted rational Cherednik algebras, etc. We show that even though our algebras have in general infinite global dimension, the graded module category is in fact a highest weight category (with infinitely many simple objects, however). Under certain conditions we are able to establish a proper tilting theory in this category and use this to show that the degree zero part of the algebra is a cellular algebra. This is joint work with Gwyn Bellamy (Glasgow).

Abstract: First hour: The Witten-Reshetikhin-Turaev knot invariants are polynomials associated to a knot in S^3 constructed using representation theory of quantum groups. A concrete combinatorial construction of these invariants is given by the Kauffman bracket skein relations. In this talk we first discuss some of the background for these constructions. We then discuss how the skein relations are related to representation varieties, and to the Poisson structure on representation varieties of topological surfaces. Second hour: The double affine Hecke algebra is a noncommutative algebra depending on parameters q and t which is associated to a Lie algebra. The DAHA has been connected to various parts of math, including symmetric polynomials, integrable systems, Hilbert schemes, and more. Frohman and Gelca showed that the skein algebra of the torus is the t=q specialization of the sl_2 DAHA. We discuss this result (and some background), and then discuss a conjecture involving the DAHA and modules coming from knot complements.

Abstract: I will describe the geometric R-matrix formalism, developed in the work of Maulik and Okounkov, that leads to the construction of a Hopf algebra $Y_Q$ acting on the equivariant cohomology of the Nakajima varieties associated to a quiver $Q$. In the first hour, I will give an introductory overview of the basic machinery of Nakajima varieties and their equivariant cohomology, which underpins the Maulik-Okounkov construction. In particular, we’ll illustrate the general definitions by focusing on the concrete example of cotangent bundles to Grassmannians. In the second hour, I’ll describe the stable basis construction in the equivariant cohomology of a symplectic variety, which is the key technical tool used to construct the geometric R-matrices. In our example of cotangent bundles to Grassmannians, we’ll see that this yields a geometric construction of the Yangian of $gl_2$. Finally, we will conclude by discussing some work in progress extending the Maulik-Okounkov construction to encompass Yangian coideal subalgebras, such as reflection equation algebras.

Abstract: This is joint work with S. Sierra. We explore the (noncommutative) geometry of representations of locally finite Lie algebras. Let L be one of these Lie algebras, and let I ⊆ U(L) be the annihilator of a locally simple L-module. We show that for each such I, there is a quiver Q so that locally simple L-modules with annihilator I are parameterized by “points” in the “noncommutative space” corresponding to the path algebra of Q. We classify the quivers that occur and along the way discover a beautiful connection to characters of the symmetric groups S_n.

Abstract: We explain two ways in which geometry can be used to produce solutions of the Yang-Baxter equation. First, we introduce Maulik and Okounkov's "stable envelope" construction of R-matrices acting in the cohomology of a symplectic variety, and describe some of the geometric properties these R-matrices enjoy. Second, we produce an R-matrix from the Hilbert scheme of points on a general surface from an intertwiner of certain highest weight Virasoro modules; for the surface C^2, this construction is due to Maulik and Okounkov. We also explain how to modify this construction to produce formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme.

Abstract: One of the earliest successes in geometric representation theory was Springer's construction of the irreducible representations of the symmetric group (or any Weyl group) in the top cohomology of fibers of a resolution of singularities of the nilpotent cone of GL(n) (or a reductive group). More recently there has been considerable progress extending these ideas to representations and sheaves with positive characteristic coefficients. Life is more complicated in positive characteristic: the category of representations is no longer semisimple, and on the geometric side this is reflected in the failure of some important geometric tools from characteristic 0 such as the decomposition theorem. In this talk I will describe work with Carl Mautner giving a picture similar to Springer theory where the role of the nilpotent cone is played by hypertoric varieties. We obtain representations of a new class of algebras which we call "Matroidal Schur algebras", which share many features with their classical cousins. In particular the categories are highest weight, and the categories for Gale dual hypertoric varieties are related by Ringel duality. In the second part of the talk I will explain some of the ideas in the proof and a conjectural geometric approach to proving that certain categories of perverse sheaves are highest weight.

Abstract: Quantum groups play a prominent role in many branches of mathematics, from gauge theory and enumerative geometry, to knot theory and quantum computing. In many cases, this is due to their tight relation with the braid groups. More specifically, to the fact that a quantum group is in the first place a Hopf algebra whose representations carry a natural action of the braid group. In the first part of the talk, I will explain how the quantum groups are in fact analytic objects, describing the monodromy of certain systems of difeerential equations arising in Lie theory. I will first review the renowned Drinfeld-Kohno theorem, describing the monodromy of the Knizhnik-Zamolodchikov equations associated to a simple Lie algebra in terms of the universal R-matrix of the corresponding quantum group. I will then present an extension of this result, providing a descrip- tion of the monodromy of the Casimir equations associated to a simple Lie algebra (in fact, to any symmetrisable Kac-Moody algebra) in terms of the quantum Weyl group operators of the corresponding quantum group. The proof relies on the notion of generalised braided category (or quasi-Coxeter category), which is to a generalised braid group what a braided monoidal category is to the standard braid group on n strands. In the second part of the talk, I will explain how Tannaka{Krein dual- ity, quantization of Lie bialgebras, dynamical KZ equations, and Hochschild cohomology in the framework of appropriate PROP categories play a funda- mental role in the proof of the monodromy theorem.

Abstract: In geometric representation theory, we typically study certain categories associated to a reductive group G (e.g. certain representations of G or g=Lie(G), D-modules on Bun_G(Curve), Character sheaves on G, ...). Often there are functors of parabolic induction and restriction going between the categories associated to G and to Levi subgroups L of G. I will explain how these functors allow us to break up our category into pieces, indexed by classes of cuspidal objects on Levi subgroup (an object is called cuspidal if it is not seen by induction from a proper Levi). We will see how things like Hecke algebras and (relative) Weyl groups naturally appear. Later, I will focus on the case of adjoint equivariant D-modules on the Lie algebra g, and indicate how this case may be generalized to other settings - mirabolic, quantum, elliptic...

Abstract: Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding planar algebras fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These planar algebras are the ones attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don't fit into any of the classical families: G_2, F_4, E_6, E_7, and E_8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-paramter family of planar algebras which introduces a variable q, and yields a new exceptional knotpolynomial. In joint work with Scott Morrison and Dylan Thurston, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.

Abstract: A quantum cluster (or quantum torus) is an algebra over C(q) with q-commuting generators. Various embeddings of quantum groups into quantum tori have been studied over the past twenty years in relation with modular doubles, quantum Gelfand-Kirillov conjecture, and construction of braided monoidal categories. In a recent paper by K. Hikami and R. Inoue, such an embedding of the quantum group U_q(sl_2) was used to relate the corresponding R-matrix with quantum cluster mutations and half-Dehn twists. I will discuss how to generalize the results of Hikami and Inoue to U_q(sl_n). The quantum group is embedded into the tensor square of the quantized categorification space of 3 flags and 3 lines in C^n, which were studied in detail in the works of V. Fock and A. Goncharov. I also plan to show how the conjugation by the R-matrix can be expressed via a sequence of cluster mutations. If time permits, I will outline a way to generalize the above construction to quantum groups of arbitrary finite type and discuss its applications to the representation theory.

Abstract: Infinite-dimensional representation theory of finite dimensional Lie algebras is a rich topic with many interesting results. One of the most beautiful pieces of this subject is a description of primitive and prime ideals of universal enveloping algebras of finite-dimensional Lie algebras, and this involves quite advanced algebraic, geometric, and combinatorial techniques. It is natural to generalize the classification of primitive and prime ideals to the setting of infinite-dimensional Lie algebras, and in my talk I will provide a description of primitive ideals of the universal enveloping algebra of sl(infinity). I hope that I will be able to explain in an understandable way algebraic and combinatorial aspects of this result.

Abstract: TBA

Abstract: I will discuss how questions on Sklyanin algebras can be solved using combinatorial techniques, namely, Groebner bases theory. Elements of homological algebra also feature in our proofs. We calculate Hilbert series, prove Koszulity, PBW, Calabi-Yau etc., depending on parameters of Sklyanin algebras. Similar methods are used for generalized Sklyanin algebras, and for other potential algebras.

Abstract: The wonderful compactification of a group plays a crucial role in several areas of geometric representation theory and related fields. The aim of this talk is to give a construction of the wonderful compactification and explain how its rich structure links the geometry of the group to the geometry of its partial flag varieties. We will describe several examples in detail. The first part of the talk will be an overview of necessary background from the representation theory of complex reductive groups.

Abstract: The wonderful compactification of a group encodes the asymptotics of matrix coefficients for the group and captures the rational degenerations of the group. In this talk, we will explain a construction of the wonderful compactification via the Vinberg semigroup which makes these properties explicit. We will then introduce quantum group versions of the Vinberg semigroup, the wonderful compactification, and the latter's stratification by G x G orbits. Our approach relies on a theory of noncommutative projective schemes, which we will review briefly.

Abstract: This will be a short reminder on the basic properties of quiver varieties. As well as giving the basic definitions, I’ll explain how one computes their dimensions, when they are smooth etc. No prior knowledge will be assumed.

Abstract: Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley-Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

Abstract: "Categorification" means replacing vector spaces with categories, in an artful way. When we categorify the notion of a ring, and a module over it, we get the notions of a tensor category, and a module category over it. Examples of these are ubiquitous throughout representation theory and algebraic geometry. If you hand a representation theorist a ring, she will ask "what are its modules?". In this talk, I'll develop some tools which you can use in case someone ever hands you a tensor category, and asks what are its module categories? It turns out that the resulting "Morita theory" for tensor categories plays a crucial role in the next talk.

Abstract: A character variety of a manifold X is a moduli space of representations of pi_1(X). It was shown by Atiyah-Bott and Goldman that character varieties of surfaces are naturally symplectic, and that character varieties of 3-manifolds define Lagrangian subvarieties in the character varieties of their boundary surfaces. In this talk, I'll explain that all this structure can be ``quantized", giving rise to a TFT which we call the quantum character TFT. We obtain in this way manifestly topological constructions of many gadgets traditionally thought of as living in the world of representation theory: quantum coordinate algebras, q-difference operator algebras, double affine Hecke algebras, etc. Quantizing the Lagrangians of different 3-manifolds gives a new approach to studying the representation theory of these objects: the quantization of a Lagrangian should be (roughly) a simple module for the quantization of the symplectic variety it lives on. The main technical ingredient in the construction and any computations with it is the Morita theory of tensor categories, as developed in the preseminar. This is joint work with Ben-Zvi, Brochier and Snyder.

Abstract: In the pre-seminar, I will give an overview of some aspects of conformal field theories and representation theory which play an important role in understanding the duality (the AdS/CFT correspondence) between strings in 10 dimensions and conformal field theories in 4 dimensions. This will include BPS states, matrix correlators, large N expansions and Schur-Weyl duality.

Abstract: Permutations subject to equivalences can be used to classify invariants of unitary group actions on polynomial functions of matrices and tensors. These equivalence classes are related to permutation centralizer algebras. One sequence of these algebras is closely related to Littlewood-Richardson coefficients. Structural properties of these algebras as well as Fourier transforms on them have applications in dualities between gauge theories and string theories. They yield results on the counting and correlators of multi-matrix invariants, relevant to the physics of super-symmetric states in the AdS/CFT correspondence. Combinatoric questions on the structure of these algebras are related to the complexity of spaces of super-symmetric states.

Abstract: The starting point for this talk is Chouinard’s theorem, which states that a module for a finite group is projective if and only if its restriction to every elementary abelian p-subgroup is projective; and Dade’s lemma, which gives an easy test for whether a module for an elementary abelian group is projective. I shall talk about analogous results for finite group schemes and finite supergroup schemes, and their consequences for the structure of the stable module category. Parts of this talk represent joint work with Iyengar, Krause and Pevtsova.

Abstract: A quick introduction to noncommutative projective geometry in the style of Artin-Tate-Van den Bergh. In this field, one studies noncommutative graded algebras with 'similar' properties to the commutative polynomial ring. Such properties can be either of homological or algebraic nature. I will talk about the classification of AS-regular algebras and define the problem I've been working on in the context of this field.

Abstract: Consider a positively graded, connected algebra A, finitely generated in degree 1, for example the polynomial ring of global dimension n. Assume that there exists some reductive group G acting on A as gradation preserving algebra automorphisms, then each degree k part decomposes as a finite sum of simple G-modules. Then a natural question is: do there exist other graded algebras B such that 1) G acts on B, with the action preserving the gradation and 2) the degree k parts of A and B are isomorphic as G-modules for each natural number k ? As one may suspect, this depends greatly on G and A itself. Some constructions and the motivating example of the 3-dimensional Sklyanin algebras will be discussed. For this example, if time permits, I will show that some additional information about these algebras can be gained by this construction.

Abstract: In this talk I am going to give a brief historic overview about the origin of commutators in group theory. Then I will pass to division rings and will show how one can obtain essential information about the structure of a division ring in terms of commutators and structures generated by commutators. Also, you will find generalization of some classic results due to Jacobson, Kaplansky and Noether about division rings.

Abstract: In recent years there has been renewed interest in the construction of finitely generated algebraic division algebras that are not finite-dimensional. This is division ring version of Kurosh Problem. There are results as local versions of Kurosh problem, for example a theorem due to Jacobson asserts: every division ring whose elements are algebraic of bounded degree over its center, is centrally finite. Recently, this result has been generalized by Jason Bell et al to left algebraic division rings over not necessarily central subfields. Using combinatorics of words, in this seminar we show the statement holds for division rings whose commutators are left algebraic over not necessarily central subfields.

Abstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,…

Abstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,…

Abstract: One of the main tools for understanding automorphisms of free groups is via the action on Culler Vogtmann Space. More recently, the geometry of this space has been the subject of intense study. We will provide an introduction to these objects as well as presenting a report on some recent joint work with Stefano Francaviglia showing that the set of "minimally displaced points" for a given automorphism is connected, and that this is enough to solve the conjugacy problem in some limited cases.

Abstract: Let G be a reductive algebraic group over a field k. The notion of a $G$-complete reducible subgroup of G was introduced by Serre; in the special case G= GLn(k), a subgroup H of G is G-completely reducible if and only if the inclusion of H in G is completely reducible in the sense of representation theory. G-complete reducibility has turned out to be an important tool for investigating the subgroup structure of simple algebraic groups. In this talk I will discuss the interplay between geometric invariant theory and the theory of G-complete reducibility.

Abstract: The Kleinian singularities make up a family of well-understood (commutative) surface singularities. In 1998, Crawley-Boevey and Holland introduced a family of algebras which may be viewed as noncommutative deformations of Kleinian singularities. Using singularity categories, I will make comparisons between the types of singularity arising in the commutative and noncommutative settings. I will also show that the "most singular" of these noncommutative deformations has a noncommutative resolution for which an analogue of the geometric McKay correspondence holds.

Abstract: The main talk concerns the search for the "Lie algebra of BPS states" associated to a preprojective algebra. In the pretalk I'll explain why, as a mathematician, one would go looking for such a thing. The evidence pointing to the existence of this algebra involves many nice results from combinatorics of representations of quivers, Donaldson-Thomas theory, and Nakajima quiver varieties. I'll try to give a geographical sketch of these other results in order to motivate the main talk.

Abstract: I'll explain the sense in which the map from the stack of finite-dimensional representations of an algebra to the coarse moduli space behaves as though it were a proper map. This turns out to be a vital piece in proving the following statement: the cohomological Hall algebra associated to the preprojective algebra of a quiver is a quantum enveloping algebra, for the strictly positive part of a new type of Kac-Moody algebra, which carries a cohomological grading. This gives a mathematical formulation for the physicists' Lie algebra of BPS states. The cohomological degree zero piece of this algebra is the positive part of the usual Kac-Moody algebra of the largest subquiver without imaginary simple roots, but even if this is the entire quiver, there's much more to this algebra than the usual Kac-Moody algebra of the quiver.

Abstract: As motivation for the work to be described in the second hour, I will discuss the classical Schur-Weyl duality for gl(n), describing the endomorphism ring of tensor powers of the vector representation in terms of the symmetric group, as well as higher Schur-Weyl duality involving more general representations and the degenerate affine Hecke algebra in place of the symmetric group. This construction has been further extended in using diagrammatic algebras to describe the representation theory of other Lie algebras and Lie superalgebras. We will also discuss some Lie supertheory towards looking at the periplectic Lie superalgebra.

Abstract: The representation theory of Lie algebras of type D is studied by Ehrig and Stroppel in an approach generalizing Arakawa and Suzuki’s work in type A. More specifically, they use affine Nazarov-Wenzl algebras, and their cyclotomic quotients, instead of the degenerate affine Hecke algebra to describe the endomorphism ring of certain representations in type D. We will discuss a signed version, or odd VW algebras living in an extension of the odd Brauer category, in a similar approach towards describing endomorphism rings of the periplectic Lie superalgebra p(n). The key ingredient will be a certain sneaky quadratic Casimir element and the corresponding Jucys-Murphy components.

Abstract: I will illustrate with many examples how one can use the representation theory of complex reductive groups to obtain combinatorial invariants of algebraic varieties equipped with an action of such a group. The main focus will be on invariants of (affine) spherical varieties.

Abstract: Spherical varieties form a remarkable class of complex algebraic varieties equipped with an action of a reductive group G, which includes toric, flag and symmetric varieties. Smooth affine spherical varieties are the local models of multiplicity free (real) Hamiltonian and quasi-Hamiltonian manifolds. A natural invariant of an affine spherical variety X is its weight monoid, which is the set of irreducible representations (or dominant weights) of G that occur in the coordinate ring of X. In the 1990s F. Knop conjectured that the weight monoid is a complete invariant for smooth affine spherical varieties, and in 2006 I. Loseu proved this conjecture. I will present joint work with G. Pezzini in which we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties. I will also discuss some applications obtained with Pezzini and K. Paulus.

Abstract: I will explain how one can describe derived categories of smooth projective varieties using full and strong exceptional collections, i.e. via an explicit finite-dimensional algebra. I will also explain the mutation of exceptional collections, and the role of the braid group. Another important ingredient in the description of a derived category is the Serre functor. It is given by Serre duality in algebraic geometry, and the Auslander-Reiten translation in the representation theory of algebras. The Serre functor induces an automorphism of the Grothendieck group, and in the case of a smooth projective surface this automorphism satisfies additional strong properties. I will review these results, and explain how it leads to the numerical classification of "noncommutative surfaces" of rank 4 due to de Thanhoffer--Van den Bergh.

Abstract: The numerical classification of "noncommutative surfaces" of rank 4 suggests the existence of a family previously not considered in the literature. By generalising Orlov's blowup formula to blowups of sheaves of maximal orders outside the ramification locus, we construct these starting from Artin--Schelter regular algebras which are finite over their center for all cases in the classification. Previously the first non-trivial case in the classification was constructed by de Thanhoffer--Presotto using noncommutative P^1-bundles. We can compare this to the blowup construction using some very classical geometry of linear systems. This comparison can be seen as a noncommutative instance of the classical isomorphism between the first Hirzebruch surface and the blowup of P^2 in a point. This is joint work with Dennis Presotto and Michel Van den Bergh.

Abstract: The Yang-Baxter equation is an important tool in theoretical physics and pure mathematics, with many applications in different domains going from condensed matter to topology. The importance of this equation led Drinfeld to ask for studying the simplest family of solutions: combinatorial or set-theoretical solutions. In this talk we review the basic theory of set-theoretical solutions, we discuss some problems and solutions and we give some application.

Abstract: We discuss the notion of a “mate” of a square in a 2-category. We will explain how it is related to base change in algebraic geometry, and that it can be understood as a homotopic condition. We then explain how this can be used to categorify the notion of Hopf algebra, and the Heisenberg double construction.

Abstract: The Hall algebra associated to a category is related to the Waldhausen S-construction in the work of Kapranov and Dyckerhoff. We explain how the higher associativity data can be extracted from this construction in a natural way, thus allowing for various higher categorical versions of Hall algebra. We then discuss a natural and systematic extension of this construction providing a bi-algebraic structure. We show how it can be used to provide a more transparent proof for the Green's theorem for the Hall algebras of hereditary categories and discuss possible extension to the higher categorical setting.

Abstract: Deformation quantization is a process that assigns to any classical mechanical system its quantum mechanical analogue. The problem can be phrased in purely algebraic terms: we would like to start with a commutative ring equipped with a Poisson bracket, and produce a noncommutative deformation of its product. A priori, the Poisson bracket only specifies the deformation to first order in the deformation parameter, but a deep theorem of Maxim Kontsevich extends the deformation to all orders in a canonical way. While the problem is algebraic, his solution is transcendental: it involves integrals over high-dimensional configuration spaces. I will give an elementary introduction to his formula, and talk about the (very few) examples in which it can actually be computed by hand.

Abstract: The integrals appearing in Kontsevich's deformation quantization formula are notoriously difficult to compute. As a result, direct calculations with the formula have so far been intractable, even in very simple examples. In forthcoming work with Peter Banks and Erik Panzer, we give an algorithm for the exact evaluation of the integrals in terms of special transcendental constants: the multiple zeta values. It allows us to calculate the formula on a computer for the first time. I will give an overview of our approach, which recasts the integration problem in purely algebraic terms, using Francis Brown's theory of single-valued multiple polylogarithms.

Abstract: A notion of prime ideal in a commutative ring can be expressed in many equivalent statements that become distinct when the ring is assumed to be noncommutative. This leads to: completely prime ideals, strictly prime ideals, strongly prime ideals, s-prime ideals, l-prime ideals, etc. Each of the aforementioned “prime” has a module analogue; and these analogues collapse to just one notion when a module is defined over a commutative ring. In this talk, I discuss the completely prime (sub)modules, their properties and torsion theories induced by the completely prime radical. Secondly, by comparing completely prime (sub)modules with prime (sub)modules, I talk about a class of 2-primal modules which is a module analogue of 2-primal rings.

Abstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.

Abstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.

Abstract: Abstract: I will explain how to study the Poisson (co)homology of a Poisson variety locally via D-modules. When there are finitely many symplectic leaves, the zeroth Poisson homology is finite-dimensional, and as an application, one has finitely many irreducible representations of every quantization. In the case that the variety is conical and admits a symplectic resolution, this conjecturally recovers the cohomology of the resolution and equips it with filtrations recording the order of vanishing of fiberwise closed differential forms on smoothings. In the case of nilpotent cones, this recovers a conjectural formula of Lusztig in terms of Kostka polynomials. In the case of smooth Poisson varieties, work in progress with Brent Pym shows that the entire Poisson cohomology is finite-dimensional when the modular filtration is finite and defines a perverse sheaf. This has applications to Feigin-Odesski Poisson structures on even-dimensional projective spaces.

Abstract: Conformal field theories, and the fusion categories derived from them, provide classes in group cohomology that generalize characteristic classes. These classes are called "anomalies", and obstruct the existence of constructing orbifold models. I will discuss two of the most charismatic groups --- the Conway group Co_0 and the Fischer--Griess Monster group M --- and explain my calculation that in both cases the anomaly has order exactly 24. The Monster calculation relies on a version of "T-duality" for finite groups which in turn relies on fundamental results about fusion categories. I will try to explain everything from the beginning, and assume no knowledge of the Monster or its cousins.

Abstract: Abstract TBA

Abstract: By a result of Ingalls and Thomas, one can think of the bounded derived category of finite dimensional representations of an ADE quiver as a categorification of non-crossing partitions of the corresponding type. The non-crossing partitions are precisely the lattice of exact localizations of the bounded derived category. I'll discuss various directions in which one can generalise this, such as the extension to doubly infinite Dynkin type A, representations over more general rings, and (time permitting) the situation for tame quivers. This is based on joint work with Gratz, with Antieau, and with Krause.

Abstract: In this talk, we recall some basic facts about the Yangian of gl(n) and, in particular, the role played by the Capelli identity in the definition of the Yangian and in the construction of the evaluation homomorphism. We then describe a similar result for sl(n), which leads to an apparently new presentation of the evaluation homomorphism in this case.

Abstract: It is well known that the universal enveloping algebra of a finite dimensional Lie algebra admits no non-trivial deformations as an algebra. In particular, there exists a (non-explicit) isomorphism between the trivial deformation of the enveloping algebra and the corresponding quantum group. An explicit description of such isomorphism was known only for sl(2). In this talk, we introduce a new realisation of the evaluation homomorphism of the Yangian of sl(n) and we use it to obtain an explicit isomorphism between classical and quantum sl(n). This is a work in progress with S. Gautam.

Abstract: We will give a gentle introduction to some of the basic invariants of singularities of rings in positive characteristic defined via the Frobenius endomorphism. In particular, we will pay close attention to F-signature and Hilbert-Kunz multiplicity, highlighting the known examples for each.

Abstract: Bertini Theorems for F-signature Abstract: In characteristic zero, it is well known that multiplier ideals and log terminal singularities satisfy Bertini-type theorems for hyperplane sections. The analogous situation in characteristic p > 0 is more complicated. While F-regular singularities satisfy Bertini, the test ideal does not. In this talk, I will describe joint work with Karl Schwede and Javier Carvajal-Rojas showing that the F-signature -- a numerical invariant of singularities that detects F-regularity -- satisfies the relevant Bertini statements for hyperplane sections. In particular, one can view this as a generalization of the corresponding results for F-regularity.

Abstract: We introduce the concepts of a Groebner basis, Hilbert series and growth of an algebra. These notions will be demonstrated on the following result. We present a simple example (4 generators and 7 relations) of a quadratic semigroup algebra of intermediate growth. The proof is obtained by computing the (infinite) reduced Groebner basis in the ideal of relations. Although the basis follows a clear and simple pattern, the corresponding set of normal words fails to form a regular language. The latter is noteworthy in its own right. The only previously known example of a quadratic algebra of intermediate growth due to Kocak is non-semigroup and is given by 14 generators and 96 quadratic relations.

Abstract: We answer a question of Ufnarovskii whether an automaton algebra must possess a generating set and an order on monomials with respect to which the reduced Groebner basis in the ideal of relations is finite. Namely, we present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Groebner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. Note that extending the ground field does not help. The proof is partially based on sensitivity of the growth of an algebra to characteristic of the ground field, which is restricted in case of finite Groebner basis.

Abstract: I will introduce a few key concepts in the theory of noncommutative projective geometry. In particular, I aim to give an idea of how one should think of a noncommutative curve/ surface. I will also describe a key example called a twisted homogeneous coordinate ring: a ring built out of geometry which plays a vital role in the theory.

Abstract: A major current goal for noncommutative projective geometers is the classification of so-called “noncommutative surfaces”. Let S denote the 3-dimensional Sklyanin algebra, then S can be thought of as the generic noncommutative surface. In recent work Rogalski, Sierra and Stafford have begun a project to classify all algebras birational to S. They successfully classify the maximal orders of the 3-Veronese subring T of S. These maximal orders can be considered as blowups at (possibly non-effective) divisors on the elliptic curve E associated to S. We are able to obtain similar results in the whole of S.

Abstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case.

Abstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case.

Abstract: I will talk about a classical problem of Halmost on almost commuting matrices and our recent result with Lukasz Grabowski.

Abstract: I will give some details about the proof of our results on almost commuting matrices that includes effective algebraic geometry and commutative algebra as well as the algebraic Ornstein Weiss principle.

Abstract: Exponential algebra extends commutative algebra by taking account of rings (particularly fields) with an exponential function. Classically there are few examples, but those, the real and the complex exponentials, are of great importance for all of mathematics. It is not even clear what should be the definition of an exponential ring, and it is certainly not at all clear what exponentially algebraic should mean. Historically Hardy (around 1911) used some tricks of the trade to get good information about zeros of one variable exponential polynomials, and Ritt, in the late 1920’s, established a quite subtle factorization theorem for one variable exponential polynomials. These in turn linked to questions about the distribution of zeros of systems of exponential polynomials. Some of these problems have remained open, and turn out to be connected both to transcendental number theory and to mathematical logic (decidability and definability issues). In the first part I will explain some basic definitions and constructions (e.g of free exponential rings),and sketch the connection to Schanuel’s Conjecture from transcendental number theory. I will also explain how logicians came to these problems, and what difference their ideas made in establishing a quite elaborate subject of exponential algebra.

Abstract: I will explain how the ideas from Part 1 connect to difficult parts of analysis and/or topology(Nevanlinna Theory, Morse Theory and Shapiro’s Conjecture from 1950 about common zeros of exponential polynomials). Serious work from Diophantine geometry is involved, due to Bombieri, Masser and Zannier.

Abstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo.

Abstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo.

Abstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542).

Abstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542).

Abstract: Suppose that we have a framework consisting of finitely many fixed-length bars connected at universal joints. Such frameworks (and variants) arise in many guises, with applications to the study of sensor networks, the matrix completion problem in statistics, robotics and protein folding. The fundamental question in rigidity theory is to determine if a framework is rigid or flexible. The standard approach in combinatorial rigidity theory is to differentiate the quadratic equations constraining the distances between joints, and work with these linear equations to determine if the framework is infinitesimally rigid or flexible. In this talk I will discuss recent progress using algebraic matroids that gives further insight into the infinitesimal theory and also provides methods for identifying special bar lengths for which a generically rigid framework is flexible. We use circuit polynomials to identify stresses, or dependence relations among the linearized distance equations and to find bar lengths which give rise to motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

Abstract: The aim of this lecture is to describe techniques that enable one to provide vector space bases for associative and Lie algebras which are given in terms of generators and defining relations.

Abstract: The aim of the talk is to present some joint results with Alexander Olshansky. When an algebra B is embedded in an algebra A then the growth functions of A produce some growth-like functions on B. Comparing these functions with the growth functions of B, one can speak about embeddings with or without distortion. We study these and related phenomena for general algebras, but the main results are in the case of associative and Lie algebras.

Abstract: We summarize part of the basic theories of Jordan algebras and (positive hermitian) Jordan triple systems in finite dimensions. This will include connections with other topics including several complex variables and homogeneous cones.

Abstract: In recent work with Les Bunce we have investigated the different ways to embed a JC*-triple as JC*-subtriple of all operators. This relates to the associative algebraic structure generated by the triple and is also reflected in the geometry of matrices over the triple. The concept of universal reversibility plays a significant role. We will describe some of this work when restricted to the finite dimensional case.

Abstract: The Hall algebra of an abelian category has structure constants coming from "counting extensions" in the category. In this talk we give a survey of some recent results involving the Hall algebra of the category of coherent sheaves on an elliptic curve. Some topics involve an explicit description of the algebra by Burban and Schiffmann and a construction of Schiffmann and Vasserot of an action on the space of symmetric functions using Hilbert schemes and double affine Hecke algebras.

Abstract: The construction of the Fukaya category of a symplectic manifold is inspired by string theory: roughly, objects are Lagrangians, morphisms are intersection points, and composition of morphisms comes from "holomorphic disks." For surfaces, a combinatorial construction of the (partially wrapped) Fukaya category was recently given by Haiden, Katzarkov, and Kontsevich. We will discuss this category and some recent results involving its (derived) Hall algebra (joint with B. Cooper).

Abstract: This talk is based on joint work with Arkady Berenstein. It frequently happens that an algebra C factors as C=AB, meaning a vector space isomorphism between C and the tensor product of its subalgebras A and B. The classical PBW theorem and its more recent incarnations -- think quantum groups and affine Hecke algebras -- are statements about algebra factorizations. Conversely, an algebra structure on the tensor product of can be established in many cases: semidirect product, braided tensor product, etc, which all fit the situations when A is an H-module algebra and B is an H-comodule algebra for some bialgebra H. We show that, quite surprisingly, any algebra factorization C=AB can be realised in this way for a suitable H: an ordinary bialgebra if the factorization is tame (which is typically the case), or a topological bialgebra in general. In particular, when C is a rational Cherednik algebra or a Kostant-Kumar nilHecke algebra, reconstructing H leads us to a Nichols algebra. DAHA and its generalizations correspond to the Hecke-Hopf algebras H, recently found by Berenstein and Kazhdan. Even in more straightforward examples of algebra factorisations, H can be a new and interesting Hopf algebra the representation theory of which begs to be explored.

Abstract: We define a notion of fast generating sets, for groups of self-homeomorphisms of a space (under composition), and apply it to the group Pl_o(I) of piecewise-linear homeomorphisms of the unit interval. As a consequence, we build some general forms of Ping-Pong Lemmas for this group, which lemmas guarantee isomorphism types for certain fg subgroups of Plo(I), based on simple combinatorial data. We also find a lemma which guarantees that some particular (unexpectedly large) set of subgroups of Pl_o(I) also embed in R. Thompson’s group F. Joint with Matt Brin and Justin Moore.

Abstract: Dimer models with boundary were introduced in joint work with King and Marsh as a natural generalisation of dimers. We use these to derive certain infinite dimensional algebras and consider idempotent subalgebras w.r.t. the boundary. The dimer models can be embedded in a surface with boundary. In the disk case, the maximal CM modules over the boundary algebra are a Frobenius category which categorifies the cluster structure of the Grassmannian.

Abstract: It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise. In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not ‘run around once’ but ‘run out’, we coined the term exodromy representation. In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine.

Abstract: In 1985 Zamolodchikov constructed a "non-linear" extension of the Virasoro algebra known as W(3)-algebra. This is the one of the first appearance of a rich class of algebraic structures, known as W-algebras, which are intimately related to physical theories with symmetry and revealed many applications in mathematics . In the first part of the talk I will review some facts about the general theory of W-algebras. Then, I will explain how to describe quantum finite and classical affine W-algebras using Lax operators. In the quantum finite case this operator satisfies a generalized Yangian identity, while in the classical affine case it is used to construct an integrable Hamiltonian hierarchy of Lax type equations. This is a joint work with A. De Sole and V.G. Kac.

Abstract: In this talk I will look at the notion of tropical critical points of the superpotential of the flag variety (in type A). A superpotential for any flag variety of general type was defined in the work of K. Rietsch, and there it was also shown how the critical points of this superpotential describe the quantum cohomology of the original flag variety. By tropicalising this superpotential, one can associate to any finite dimension representation of the group SLn, a family of polytopes indexed by the reduced expressions for the longest element of the Weyl group. I will then show how looking at the tropical critical points of the superpotential gives a distinguished point in each of these polytopes, and will also interpret this point via a construction coming just from the original flag variety.

Abstract: It has long been expected, and is now proved in many important cases, that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered. I will outline an approach to this question using the ideas of noncommutative projective geometry, from which we see that the correct object to study is a groupoid, rather than a group, and maps in this groupoid are the replacement for automorphisms. I will illustrate this with the example of quantum projective space. This is joint work with Nicholas Cooney (Clermont-Ferrand).

Abstract: We define the affine VW supercategory sW, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group. The main obstacle was the lack of a quadratic Casimir element for p(n). When M is the trivial representation, the action factors through the action of the previously known Brauer supercategory sBr. Our main result is an explicit basis theorem for the morphism sW, and as a consequence we recover the basis theorem for sBr. The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation. This is joint work with Zajj Daugherty, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova and Catharina Stroppel, arising from the WINART workshop.

Abstract: The classical Dixmier-Moeglin equivalence for noetherian algebras studies when three seemingly distinct notions for prime ideals are in fact equivalent (more precisely, the notions of locally-closed, primitive, and rational). This equivalence is known to hold for a large class of algebras (including commutative algebras, and quantised coordinate rings). In the last four years, there has been applications of the model theory of differential fields that yield new examples where the equivalence does not hold, and, more recently, also establish the equivalence for certain families of Hopf-Ore extensions. In this talk, I will give an overview of how this connection between "model theory" and "the Dixmier-Moeglin equivalence" arises. This will cover several results obtained jointly with Jason Bell, Stephane Launois, and Rahim Moosa.

Abstract: Nakajima's quiver varieties form an important class of algebraic symplectic varieties. A quiver variety comes naturally equipped with certain “tautological vector bundles”; I will explain joint work with McGerty that shows that the cohomology ring of the quiver variety is generated by the Chern classes of the tautological bundles. Analogous results (work in preparation with McGerty) also hold for the Crawley-Boevey—Shaw “multiplicative quiver varieties,’’ in particular for twisted character varieties; and the cohomology results in both cases generalize to other cohomology theories, derived categories, etc. I hope to explain the main ideas behind the proofs of such theorems and how they form part of a general pattern in noncommutative geometry.

Abstract: I will report on joint work with D. Wyss and P. Ziegler. We prove a conjecture by Hausel-Thaddeus which predicts an agreement of appropriately defined Hodge numbers for certain moduli spaces of Higgs bundles over the complex numbers. Despite the complex-analytic nature of the statement our proof is entirely arithmetic. If time permits I will also discuss the connection to the fundamental lemma.

Abstract: The theory of quantum symmetric pairs provides coideal subalgebras of quantized enveloping algebras which are quantum group analogs of Lie subalgebras fixed under an involution. The finite dimensional representations of a quantized enveloping algebra form a braided monoidal category C, and the finite dimensional representations of any coideal subalgebra form a module category over C. In this talk I will discuss the notion of a braided module category over C as introduced by A. Brochier in 2013, and I will explain how quantum symmetric pairs provide examples. Time permitting, I will also indicate how the underlying universal K-matrix can be employed to describe a basis of the centre of quantum symmetric pair coideal subalgebras. This simplifies joint work with G. Letzter from 2006.

Abstract: For L/K a finite Galois extension of fields with Galois group G the existence of a normal basis implies that L is a free K[G]-module of rank one. In general there can be other Hopf algebras acting on L with similar properties, namely those which endow L/K with a Hopf-Galois structure. Hopf-Galois theory was initially introduced in 1969 by S. Chase and M. Sweedler and has applications in Galois module theory. On the other hand, the Yang-Baxter equation is a matrix equation for the linear automorphisms of the tensor product of a vector space with itself. The Yang-Baxter equation is one of the important equations in quantum group theory, which has applications in mathematical physics. In 1992 V. Drinfeld suggested studying the set-theoretic version of this equation as a simpler way of solving some instances of it. Currently, the classification of Hopf-Galois structures and the set-theoretic solutions of the Yang-Baxter equation are among important topics of research. In this talk we will explain how the study of Hopf-Galois theory and the Yang-Baxter equation came to be connected via algebraic objects called skew braces. Then we will explain how one can classify and study the Hopf-Galois structures and skew braces of order p^3 for a prime number p.

Abstract: We define and study Hall algebra structures on the (co)homology of the moduli stacks of coherent sheaves and Higgs sheaves on compact Riemann surfaces. We provide some generation and torsion-freeness results in both cases, and an algebraic presentation in the case of coherent sheaves. This is joint work with F. Sala, and E. Vasserot respectively.

Abstract: Fusion categories are tensor categories that look much like the category of complex representations of a finite group: they have duals, are semisimple, and have finitely many simple objects. In addition to finite groups, the main source of examples are the semisimplified quantum groups at roots of unity. Moore and Seiberg asked whether quantum group categories might explain all fusion categories. The goal of this talk is to survey the current state of knowledge about ''exceptional'' fusion categories which don't seem to come from groups or quantum groups. In a sense this talk will be more like an experimental physics talk, in that one is searching for ''new particles'' in various regimes (e.g. ''low index subfactors'') and seeing what you can find. The punchline is that we know one new large family of fusion categories (the Izumi quadratic categories) and four isolated examples (the Extended Haagerup Subfactors). This will include some of my own work joint with Bigelow, Grossman, Izumi, Morrison, Penneys, Peters, and others, but also summarize the work of many other people (especially Asaeda, Bisch, Ocneanu, Haagerup, Izumi, Jones, and Popa working in Subfactor theory).

Abstract: An important theorem by Kohno, Drinfeld and Kazhdan-Lusztig states that the explicit representations of braid groups obtained from the representation theory of quantum groups compute the monodromy of the so-called KZ equation in conformal field theory. Remarkably this connection can also be interpreted as a quantization of the action by isomonodromy of the mapping class group on the moduli space of flat connections on a Riemann spheres with several punctures. In this talk I will sketch some recent progresses towards a higher genus version of this result, computing the monodromy of an analog of the KZ connection on the moduli spaces of Riemann surfaces in terms of a certain canonical quantization of the character varieties of those surfaces. This can be interpreted as a quantization of the symplectic nature and the mapping class group equivariance of the Riemann-Hilbert correspondence. This is partly based on joint work with D. Ben-Zvi, D. Jordan and N. Snyder.

Abstract: I will review the work done over the last few years on categorifications of the Knizhnik–Zamolodchikov connection via a differential crossed module of 2-chord diagrams. Possible applications to higher category theory and to the topology of knotted surfaces in the 4-sphere will be explored. All algebraic and differential-geometrical background will be carefully explained.

Abstract: I will talk about an ongoing project with Sachin Gautam aimed at computing the monodromy of differential and difference equations associated to Yangians. As corollaries, one obtains a meromorphic braided tensor equivalence between finite-dimensional representations of Yangians and quantum loop algebras, as well as a classification of finite-dimensional representations of elliptic quantum groups.

Abstract: The building blocks of ring theory are finite dimensional central simple algebras, governed by the Brauer group of the base field. Stretching the theory to infinity, we consider the class of algebras which are locally central simple, such as "supernatural" matrix algebras, and some infinite dimensional algebraic division algebras. We develop a theory for this class, generalizing the basic notions from the Brauer group, such as a supernatural degree and supernatural matrix degree, and define a Brauer monoid for countably generated locally central simple algebras. Examples such as infinite dimensional Clifford algebras and infinite dimensional crossed products serve to demonstrate the limitations of the new theory. This is based on joint work with Eli Matzri and our PhD students Shira Gilat and Tamar Bar-On.

Abstract: Given G a finite Abelian subgroup of SL_3(C) there is an associated coloured planar partition counting problem, which may be approached via studying Euler characteristics of moduli of sheaves on a crepant resolution of C^3/G. If G is trivial, this gives the McMahon partition function counting uncoloured planar partitions. A prediction from DT theory is that this partition function should be determined by a much sparser one, obtained by dividing by the McMahon partition function e times, where e is the Euler characteristic of any crepant resolution. I will discuss a recent conjecture with Szendroi and Ongaro, regarding this "reduced" partition function, namely, that it has only positive coefficients. The approach I will discuss for proving this conjecture is somewhat algebraic, and is an upgrade of the famous PT/DT correspondence to cohomology. Namely, the route to proving the conjecture consists of showing that the unreduced partition function is a module over a certain quantum group (the degree zero cohomological Hall algebra of sheaves on the resolution), where this quantum group has characteristic function given by the required power of the McMahon partition function. Freeness of this module then provides a categorification of the reduced partition function - it is the characteristic function of the set of generators for this module.

Abstract: In the talk we describe a natural solution to the quantum Yang-Baxter equation associated to the equivariant cohomology of the Grassmanian manifolds and study the appropriate quantum integrable system. We discuss the connection of the features of this quantum integrable system to Schubert calculus.

Abstract: In the 1960s, Deligne and Mumford compactified the moduli space of smooth curves by adding nodal curves at the boundary. We will look at the analogous question of compactifying the moduli space of weighted projective lines. Our interest is mainly in the various commutative and noncommutative degenerations that can naturally arise. This talk is about preliminary work done in this direction with Abdelgadir, Okawa and Ueda.

Abstract: The existence of a grading on a ring often makes computations a lot easier. In particular this is true for the computation of homotopy invariants. For example one can readily compute such invariants for the stable categories of graded modules over connected graded self-injective algebras. Using work of Tabuada, we will show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. This is based on joint work with Greg Stevenson.

Abstract: A configuration space of $k$ framed flags in an $N$-dimensional space is a pre-symplectic manifold admitting distinguished coordinates constructed as wedge products. If we take infinite $k$ but impose a symmetry with respect to a certain action of the fundamental group of a surface we obtain the coordinates on the (framed) $SL(N)$ character variety. If we take infinite $N$, but impose a symmetry with respect to a group $\mathbb{Z}^2$ we obtain coordinates on the phase space of a Goncharov-Kenyon (GK) integrable system - generalization of many known integrable systems arising from Poisson-Lie groups. This phase space can be identified with the space of planar curves provided with a line bundle. The GK integrable system admits continuous and discrete commuting flows. (The simplest example of the latter is the Poncelet porism). In coordinates this discrete flow corresponds to birational transformations composed of elementary ones called cluster mutations. We will give explicit formula of cluster coordinates, discuss stable points of these discrete flows and if time permits explain what the $\tau$ functions of Sato is and show that our cluster coordinates special values of this function.

Abstract: Let g be a finite dimenional simple complex Lie algebra. Following M. Noumi et al. (1995) and G. Letzter (1999) a fixed-point subalgebra of g with respect to an involutive automorphism can be quantized, which yields a coideal subalgebra B of U_q(g). This was re-engineered and generalized to arbitrary quantized Kac-Moody algebras by S. Kolb (2014). More recently, M. Balagović and S. Kolb showed that, for g of finite type, B is quasitriangular with respect to the category of finite-dimensional representations of U_q(g). As a consequence, associated to B there is a "universal solution" K of the reflection equation (4-braid relation). This entire story works in a more general setting, where the relevant subalgebra of g is no longer the fixed-point subalgebra of a semisimple automorphism. The underlying combinatorial data, dubbed "generalized Satake diagrams", arose previously in work by A. Heck on the classification of involutive automorphisms of root systems. Conjecturally, for g of finite type, this classifies all quasitriangular coideal subalgebras. Joint work with Vidas Regelskis (arXiv:1807.02388).

Abstract: The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown.

Abstract: The classical open relativistic Toda chain is a well-known integrable Hamiltonian system which appears in various different contexts in Lie theory and mathematical physics. As was observed by Gekhtman-Shapiro-Vainshtein, the phase space of the relativistic Toda chain admits the additional structure of a cluster variety. I will explain how this cluster structure can also be used to analyze the quantization of the relativistic Toda chain. In particular, we will see that the Baxter Q-operator for the quantum system can be realized as a sequence of quantum cluster mutations, which allows us to obtain a Givental-type integral representation of the Toda eigenvectors, the q-Whittaker functions. Joint work with Alexander Shapiro.

Abstract: In this talk I will show that q-W algebras, which are quantum group analogues of W-algebras, belong to the class of the so-called Mickelsson algebras. They can be described in terms of certain analogues of Zhelobenko operators.

Abstract: Laplace-Runge-Lenz vector represents hidden symmetry of Coulomb problem (equivalently, hydrogen atom), which is so(4). I am going to discuss its generalisation for the Dunkl settings in which a Coxeter group is present. The corresponding model is related to Calogero-Moser system, and the arising symmetry algebra is related to the Dunkl angular moment subalgebra of the rational Cherednik algebra. The talk is based on joint works with T. Hakobyan and A. Nersessian.

Abstract: Associative Clifford algebras have long been in use in the algebaic theory of quadratic forms, as well as in geometry and physics. Attempts to formualate the Rost invariant for quadratic forms led to the definition of the alternative Clifford algebra by Musgrave. We describe the structure of the alternative Clifford algebra of a ternary quadratic form, and present some other preliminary results and open problems. This talk is based on join work with Uzi Vishne.

Abstract: The theory of cluster algebras gives a concrete and explicit way to quantize character varieties for a group G and a surface S. In order to do this, one must show that character varieties admit natural cluster structures. It turns out to be enough to carry this out in the case of a disc with marked points. I will briefly explain the theory of cluster algebras and then describe a general procedure for constructing the cluster structure on Conf_n, the space of configurations of n flags for the group G. The key step will be to show that cluster variables can be realized in terms of tensor invariants of G.

Abstract: Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathcal{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathcal{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow.

Abstract: In this talk I will articulate and contextualize the following sequence of results. 1) The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group. 2) Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories. 3) In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.

Abstract: Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. In this talk we will introduce the idealizer and describe some interesting results about how the noetherianity of these subrings is closely linked to the orbit of I under the G-action. We will also give examples to show how this works in practice.

Abstract: From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. This is joint work with Wei Hu.

Abstract: Given a Fano variety X, Dubrovin's conjecture relates semisimplicity of the big quantum cohomology ring of X to the existence of a full exceptional collection in the derived category of coherent sheaves on X. One of the goals of our joint work with C.Pech, R.Gonzales, and N. Perrin is to establish Dubrovin's conjecture for the varieties in the title. In this talk, I will focus on the derived category side of the conjecture, and try to explain a connection to categorical joins that have recently been introduced by A.Kuznetsov and A.Perry. This is based on https://arxiv.org/abs/1803.05063.

Abstract: We show that behind any Lie-Rinehart algebra, and in particular behind any singular foliation or behind any affine variety, there is a canonical (homotopy class) of Lie-infinity algebroid (also called "dg-manifolds" or "Q-manifolds"). We are able to give an explicit construction. Also, we shall try to explain the algebraic and geometrical meanings of this higher structure. Joint works with Sylvain Lavau and Thomas Strobl.

Abstract: R-matrices arising from quantum affine algebras may be used to define the transfer matrix of a closed quantum spin chain. The 'prime directive' of the field of quantum integrable systems is to find the eigenvectors and eigenvalues of this transfer matrix. A powerful approach involves first constructing a new `Q-operator' defined as a trace over an infinite-dimensional module of a Borel subalgebra of the quantum affine algebra. A short exact sequence for infinite-dimensional modules then leads to a functional relation for simultaneous eigenvalues of the transfer matrix and Q-operator. These functional relations may then be solved exactly. In this talk I will summarise this approach and then extend it to 'open' quantum systems. The transfer matrix of an open system involves both the quantum affine algebra R-matrix and a solution of Cherednik's reflection equation associated with a coideal subalgebra. I will give a new construction of the Q-operator for such open systems and derive functional relations. This is joint work with Bart Vlaar.

Abstract: In this talk I will consider the Deligne category Rep((δ)) and its module category. The main point is to show that it categorifies a certain Fock space for a coideal subalgebra inside a quantum group. As an application we obtain a connection between decomposition numbers in Brauer centralizer algebras and Kazdhan-Lusztig polynomials of type D. If time allows we will also mention the relevance of this construction to the representation theory of Lie superalgebras.

Abstract: Brauer algebras B_n(\delta) are finite dimensional algebras introduced by Richard Brauer in order to study the n-th tensor power of the defining representations of the orthogonal and symplectic groups. They play the same role that the group algebras of the symmetric groups do for the representation theory of the general linear groups in the classical Schur-Weyl duality. We will discuss two different constructions which show that the Brauer algebras are graded cellular algebras and then show that they define the same gradings on B_n(\delta).

Abstract: In the 1960s Magnus conjectured that two one-relator groups and are isomorphic if and only if they "obviously" are. Although counter-examples were found in the 1970s, there exist important sub-classes of one-relator groups where the conjecture does hold. Indeed, the conjecture holds for almost all one-relator groups! I will unpack the word "obviously" in the previous paragraph, and I will explain the known counter-examples to the conjecture. All these counter-examples are non-hyperbolic. I will end the talk by giving a hyperbolic counter-example, as well as some positive results.

Abstract: Given a regular sequence (f1,..,fn) in a commutative K algebra R, we will study the homotopy abelianity (over K and over R) of the differential graded Lie algebra of endomorphisms of the Koszul resolution of the regular sequence. At the end, we will briefly discuss how the result give an annihilation theorem for obstructions to deformations of lci ideal sheaves.

Abstract: Many groups of geometric interest present an interesting dichotomy at the level of their subgroups: their subgroups are either very large (they contain a free subgroup) or very small (they are virtually abelian). In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove such a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Abstract: I will discuss the theory of 3d partitions, the counting of the number of ways to arrange boxes in the corner of a room (as opposed to boxes in a Young diagram). In applications to algebra and geometry, one introduces a colour scheme to these boxes, and refines the partition function to take account of the number of boxes of each colour. I will report on some recent work with Ongaro and Szendroi on the resulting partition functions, as well as q-refinements and positivity conjectures.

Abstract: For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

Abstract: Trigonometric Cherednik operators are a remarkable commutative family of commuting difference-reflection operators, arising in the basic representation of the double affine Hecke algebra of a root system R. There exists also their elliptic version, due to Komori and Hikami. I will show how to use these operators to construct Lax pairs (previously unknown beyond type A) for the Ruijsenaars models for all root systems. I will then explain how one can modify the elliptic R-matrices of Komori—Hikami and construct new generalisations of the Ruijsenaars models. Partly based on arXiv:1804.01766.

Abstract: In 1990s John H. Conway proposed "topographic" approach to describe the values of the binary quadratic forms, which can be applied also to the description of the celebrated Markov triples, featured in many areas of mathematics, including algebraic and hyperbolic geometry, theory of Frobenius manifolds and quiver mutations. In the talk I will review Conway’s approach from the point of view of the theory of two-valued groups. The first important examples of such groups were discovered by Buchstaber and Novikov in algebraic topology, which was developed further by Buchstaber and Rees. I will explain some classification results in the theory of two-valued groups, which emphasize again the role of the group PGL(2,Z) and present a novel view on the results of Conway, Markov and Mordell. The talk is based on a joint work with V.M. Buchstaber.

Abstract: In this project we give a construction of a system of Darboux-type coordinates on the space of unipotent upper triangular bilinear forms equipped with the Poisson structure discovered by A. Bondal in 1995. Some special cases of low dimensional symplectic leaves were identified earlier by L. Chekhov and M. Mazzocco with the Poisson algebras of hyperbolic length functions where Darboux-type coordinates are obtained by hyperbolic lengths of special system of loops. Utilizing the construction of Fock-Goncharov coordinates for flat SL_N connections on the disc with 3 marked points on the boundary, we compute Darboux-type coordinates for the maximal symplectic leaves. This is a joint work with L. Chekhov.

Abstract: In the study of cluster algebras, a combinatorial tool named scattering diagram plays an important role. In this talk, we will investigate the relation between the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich and the stability scattering diagram of Bridgeland. We will also discuss how scattering diagrams can be used to study the Donaldson-Thomas invariants of a quiver with potential.

Abstract: For a compact Lie group G, the classical Peter-Weyl Theorem states that the regular representation of G on L^2(G) decomposes as the direct sum of its irreducible unitary representations. These results were generalized for real reductive groups by Harish-Chandra, and for compact quantum groups by Woronowicz, at the same time, the case of non-compact quantum groups remained open. In this talk I will explain the Peter-Weyl Theorem for split real quantum groups of type A_n. I will discuss ingredients necessary to formulate and prove the theorem, including the GNS representations of C*-algebras, quantum parallel transports, and cluster realization of positive representations. This is a joint work with Gus Schrader and Alexander Shapiro.

Abstract: We will review the main structures used for renormalising singular SPDEs. Indeed, the solutions of these equations are described using stochastic processes indexed by decorated trees. Hopf algebras techniques have been successful for recentering these objects and proving their convergence.

Abstract: It is known that many (upper) cluster algebras possess very different good bases which are parametrized by the tropical points of Langlands dual cluster varieties. For any given injective reachable upper cluster algebra, we describe all of its bases parametrized by the tropical points. In addition, we obtain the existence of the generic bases for such upper cluster algebras. Our results apply to many cluster algebras arising from representation theory, including quantized enveloping algebras, quantum affine algebras, double Bruthat cells, etc.

Abstract: The singularity category (or stable derived category) was introduced by Buchweitz in 1986 and rediscovered in a geometric context by Orlov in 2003. It measures the failure of regularity of an algebra or scheme. Following Buchweitz, one defines the Tate-Hochschild cohomology of an algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. In recent work, Zhengfang Wang has shown that Tate-Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber bracket in cohomology and a B-infinity structure at the cochain level. This suggests that Tate-Hochschild cohomology might be isomorphic to the classical Hochschild cohomology of a (differential graded) category, in analogy with a theorem of Lowen-Van den Bergh in the classical case. We show that indeed, at least as a graded algebra, Tate-Hochschild cohomology is the classical Hochschild cohomology of the singularity category with its canonical dg enhancement. In joint work with Zheng Hua, we have applied this to prove a weakened version of a conjecture by Donovan-Wemyss on the reconstruction of a (complete, isolated) compound Du Val singularity from its contraction algebra, i.e. the algebra representing the non commutative deformations of the exceptional fiber of a resolution.

Abstract: Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\kappa$ such that $q=\exp(\frac{\pi i}{\kappa})$. We study the limit of the Drinfeld-Kohno correspondence when $\kappa\to 0$ along the imaginary line. Namely, on the KZ side this limit is the Gaudin integrable magnet chain, while on the quantum group side the limit is the tensor product of $\mathfrak{g}$-crystals. The limit of the braid group action by the monodromy of KZ equation is the action of the fundamental group of the Deligne-Mumford space of real stable rational curves with marked points (called cactus group) on the set of eigenlines for Gaudin Hamiltonians (given by algebraic Bethe ansatz). On the quantum group side, the cactus group acts by crystal commutors on the tensor product of $\mathfrak{g}$-crystals. We construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding tensor product of $\mathfrak{g}$-crystals, which preserves the natural cactus group action on these sets. If time allows I will also dicuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells. This is a joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes (https://arxiv.org/abs/1708.05105)

Abstract: In this talk I will describe how to make sense of the function (1+t)^x over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz.

Abstract: Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. Let $\cal S$ be a class of groups and let $f_{\cal S}(n)$ be the number of isomorphism classes of groups in $\cal S$, of order $n$. Some interesting classes that have been studied are the class of soluble groups, varieties of groups, $p$-groups and $A$-groups. Finite $A$-groups are those with abelian Sylow subgroups. In 1993, L Pyber proved a result, which bettered a conjecture, when he showed that $ f(n) \leq n^{\frac{2}{27}{\mu(n)}^2^2 + O({\mu(n)}^^{\frac{5}{3}})}$ where $\mu(n)$ denotes the maximum $\alpha$, such that $p^{\alpha}$ divides $n$ for any prime $p$. While Pyber's upper bound has the correct leading term, it is certainly not the case for the error term. The key to this puzzle may lie in deeper investigation of $A$-groups and varieties of $A$-groups. We present results concerned with asymptotic bounds for $f_{\cal S}(n)$ when ${\cal S}$ is a variety of $A$-groups, namely, ${\mathfrak U} = {\mathfrak A}_p{\mathfrak A}_q$, and ${\mathfrak V} = {\mathfrak A}_p{\mathfrak A}_q \vee {\mathfrak A}_q{\mathfrak A}_p$ and discuss open questions in this area. The talk will be self-contained and should be accessible to anyone with a basic knowledge of group theory.

Abstract: In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a Calabi-Yau category. The main difficulty to be overcome in that paper was dealing with the fact that TCFT's constructed form Calabi-Yau categories are always required to have at least one input. This problem was originally solved in a non-constructive fashion using dg-Weyl algebras and associated Fock spaces. In my talk I shall describe recent work on giving a new definition of Costello's invariants. We bypass the dg-Weyl algebra approach completely. Instead we use a Koszul resolution of the space of Sigma_n-invariant chains on M_{g,n}. This approach involves no choices, and makes the new invariants amenable to explicit computer calculations. I willl list some of the higher genus invariants that we computed; they agree with predictions from mirror symmetry. This talk is based on joint works with Junwu Tu and with Kevin Costello.

Abstract: Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk, we review the main problems related to Nichols algebras and we discuss some classification theorems. The talk is mainly based on joints works with I. Heckenberger.

Abstract: Skein modules are a natural extension of the Jones polynomial to 3-manifolds. In the case where the manifold is a thickened surface, they naturally come with a (usually) non-commutative multiplicative structure. I'll review basic definitions, formulas and conjectures about these skein modules, before discussing their categorification by foams. We will study simplicity properties, and explain the difference between the torus, where I will give an almost-proven categorified Frohman-Gelca formula, and other surfaces, where we will discuss reformulations of the Fock-Goncharov-Thurston positivity conjecture. This is joint with P. Wedrich.

Abstract: Given a family of principal bundles on an elliptic curve, a natural problem is to describe the locus of unstable bundles, and the fibres of the map from its complement to the coarse moduli space of semistable bundles. While the coarse moduli space map can be somewhat difficult to understand directly, it has a simultaneous resolution of singularities at the level of stacks, called the elliptic Grothendieck-Springer resolution, which is often easier to analyse in practice. After explaining this general machinery and its extension from semistable to unstable bundles, I will give some examples of families for which the unstable part of the resolution can be computed explicitly, from which a complete description of the entire picture miraculously follows.

Abstract: Morning Schedule: @10:00 Speaker: Minhyong Kim (University of Oxford) Title: Principal Bundles in Diophantine Geometry. Abstract: Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments. * @11:30 Speaker: James Lucietti (MPI Bonn) Title: Black holes in higher dimensions. Abstract: Black holes are one of the most remarkable predictions of Einstein's theory of General Relativity. I will give an overview of general results which constrain the topology and geometry of black holes in four and higher-dimensional spacetimes. While in four dimensions this leads to the celebrated black hole uniqueness theorem, in higher-dimensions this is not the case and their classification remains an open problem. I will describe recent progress in the classification of higher-dimensional black holes. *

Abstract: Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to their wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to K-theoretic fixed point classes of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present the result that, in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra---a first step towards developing a notion of 'wreath Macdonald operators'.

Abstract: A public lecture by Dr. Laura Ciobanu on the mathematics behind the languages we speak, and their connections to computer science. https://www.eventbrite.co.uk/e/mathematics-grammars-and-babies-tickets-72088389313?aff=eand

Abstract: Double brackets were introduced by M. Van den Bergh in his successful attempt to understand the Poisson geometry of (multiplicative) quiver varieties directly at the level of the path algebra of quivers. I will begin with a review of the basics of this theory and its relation to usual geometric structures. I will then move on to the properties of double brackets that can be used to study integrable systems. As a first application, I will explain how the double Poisson bracket on the path algebra of an extended Jordan (or one-loop) quiver can be used to easily derive integrable systems of Calogero-Moser type. As a second application, I will explain the corresponding relation between double quasi-Poisson brackets and Ruijsenaars-Schneider systems based on recent works with O. Chalykh (Leeds).

Abstract: The first half of the talk, which will be very algebraic, will describe intersection hyperplane arrangements inside Tits cones of (both finite and affine) Coxeter groups. I will demonstrate the local wall crossing rules, compute some examples, and state the classification of such hyperplane arrangements in small dimension. For example, these give precisely 16 tilings of the plane, with only 3 being the "traditional" Coxeter tilings. The new hyperplane arrangements turn out to be quite fundamental: they (1) classify all noncommutative resolutions for cDV singularities, (2) classify tilting theory for (contracted) projective algebras, and (3) also have many algebraic-geometric consequences. One is that they describe the stability manifold for an arbitrary 3-fold flop X --> Spec R, where X can have terminal singularities. Another is that they describe a large part of the autoequivalence group. Another is that they give lower bounds for certain curve-counting GV invariants. Parts of the talk are joint with Iyama, parts with Hirano, and parts with Donovan.

Abstract: Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

Abstract: https://www.maths.ed.ac.uk/~aappel/artin56.html

Abstract: https://www.maths.ed.ac.uk/~aappel/artin56.html

Abstract: Quiver Schur algebras are a generalization of Khovanov-Lauda-Rouquier algebras, well known for their role in the categorification of quantum groups. In this talk I will discuss their basic structural properties, as well as their connection to the cohomological Hall algebras defined by Kontsevich and Soibelman.

Abstract: Flag varieties have well known stratifications by Schubert cells. Intersecting two such stratifications corresponding to opposite Borel subgroups, one obtains finer stratifications which play a role in describing the totally nonnegative part of the flag varieties (in the sense of Lusztig). In joint work with Geiss and Schröer, we studied Frobenius categories attached to Schubert cells of flag varieties of type A,D,E, yielding cluster algebra structures on their coordinate rings. After recalling this, I will explain how one can extend this work to the above finer stratifications of the flag varieties.

Abstract: The free loop group of a compact Lie group has a very interesting representation theory, related to modular forms, integrable systems, quantum groups, vertex algebras, string theory... Of particular interest is the fusion product of representations. The representation theory of the *based* loop groups have never been considered before (even though individual examples of representations have been considered in disguise). We will explain how to extend the fusion product to representations of the based loop groups, and how to recover the category of representations of the free loop group from the category of representations of the based loop group. At last, I will explain in what sense I expect the representation theory of based loop groups to be wild (unlike that of the based loop groups), and which representations one might have a hope to classify.

Abstract: The homogeneous coordinate ring $\mathbb{C}[Gr(k,n)]$ of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmanniancluster category $C(k,n)$, as shown by Jensen-King-Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category ``$C(2,\infty)$''and how to model triangulations of a regular ``$\infty$-gon''. In this talk we show how the category $CM(R)$ of maximal Cohen-Macaulay modules over the coordinate ring $R$ for the $A_{\infty}$-curve allows us to construct triangulations of the $\infty$-gon, making use of the language of Grassmannian cluster categories. This is joint work with J. August, M. Cheung, S. Gratz, and S. Schroll.

Abstract: In this talk I will survey the results obtained in my joint paper with Francesco Sala, arXiv 1903.07253. Using techniques from derived geometry, I will explain how to find a natural categorification of K-theoretical Hall algebras associated to 2-dimensional objects. Among the examples, I will discuss the Hall algebra attached to a surface and the ones attached to Higgs bundles and flat vector bundles on a curve. If time permits, I will sketch how to obtain the categorification of certain natural representations.

Abstract: Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw. They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. Inthis talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part. Time permitting we may discuss the relation of this resultto a conjecture of Hausel.

Abstract: The universal elliptic KZB connection has a twisted (or cyclotomic) counterpart. This is a flat connection defined on a G-principal bundle over the moduli space of elliptic curves with n marked points and a (M; N)-level structure. Here the Lie algebra associated to G is constructed from a twisted elliptic Kohno-Drinfeld Lie algebra, the Lie algebra sl2, and a twisted derivation algebra controlling the algebraic information of some modular forms. After presenting this connection I will retrieve an ellipsitomic (or twisted elliptic) KZB associator from its monodromy and I will relate it to a connection associated to classical dynamical r-matrices with spectral parameter, following the work of Etingof and Schiffmann. Some parts of the results come from a joint work with Damien Calaque.

Abstract:

In mirror symmetry for non-Calabi-Yau varieties, a big role is played by Laurent polynomials. For example a kind of mirror object that first appeared in works of Batyrev and Givental on  smooth projective toric varieties, is given by an explicitLaurent polynomial living on the dual torus. Namely this Laurent polynomial is read off from the rays of the fan defining the toric variety. In this talk I will explain a joint work with Jamie Judd about Laurent polynomials over a field of generalised Puiseauxseries (Novikov field). Namely our result characterises the positive Laurent polynomials which have a unique positive critical point, as those whose Newton polytope has 0 in their interior. One application of this result using mirror symmetry and work of Fukaya,Oh, Ohta, Ono, and Woodward, is to the construction of non-dispaceable Lagrangians in smoth toric manifolds (resp. orbifolds).

Abstract: Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convexlattice polytopes. In this talk, I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifications of cluster varieties. Time permitting,I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrinsic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work withMan-Wai Cheung and Alfredo Nájera Chávez.

Abstract: The exotic nilpotent cone as defined by Kato gives a 'TypeA-like' Springer correspondence for Type C. In particular, there is a bijection between the symplectic group orbits on the exotic nilpotent cone and the irreducible representations of the Weyl group of Type C. In this talk, I will outline the various geometric and combinatorial results that follow from this. These results are joint work with Vinoth Nanadakumar and Daniele Rosso, and Arik Wilbert.


---------------------------


Instructions for joining the seminar:

This seminar will be hosted on Microsoft Teams. All attendees will need to join the University of Edinburgh team "Web MAXIMALS" before the seminar. University of Edinburgh students and staff should be able to add themselves by opening Microsoft Teams in Office365, clicking on "Join or create team" under the "Teams" tab, and searching for "Web MAXIMALS". External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added.

Abstract:
Real reflection groups are finite groups of real matrices generated by reflections, and they include several known families of groups, such as symmetric groups and dihedral groups. Their Hecke algebras appear as endomorphism algebras in the study of finite reductive groups. Complex reflection groups generalize real reflection groups in a natural way and their Hecke algebras were defined by Broué, Malle and Rouquier twenty years ago. However, many basic properties of Hecke algebras associated with real reflection groups were simply conjectured in the complex case. In this talk we will discuss some of the most fundamental conjectures, their state of art and our contribution towards their proof.


Instructions for joining the seminar:
This seminar will be hosted on Microsoft Teams. The Web MAXIMALS team can be accessed at https://teams.microsoft.com/l/channel/19%3ab3e8e7d9dfbd4fbd96e6e087666744e5%40thread.tacv2/General?groupId=00457dab-4343-4bb3-b941-077c303454fb&tenantId=2e9f06b0-1669-4589-8789-10a06934dc61. External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added. Please note that this seminar will be recorded unless objections are raised with the organisers beforehand.

Abstract: Your booking is for Sighthill HWRC, Bankhead Avenue, EH11 4EA. If you can no longer attend your booking then please cancel using the link provided in your confirmation email.

Abstract: Your booking is for Sighthill HWRC, Bankhead Avenue, EH11 4EA. If you can no longer attend your booking then please cancel using the link provided in your confirmation email.

Abstract:
Real reflection groups are finite groups of real matrices generated by reflections, and they include several known families of groups, such as symmetric groups and dihedral groups. Their Hecke algebras appear as endomorphism algebras in the study of finite reductive groups. Complex reflection groups generalize real reflection groups in a natural way and their Hecke algebras were defined by Broué, Malle and Rouquier twenty years ago. However, many basic properties of Hecke algebras associated with real reflection groups were simply conjectured in the complex case. In this talk we will discuss some of the most fundamental conjectures, their state of art and our contribution towards their proof.


Instructions for joining the seminar:
This seminar will be hosted on Microsoft Teams. The Web MAXIMALS team can be accessed at https://teams.microsoft.com/l/channel/19%3ab3e8e7d9dfbd4fbd96e6e087666744e5%40thread.tacv2/General?groupId=00457dab-4343-4bb3-b941-077c303454fb&tenantId=2e9f06b0-1669-4589-8789-10a06934dc61. External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added. Please note that this seminar will be recorded unless objections are raised with the organisers beforehand.

Abstract: The exotic nilpotent cone as defined by Kato gives a 'TypeA-like' Springer correspondence for Type C. In particular, there is a bijection between the symplectic group orbits on the exotic nilpotent cone and the irreducible representations of the Weyl group of Type C. In this talk, I will outline the various geometric and combinatorial results that follow from this. These results are joint work with Vinoth Nanadakumar and Daniele Rosso, and Arik Wilbert.


---------------------------


Instructions for joining the seminar:

This seminar will be hosted on Microsoft Teams. All attendees will need to join the University of Edinburgh team "Web MAXIMALS" before the seminar. University of Edinburgh students and staff should be able to add themselves by opening Microsoft Teams in Office365, clicking on "Join or create team" under the "Teams" tab, and searching for "Web MAXIMALS". External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added.

Abstract: Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convexlattice polytopes. In this talk, I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifications of cluster varieties. Time permitting,I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrinsic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work withMan-Wai Cheung and Alfredo Nájera Chávez.

Abstract:

In mirror symmetry for non-Calabi-Yau varieties, a big role is played by Laurent polynomials. For example a kind of mirror object that first appeared in works of Batyrev and Givental on  smooth projective toric varieties, is given by an explicitLaurent polynomial living on the dual torus. Namely this Laurent polynomial is read off from the rays of the fan defining the toric variety. In this talk I will explain a joint work with Jamie Judd about Laurent polynomials over a field of generalised Puiseauxseries (Novikov field). Namely our result characterises the positive Laurent polynomials which have a unique positive critical point, as those whose Newton polytope has 0 in their interior. One application of this result using mirror symmetry and work of Fukaya,Oh, Ohta, Ono, and Woodward, is to the construction of non-dispaceable Lagrangians in smoth toric manifolds (resp. orbifolds).

Abstract: The universal elliptic KZB connection has a twisted (or cyclotomic) counterpart. This is a flat connection defined on a G-principal bundle over the moduli space of elliptic curves with n marked points and a (M; N)-level structure. Here the Lie algebra associated to G is constructed from a twisted elliptic Kohno-Drinfeld Lie algebra, the Lie algebra sl2, and a twisted derivation algebra controlling the algebraic information of some modular forms. After presenting this connection I will retrieve an ellipsitomic (or twisted elliptic) KZB associator from its monodromy and I will relate it to a connection associated to classical dynamical r-matrices with spectral parameter, following the work of Etingof and Schiffmann. Some parts of the results come from a joint work with Damien Calaque.

Abstract: Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw. They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. Inthis talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part. Time permitting we may discuss the relation of this resultto a conjecture of Hausel.

Abstract: In this talk I will survey the results obtained in my joint paper with Francesco Sala, arXiv 1903.07253. Using techniques from derived geometry, I will explain how to find a natural categorification of K-theoretical Hall algebras associated to 2-dimensional objects. Among the examples, I will discuss the Hall algebra attached to a surface and the ones attached to Higgs bundles and flat vector bundles on a curve. If time permits, I will sketch how to obtain the categorification of certain natural representations.

Abstract: The homogeneous coordinate ring $\mathbb{C}[Gr(k,n)]$ of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmanniancluster category $C(k,n)$, as shown by Jensen-King-Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category ``$C(2,\infty)$''and how to model triangulations of a regular ``$\infty$-gon''. In this talk we show how the category $CM(R)$ of maximal Cohen-Macaulay modules over the coordinate ring $R$ for the $A_{\infty}$-curve allows us to construct triangulations of the $\infty$-gon, making use of the language of Grassmannian cluster categories. This is joint work with J. August, M. Cheung, S. Gratz, and S. Schroll.

Abstract: The free loop group of a compact Lie group has a very interesting representation theory, related to modular forms, integrable systems, quantum groups, vertex algebras, string theory... Of particular interest is the fusion product of representations. The representation theory of the *based* loop groups have never been considered before (even though individual examples of representations have been considered in disguise). We will explain how to extend the fusion product to representations of the based loop groups, and how to recover the category of representations of the free loop group from the category of representations of the based loop group. At last, I will explain in what sense I expect the representation theory of based loop groups to be wild (unlike that of the based loop groups), and which representations one might have a hope to classify.

Abstract: Flag varieties have well known stratifications by Schubert cells. Intersecting two such stratifications corresponding to opposite Borel subgroups, one obtains finer stratifications which play a role in describing the totally nonnegative part of the flag varieties (in the sense of Lusztig). In joint work with Geiss and Schröer, we studied Frobenius categories attached to Schubert cells of flag varieties of type A,D,E, yielding cluster algebra structures on their coordinate rings. After recalling this, I will explain how one can extend this work to the above finer stratifications of the flag varieties.

Abstract: Quiver Schur algebras are a generalization of Khovanov-Lauda-Rouquier algebras, well known for their role in the categorification of quantum groups. In this talk I will discuss their basic structural properties, as well as their connection to the cohomological Hall algebras defined by Kontsevich and Soibelman.

Abstract: https://www.maths.ed.ac.uk/~aappel/artin56.html

Abstract: https://www.maths.ed.ac.uk/~aappel/artin56.html

Abstract: Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

Abstract: The first half of the talk, which will be very algebraic, will describe intersection hyperplane arrangements inside Tits cones of (both finite and affine) Coxeter groups. I will demonstrate the local wall crossing rules, compute some examples, and state the classification of such hyperplane arrangements in small dimension. For example, these give precisely 16 tilings of the plane, with only 3 being the "traditional" Coxeter tilings. The new hyperplane arrangements turn out to be quite fundamental: they (1) classify all noncommutative resolutions for cDV singularities, (2) classify tilting theory for (contracted) projective algebras, and (3) also have many algebraic-geometric consequences. One is that they describe the stability manifold for an arbitrary 3-fold flop X --> Spec R, where X can have terminal singularities. Another is that they describe a large part of the autoequivalence group. Another is that they give lower bounds for certain curve-counting GV invariants. Parts of the talk are joint with Iyama, parts with Hirano, and parts with Donovan.

Abstract: Double brackets were introduced by M. Van den Bergh in his successful attempt to understand the Poisson geometry of (multiplicative) quiver varieties directly at the level of the path algebra of quivers. I will begin with a review of the basics of this theory and its relation to usual geometric structures. I will then move on to the properties of double brackets that can be used to study integrable systems. As a first application, I will explain how the double Poisson bracket on the path algebra of an extended Jordan (or one-loop) quiver can be used to easily derive integrable systems of Calogero-Moser type. As a second application, I will explain the corresponding relation between double quasi-Poisson brackets and Ruijsenaars-Schneider systems based on recent works with O. Chalykh (Leeds).

Abstract: A public lecture by Dr. Laura Ciobanu on the mathematics behind the languages we speak, and their connections to computer science. https://www.eventbrite.co.uk/e/mathematics-grammars-and-babies-tickets-72088389313?aff=eand

Abstract: Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to their wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to K-theoretic fixed point classes of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present the result that, in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra---a first step towards developing a notion of 'wreath Macdonald operators'.

Abstract: Morning Schedule: @10:00 Speaker: Minhyong Kim (University of Oxford) Title: Principal Bundles in Diophantine Geometry. Abstract: Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments. * @11:30 Speaker: James Lucietti (MPI Bonn) Title: Black holes in higher dimensions. Abstract: Black holes are one of the most remarkable predictions of Einstein's theory of General Relativity. I will give an overview of general results which constrain the topology and geometry of black holes in four and higher-dimensional spacetimes. While in four dimensions this leads to the celebrated black hole uniqueness theorem, in higher-dimensions this is not the case and their classification remains an open problem. I will describe recent progress in the classification of higher-dimensional black holes. *

Abstract: Given a family of principal bundles on an elliptic curve, a natural problem is to describe the locus of unstable bundles, and the fibres of the map from its complement to the coarse moduli space of semistable bundles. While the coarse moduli space map can be somewhat difficult to understand directly, it has a simultaneous resolution of singularities at the level of stacks, called the elliptic Grothendieck-Springer resolution, which is often easier to analyse in practice. After explaining this general machinery and its extension from semistable to unstable bundles, I will give some examples of families for which the unstable part of the resolution can be computed explicitly, from which a complete description of the entire picture miraculously follows.

Abstract: Skein modules are a natural extension of the Jones polynomial to 3-manifolds. In the case where the manifold is a thickened surface, they naturally come with a (usually) non-commutative multiplicative structure. I'll review basic definitions, formulas and conjectures about these skein modules, before discussing their categorification by foams. We will study simplicity properties, and explain the difference between the torus, where I will give an almost-proven categorified Frohman-Gelca formula, and other surfaces, where we will discuss reformulations of the Fock-Goncharov-Thurston positivity conjecture. This is joint with P. Wedrich.

Abstract: Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk, we review the main problems related to Nichols algebras and we discuss some classification theorems. The talk is mainly based on joints works with I. Heckenberger.

Abstract: In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a Calabi-Yau category. The main difficulty to be overcome in that paper was dealing with the fact that TCFT's constructed form Calabi-Yau categories are always required to have at least one input. This problem was originally solved in a non-constructive fashion using dg-Weyl algebras and associated Fock spaces. In my talk I shall describe recent work on giving a new definition of Costello's invariants. We bypass the dg-Weyl algebra approach completely. Instead we use a Koszul resolution of the space of Sigma_n-invariant chains on M_{g,n}. This approach involves no choices, and makes the new invariants amenable to explicit computer calculations. I willl list some of the higher genus invariants that we computed; they agree with predictions from mirror symmetry. This talk is based on joint works with Junwu Tu and with Kevin Costello.

Abstract: Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. Let $\cal S$ be a class of groups and let $f_{\cal S}(n)$ be the number of isomorphism classes of groups in $\cal S$, of order $n$. Some interesting classes that have been studied are the class of soluble groups, varieties of groups, $p$-groups and $A$-groups. Finite $A$-groups are those with abelian Sylow subgroups. In 1993, L Pyber proved a result, which bettered a conjecture, when he showed that $ f(n) \leq n^{\frac{2}{27}{\mu(n)}^2^2 + O({\mu(n)}^^{\frac{5}{3}})}$ where $\mu(n)$ denotes the maximum $\alpha$, such that $p^{\alpha}$ divides $n$ for any prime $p$. While Pyber's upper bound has the correct leading term, it is certainly not the case for the error term. The key to this puzzle may lie in deeper investigation of $A$-groups and varieties of $A$-groups. We present results concerned with asymptotic bounds for $f_{\cal S}(n)$ when ${\cal S}$ is a variety of $A$-groups, namely, ${\mathfrak U} = {\mathfrak A}_p{\mathfrak A}_q$, and ${\mathfrak V} = {\mathfrak A}_p{\mathfrak A}_q \vee {\mathfrak A}_q{\mathfrak A}_p$ and discuss open questions in this area. The talk will be self-contained and should be accessible to anyone with a basic knowledge of group theory.

Abstract: In this talk I will describe how to make sense of the function (1+t)^x over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz.

Abstract: Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\kappa$ such that $q=\exp(\frac{\pi i}{\kappa})$. We study the limit of the Drinfeld-Kohno correspondence when $\kappa\to 0$ along the imaginary line. Namely, on the KZ side this limit is the Gaudin integrable magnet chain, while on the quantum group side the limit is the tensor product of $\mathfrak{g}$-crystals. The limit of the braid group action by the monodromy of KZ equation is the action of the fundamental group of the Deligne-Mumford space of real stable rational curves with marked points (called cactus group) on the set of eigenlines for Gaudin Hamiltonians (given by algebraic Bethe ansatz). On the quantum group side, the cactus group acts by crystal commutors on the tensor product of $\mathfrak{g}$-crystals. We construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding tensor product of $\mathfrak{g}$-crystals, which preserves the natural cactus group action on these sets. If time allows I will also dicuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells. This is a joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes (https://arxiv.org/abs/1708.05105)

Abstract: The singularity category (or stable derived category) was introduced by Buchweitz in 1986 and rediscovered in a geometric context by Orlov in 2003. It measures the failure of regularity of an algebra or scheme. Following Buchweitz, one defines the Tate-Hochschild cohomology of an algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. In recent work, Zhengfang Wang has shown that Tate-Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber bracket in cohomology and a B-infinity structure at the cochain level. This suggests that Tate-Hochschild cohomology might be isomorphic to the classical Hochschild cohomology of a (differential graded) category, in analogy with a theorem of Lowen-Van den Bergh in the classical case. We show that indeed, at least as a graded algebra, Tate-Hochschild cohomology is the classical Hochschild cohomology of the singularity category with its canonical dg enhancement. In joint work with Zheng Hua, we have applied this to prove a weakened version of a conjecture by Donovan-Wemyss on the reconstruction of a (complete, isolated) compound Du Val singularity from its contraction algebra, i.e. the algebra representing the non commutative deformations of the exceptional fiber of a resolution.

Abstract: It is known that many (upper) cluster algebras possess very different good bases which are parametrized by the tropical points of Langlands dual cluster varieties. For any given injective reachable upper cluster algebra, we describe all of its bases parametrized by the tropical points. In addition, we obtain the existence of the generic bases for such upper cluster algebras. Our results apply to many cluster algebras arising from representation theory, including quantized enveloping algebras, quantum affine algebras, double Bruthat cells, etc.

Abstract: We will review the main structures used for renormalising singular SPDEs. Indeed, the solutions of these equations are described using stochastic processes indexed by decorated trees. Hopf algebras techniques have been successful for recentering these objects and proving their convergence.

Abstract: For a compact Lie group G, the classical Peter-Weyl Theorem states that the regular representation of G on L^2(G) decomposes as the direct sum of its irreducible unitary representations. These results were generalized for real reductive groups by Harish-Chandra, and for compact quantum groups by Woronowicz, at the same time, the case of non-compact quantum groups remained open. In this talk I will explain the Peter-Weyl Theorem for split real quantum groups of type A_n. I will discuss ingredients necessary to formulate and prove the theorem, including the GNS representations of C*-algebras, quantum parallel transports, and cluster realization of positive representations. This is a joint work with Gus Schrader and Alexander Shapiro.

Abstract: In the study of cluster algebras, a combinatorial tool named scattering diagram plays an important role. In this talk, we will investigate the relation between the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich and the stability scattering diagram of Bridgeland. We will also discuss how scattering diagrams can be used to study the Donaldson-Thomas invariants of a quiver with potential.

Abstract: In this project we give a construction of a system of Darboux-type coordinates on the space of unipotent upper triangular bilinear forms equipped with the Poisson structure discovered by A. Bondal in 1995. Some special cases of low dimensional symplectic leaves were identified earlier by L. Chekhov and M. Mazzocco with the Poisson algebras of hyperbolic length functions where Darboux-type coordinates are obtained by hyperbolic lengths of special system of loops. Utilizing the construction of Fock-Goncharov coordinates for flat SL_N connections on the disc with 3 marked points on the boundary, we compute Darboux-type coordinates for the maximal symplectic leaves. This is a joint work with L. Chekhov.

Abstract: In 1990s John H. Conway proposed "topographic" approach to describe the values of the binary quadratic forms, which can be applied also to the description of the celebrated Markov triples, featured in many areas of mathematics, including algebraic and hyperbolic geometry, theory of Frobenius manifolds and quiver mutations. In the talk I will review Conway’s approach from the point of view of the theory of two-valued groups. The first important examples of such groups were discovered by Buchstaber and Novikov in algebraic topology, which was developed further by Buchstaber and Rees. I will explain some classification results in the theory of two-valued groups, which emphasize again the role of the group PGL(2,Z) and present a novel view on the results of Conway, Markov and Mordell. The talk is based on a joint work with V.M. Buchstaber.

Abstract: Trigonometric Cherednik operators are a remarkable commutative family of commuting difference-reflection operators, arising in the basic representation of the double affine Hecke algebra of a root system R. There exists also their elliptic version, due to Komori and Hikami. I will show how to use these operators to construct Lax pairs (previously unknown beyond type A) for the Ruijsenaars models for all root systems. I will then explain how one can modify the elliptic R-matrices of Komori—Hikami and construct new generalisations of the Ruijsenaars models. Partly based on arXiv:1804.01766.

Abstract: For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

Abstract: I will discuss the theory of 3d partitions, the counting of the number of ways to arrange boxes in the corner of a room (as opposed to boxes in a Young diagram). In applications to algebra and geometry, one introduces a colour scheme to these boxes, and refines the partition function to take account of the number of boxes of each colour. I will report on some recent work with Ongaro and Szendroi on the resulting partition functions, as well as q-refinements and positivity conjectures.

Abstract: Many groups of geometric interest present an interesting dichotomy at the level of their subgroups: their subgroups are either very large (they contain a free subgroup) or very small (they are virtually abelian). In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove such a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Abstract: Given a regular sequence (f1,..,fn) in a commutative K algebra R, we will study the homotopy abelianity (over K and over R) of the differential graded Lie algebra of endomorphisms of the Koszul resolution of the regular sequence. At the end, we will briefly discuss how the result give an annihilation theorem for obstructions to deformations of lci ideal sheaves.

Abstract: In the 1960s Magnus conjectured that two one-relator groups and are isomorphic if and only if they "obviously" are. Although counter-examples were found in the 1970s, there exist important sub-classes of one-relator groups where the conjecture does hold. Indeed, the conjecture holds for almost all one-relator groups! I will unpack the word "obviously" in the previous paragraph, and I will explain the known counter-examples to the conjecture. All these counter-examples are non-hyperbolic. I will end the talk by giving a hyperbolic counter-example, as well as some positive results.

Abstract: Brauer algebras B_n(\delta) are finite dimensional algebras introduced by Richard Brauer in order to study the n-th tensor power of the defining representations of the orthogonal and symplectic groups. They play the same role that the group algebras of the symmetric groups do for the representation theory of the general linear groups in the classical Schur-Weyl duality. We will discuss two different constructions which show that the Brauer algebras are graded cellular algebras and then show that they define the same gradings on B_n(\delta).

Abstract: In this talk I will consider the Deligne category Rep((δ)) and its module category. The main point is to show that it categorifies a certain Fock space for a coideal subalgebra inside a quantum group. As an application we obtain a connection between decomposition numbers in Brauer centralizer algebras and Kazdhan-Lusztig polynomials of type D. If time allows we will also mention the relevance of this construction to the representation theory of Lie superalgebras.

Abstract: R-matrices arising from quantum affine algebras may be used to define the transfer matrix of a closed quantum spin chain. The 'prime directive' of the field of quantum integrable systems is to find the eigenvectors and eigenvalues of this transfer matrix. A powerful approach involves first constructing a new `Q-operator' defined as a trace over an infinite-dimensional module of a Borel subalgebra of the quantum affine algebra. A short exact sequence for infinite-dimensional modules then leads to a functional relation for simultaneous eigenvalues of the transfer matrix and Q-operator. These functional relations may then be solved exactly. In this talk I will summarise this approach and then extend it to 'open' quantum systems. The transfer matrix of an open system involves both the quantum affine algebra R-matrix and a solution of Cherednik's reflection equation associated with a coideal subalgebra. I will give a new construction of the Q-operator for such open systems and derive functional relations. This is joint work with Bart Vlaar.

Abstract: We show that behind any Lie-Rinehart algebra, and in particular behind any singular foliation or behind any affine variety, there is a canonical (homotopy class) of Lie-infinity algebroid (also called "dg-manifolds" or "Q-manifolds"). We are able to give an explicit construction. Also, we shall try to explain the algebraic and geometrical meanings of this higher structure. Joint works with Sylvain Lavau and Thomas Strobl.

Abstract: Given a Fano variety X, Dubrovin's conjecture relates semisimplicity of the big quantum cohomology ring of X to the existence of a full exceptional collection in the derived category of coherent sheaves on X. One of the goals of our joint work with C.Pech, R.Gonzales, and N. Perrin is to establish Dubrovin's conjecture for the varieties in the title. In this talk, I will focus on the derived category side of the conjecture, and try to explain a connection to categorical joins that have recently been introduced by A.Kuznetsov and A.Perry. This is based on https://arxiv.org/abs/1803.05063.

Abstract: From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. This is joint work with Wei Hu.

Abstract: Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. In this talk we will introduce the idealizer and describe some interesting results about how the noetherianity of these subrings is closely linked to the orbit of I under the G-action. We will also give examples to show how this works in practice.

Abstract: In this talk I will articulate and contextualize the following sequence of results. 1) The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group. 2) Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories. 3) In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.

Abstract: Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathcal{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathcal{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow.

Abstract: The theory of cluster algebras gives a concrete and explicit way to quantize character varieties for a group G and a surface S. In order to do this, one must show that character varieties admit natural cluster structures. It turns out to be enough to carry this out in the case of a disc with marked points. I will briefly explain the theory of cluster algebras and then describe a general procedure for constructing the cluster structure on Conf_n, the space of configurations of n flags for the group G. The key step will be to show that cluster variables can be realized in terms of tensor invariants of G.

Abstract: Associative Clifford algebras have long been in use in the algebaic theory of quadratic forms, as well as in geometry and physics. Attempts to formualate the Rost invariant for quadratic forms led to the definition of the alternative Clifford algebra by Musgrave. We describe the structure of the alternative Clifford algebra of a ternary quadratic form, and present some other preliminary results and open problems. This talk is based on join work with Uzi Vishne.

Abstract: Laplace-Runge-Lenz vector represents hidden symmetry of Coulomb problem (equivalently, hydrogen atom), which is so(4). I am going to discuss its generalisation for the Dunkl settings in which a Coxeter group is present. The corresponding model is related to Calogero-Moser system, and the arising symmetry algebra is related to the Dunkl angular moment subalgebra of the rational Cherednik algebra. The talk is based on joint works with T. Hakobyan and A. Nersessian.

Abstract: In this talk I will show that q-W algebras, which are quantum group analogues of W-algebras, belong to the class of the so-called Mickelsson algebras. They can be described in terms of certain analogues of Zhelobenko operators.

Abstract: The classical open relativistic Toda chain is a well-known integrable Hamiltonian system which appears in various different contexts in Lie theory and mathematical physics. As was observed by Gekhtman-Shapiro-Vainshtein, the phase space of the relativistic Toda chain admits the additional structure of a cluster variety. I will explain how this cluster structure can also be used to analyze the quantization of the relativistic Toda chain. In particular, we will see that the Baxter Q-operator for the quantum system can be realized as a sequence of quantum cluster mutations, which allows us to obtain a Givental-type integral representation of the Toda eigenvectors, the q-Whittaker functions. Joint work with Alexander Shapiro.

Abstract: The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown.

Abstract: Let g be a finite dimenional simple complex Lie algebra. Following M. Noumi et al. (1995) and G. Letzter (1999) a fixed-point subalgebra of g with respect to an involutive automorphism can be quantized, which yields a coideal subalgebra B of U_q(g). This was re-engineered and generalized to arbitrary quantized Kac-Moody algebras by S. Kolb (2014). More recently, M. Balagović and S. Kolb showed that, for g of finite type, B is quasitriangular with respect to the category of finite-dimensional representations of U_q(g). As a consequence, associated to B there is a "universal solution" K of the reflection equation (4-braid relation). This entire story works in a more general setting, where the relevant subalgebra of g is no longer the fixed-point subalgebra of a semisimple automorphism. The underlying combinatorial data, dubbed "generalized Satake diagrams", arose previously in work by A. Heck on the classification of involutive automorphisms of root systems. Conjecturally, for g of finite type, this classifies all quasitriangular coideal subalgebras. Joint work with Vidas Regelskis (arXiv:1807.02388).

Abstract: A configuration space of $k$ framed flags in an $N$-dimensional space is a pre-symplectic manifold admitting distinguished coordinates constructed as wedge products. If we take infinite $k$ but impose a symmetry with respect to a certain action of the fundamental group of a surface we obtain the coordinates on the (framed) $SL(N)$ character variety. If we take infinite $N$, but impose a symmetry with respect to a group $\mathbb{Z}^2$ we obtain coordinates on the phase space of a Goncharov-Kenyon (GK) integrable system - generalization of many known integrable systems arising from Poisson-Lie groups. This phase space can be identified with the space of planar curves provided with a line bundle. The GK integrable system admits continuous and discrete commuting flows. (The simplest example of the latter is the Poncelet porism). In coordinates this discrete flow corresponds to birational transformations composed of elementary ones called cluster mutations. We will give explicit formula of cluster coordinates, discuss stable points of these discrete flows and if time permits explain what the $\tau$ functions of Sato is and show that our cluster coordinates special values of this function.

Abstract: The existence of a grading on a ring often makes computations a lot easier. In particular this is true for the computation of homotopy invariants. For example one can readily compute such invariants for the stable categories of graded modules over connected graded self-injective algebras. Using work of Tabuada, we will show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. This is based on joint work with Greg Stevenson.

Abstract: In the 1960s, Deligne and Mumford compactified the moduli space of smooth curves by adding nodal curves at the boundary. We will look at the analogous question of compactifying the moduli space of weighted projective lines. Our interest is mainly in the various commutative and noncommutative degenerations that can naturally arise. This talk is about preliminary work done in this direction with Abdelgadir, Okawa and Ueda.

Abstract: In the talk we describe a natural solution to the quantum Yang-Baxter equation associated to the equivariant cohomology of the Grassmanian manifolds and study the appropriate quantum integrable system. We discuss the connection of the features of this quantum integrable system to Schubert calculus.

Abstract: Given G a finite Abelian subgroup of SL_3(C) there is an associated coloured planar partition counting problem, which may be approached via studying Euler characteristics of moduli of sheaves on a crepant resolution of C^3/G. If G is trivial, this gives the McMahon partition function counting uncoloured planar partitions. A prediction from DT theory is that this partition function should be determined by a much sparser one, obtained by dividing by the McMahon partition function e times, where e is the Euler characteristic of any crepant resolution. I will discuss a recent conjecture with Szendroi and Ongaro, regarding this "reduced" partition function, namely, that it has only positive coefficients. The approach I will discuss for proving this conjecture is somewhat algebraic, and is an upgrade of the famous PT/DT correspondence to cohomology. Namely, the route to proving the conjecture consists of showing that the unreduced partition function is a module over a certain quantum group (the degree zero cohomological Hall algebra of sheaves on the resolution), where this quantum group has characteristic function given by the required power of the McMahon partition function. Freeness of this module then provides a categorification of the reduced partition function - it is the characteristic function of the set of generators for this module.

Abstract: The building blocks of ring theory are finite dimensional central simple algebras, governed by the Brauer group of the base field. Stretching the theory to infinity, we consider the class of algebras which are locally central simple, such as "supernatural" matrix algebras, and some infinite dimensional algebraic division algebras. We develop a theory for this class, generalizing the basic notions from the Brauer group, such as a supernatural degree and supernatural matrix degree, and define a Brauer monoid for countably generated locally central simple algebras. Examples such as infinite dimensional Clifford algebras and infinite dimensional crossed products serve to demonstrate the limitations of the new theory. This is based on joint work with Eli Matzri and our PhD students Shira Gilat and Tamar Bar-On.

Abstract: I will talk about an ongoing project with Sachin Gautam aimed at computing the monodromy of differential and difference equations associated to Yangians. As corollaries, one obtains a meromorphic braided tensor equivalence between finite-dimensional representations of Yangians and quantum loop algebras, as well as a classification of finite-dimensional representations of elliptic quantum groups.

Abstract: I will review the work done over the last few years on categorifications of the Knizhnik–Zamolodchikov connection via a differential crossed module of 2-chord diagrams. Possible applications to higher category theory and to the topology of knotted surfaces in the 4-sphere will be explored. All algebraic and differential-geometrical background will be carefully explained.

Abstract: An important theorem by Kohno, Drinfeld and Kazhdan-Lusztig states that the explicit representations of braid groups obtained from the representation theory of quantum groups compute the monodromy of the so-called KZ equation in conformal field theory. Remarkably this connection can also be interpreted as a quantization of the action by isomonodromy of the mapping class group on the moduli space of flat connections on a Riemann spheres with several punctures. In this talk I will sketch some recent progresses towards a higher genus version of this result, computing the monodromy of an analog of the KZ connection on the moduli spaces of Riemann surfaces in terms of a certain canonical quantization of the character varieties of those surfaces. This can be interpreted as a quantization of the symplectic nature and the mapping class group equivariance of the Riemann-Hilbert correspondence. This is partly based on joint work with D. Ben-Zvi, D. Jordan and N. Snyder.

Abstract: Fusion categories are tensor categories that look much like the category of complex representations of a finite group: they have duals, are semisimple, and have finitely many simple objects. In addition to finite groups, the main source of examples are the semisimplified quantum groups at roots of unity. Moore and Seiberg asked whether quantum group categories might explain all fusion categories. The goal of this talk is to survey the current state of knowledge about ''exceptional'' fusion categories which don't seem to come from groups or quantum groups. In a sense this talk will be more like an experimental physics talk, in that one is searching for ''new particles'' in various regimes (e.g. ''low index subfactors'') and seeing what you can find. The punchline is that we know one new large family of fusion categories (the Izumi quadratic categories) and four isolated examples (the Extended Haagerup Subfactors). This will include some of my own work joint with Bigelow, Grossman, Izumi, Morrison, Penneys, Peters, and others, but also summarize the work of many other people (especially Asaeda, Bisch, Ocneanu, Haagerup, Izumi, Jones, and Popa working in Subfactor theory).

Abstract: We define and study Hall algebra structures on the (co)homology of the moduli stacks of coherent sheaves and Higgs sheaves on compact Riemann surfaces. We provide some generation and torsion-freeness results in both cases, and an algebraic presentation in the case of coherent sheaves. This is joint work with F. Sala, and E. Vasserot respectively.

Abstract: For L/K a finite Galois extension of fields with Galois group G the existence of a normal basis implies that L is a free K[G]-module of rank one. In general there can be other Hopf algebras acting on L with similar properties, namely those which endow L/K with a Hopf-Galois structure. Hopf-Galois theory was initially introduced in 1969 by S. Chase and M. Sweedler and has applications in Galois module theory. On the other hand, the Yang-Baxter equation is a matrix equation for the linear automorphisms of the tensor product of a vector space with itself. The Yang-Baxter equation is one of the important equations in quantum group theory, which has applications in mathematical physics. In 1992 V. Drinfeld suggested studying the set-theoretic version of this equation as a simpler way of solving some instances of it. Currently, the classification of Hopf-Galois structures and the set-theoretic solutions of the Yang-Baxter equation are among important topics of research. In this talk we will explain how the study of Hopf-Galois theory and the Yang-Baxter equation came to be connected via algebraic objects called skew braces. Then we will explain how one can classify and study the Hopf-Galois structures and skew braces of order p^3 for a prime number p.

Abstract: The theory of quantum symmetric pairs provides coideal subalgebras of quantized enveloping algebras which are quantum group analogs of Lie subalgebras fixed under an involution. The finite dimensional representations of a quantized enveloping algebra form a braided monoidal category C, and the finite dimensional representations of any coideal subalgebra form a module category over C. In this talk I will discuss the notion of a braided module category over C as introduced by A. Brochier in 2013, and I will explain how quantum symmetric pairs provide examples. Time permitting, I will also indicate how the underlying universal K-matrix can be employed to describe a basis of the centre of quantum symmetric pair coideal subalgebras. This simplifies joint work with G. Letzter from 2006.

Abstract: I will report on joint work with D. Wyss and P. Ziegler. We prove a conjecture by Hausel-Thaddeus which predicts an agreement of appropriately defined Hodge numbers for certain moduli spaces of Higgs bundles over the complex numbers. Despite the complex-analytic nature of the statement our proof is entirely arithmetic. If time permits I will also discuss the connection to the fundamental lemma.

Abstract: Nakajima's quiver varieties form an important class of algebraic symplectic varieties. A quiver variety comes naturally equipped with certain “tautological vector bundles”; I will explain joint work with McGerty that shows that the cohomology ring of the quiver variety is generated by the Chern classes of the tautological bundles. Analogous results (work in preparation with McGerty) also hold for the Crawley-Boevey—Shaw “multiplicative quiver varieties,’’ in particular for twisted character varieties; and the cohomology results in both cases generalize to other cohomology theories, derived categories, etc. I hope to explain the main ideas behind the proofs of such theorems and how they form part of a general pattern in noncommutative geometry.

Abstract: The classical Dixmier-Moeglin equivalence for noetherian algebras studies when three seemingly distinct notions for prime ideals are in fact equivalent (more precisely, the notions of locally-closed, primitive, and rational). This equivalence is known to hold for a large class of algebras (including commutative algebras, and quantised coordinate rings). In the last four years, there has been applications of the model theory of differential fields that yield new examples where the equivalence does not hold, and, more recently, also establish the equivalence for certain families of Hopf-Ore extensions. In this talk, I will give an overview of how this connection between "model theory" and "the Dixmier-Moeglin equivalence" arises. This will cover several results obtained jointly with Jason Bell, Stephane Launois, and Rahim Moosa.

Abstract: We define the affine VW supercategory sW, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group. The main obstacle was the lack of a quadratic Casimir element for p(n). When M is the trivial representation, the action factors through the action of the previously known Brauer supercategory sBr. Our main result is an explicit basis theorem for the morphism sW, and as a consequence we recover the basis theorem for sBr. The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation. This is joint work with Zajj Daugherty, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova and Catharina Stroppel, arising from the WINART workshop.

Abstract: It has long been expected, and is now proved in many important cases, that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered. I will outline an approach to this question using the ideas of noncommutative projective geometry, from which we see that the correct object to study is a groupoid, rather than a group, and maps in this groupoid are the replacement for automorphisms. I will illustrate this with the example of quantum projective space. This is joint work with Nicholas Cooney (Clermont-Ferrand).

Abstract: In this talk I will look at the notion of tropical critical points of the superpotential of the flag variety (in type A). A superpotential for any flag variety of general type was defined in the work of K. Rietsch, and there it was also shown how the critical points of this superpotential describe the quantum cohomology of the original flag variety. By tropicalising this superpotential, one can associate to any finite dimension representation of the group SLn, a family of polytopes indexed by the reduced expressions for the longest element of the Weyl group. I will then show how looking at the tropical critical points of the superpotential gives a distinguished point in each of these polytopes, and will also interpret this point via a construction coming just from the original flag variety.

Abstract: In 1985 Zamolodchikov constructed a "non-linear" extension of the Virasoro algebra known as W(3)-algebra. This is the one of the first appearance of a rich class of algebraic structures, known as W-algebras, which are intimately related to physical theories with symmetry and revealed many applications in mathematics . In the first part of the talk I will review some facts about the general theory of W-algebras. Then, I will explain how to describe quantum finite and classical affine W-algebras using Lax operators. In the quantum finite case this operator satisfies a generalized Yangian identity, while in the classical affine case it is used to construct an integrable Hamiltonian hierarchy of Lax type equations. This is a joint work with A. De Sole and V.G. Kac.

Abstract: It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise. In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not ‘run around once’ but ‘run out’, we coined the term exodromy representation. In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine.

Abstract: Dimer models with boundary were introduced in joint work with King and Marsh as a natural generalisation of dimers. We use these to derive certain infinite dimensional algebras and consider idempotent subalgebras w.r.t. the boundary. The dimer models can be embedded in a surface with boundary. In the disk case, the maximal CM modules over the boundary algebra are a Frobenius category which categorifies the cluster structure of the Grassmannian.

Abstract: We define a notion of fast generating sets, for groups of self-homeomorphisms of a space (under composition), and apply it to the group Pl_o(I) of piecewise-linear homeomorphisms of the unit interval. As a consequence, we build some general forms of Ping-Pong Lemmas for this group, which lemmas guarantee isomorphism types for certain fg subgroups of Plo(I), based on simple combinatorial data. We also find a lemma which guarantees that some particular (unexpectedly large) set of subgroups of Pl_o(I) also embed in R. Thompson’s group F. Joint with Matt Brin and Justin Moore.

Abstract: This talk is based on joint work with Arkady Berenstein. It frequently happens that an algebra C factors as C=AB, meaning a vector space isomorphism between C and the tensor product of its subalgebras A and B. The classical PBW theorem and its more recent incarnations -- think quantum groups and affine Hecke algebras -- are statements about algebra factorizations. Conversely, an algebra structure on the tensor product of can be established in many cases: semidirect product, braided tensor product, etc, which all fit the situations when A is an H-module algebra and B is an H-comodule algebra for some bialgebra H. We show that, quite surprisingly, any algebra factorization C=AB can be realised in this way for a suitable H: an ordinary bialgebra if the factorization is tame (which is typically the case), or a topological bialgebra in general. In particular, when C is a rational Cherednik algebra or a Kostant-Kumar nilHecke algebra, reconstructing H leads us to a Nichols algebra. DAHA and its generalizations correspond to the Hecke-Hopf algebras H, recently found by Berenstein and Kazhdan. Even in more straightforward examples of algebra factorisations, H can be a new and interesting Hopf algebra the representation theory of which begs to be explored.

Abstract: The construction of the Fukaya category of a symplectic manifold is inspired by string theory: roughly, objects are Lagrangians, morphisms are intersection points, and composition of morphisms comes from "holomorphic disks." For surfaces, a combinatorial construction of the (partially wrapped) Fukaya category was recently given by Haiden, Katzarkov, and Kontsevich. We will discuss this category and some recent results involving its (derived) Hall algebra (joint with B. Cooper).

Abstract: The Hall algebra of an abelian category has structure constants coming from "counting extensions" in the category. In this talk we give a survey of some recent results involving the Hall algebra of the category of coherent sheaves on an elliptic curve. Some topics involve an explicit description of the algebra by Burban and Schiffmann and a construction of Schiffmann and Vasserot of an action on the space of symmetric functions using Hilbert schemes and double affine Hecke algebras.

Abstract: In recent work with Les Bunce we have investigated the different ways to embed a JC*-triple as JC*-subtriple of all operators. This relates to the associative algebraic structure generated by the triple and is also reflected in the geometry of matrices over the triple. The concept of universal reversibility plays a significant role. We will describe some of this work when restricted to the finite dimensional case.

Abstract: We summarize part of the basic theories of Jordan algebras and (positive hermitian) Jordan triple systems in finite dimensions. This will include connections with other topics including several complex variables and homogeneous cones.

Abstract: The aim of the talk is to present some joint results with Alexander Olshansky. When an algebra B is embedded in an algebra A then the growth functions of A produce some growth-like functions on B. Comparing these functions with the growth functions of B, one can speak about embeddings with or without distortion. We study these and related phenomena for general algebras, but the main results are in the case of associative and Lie algebras.

Abstract: The aim of this lecture is to describe techniques that enable one to provide vector space bases for associative and Lie algebras which are given in terms of generators and defining relations.

Abstract: Suppose that we have a framework consisting of finitely many fixed-length bars connected at universal joints. Such frameworks (and variants) arise in many guises, with applications to the study of sensor networks, the matrix completion problem in statistics, robotics and protein folding. The fundamental question in rigidity theory is to determine if a framework is rigid or flexible. The standard approach in combinatorial rigidity theory is to differentiate the quadratic equations constraining the distances between joints, and work with these linear equations to determine if the framework is infinitesimally rigid or flexible. In this talk I will discuss recent progress using algebraic matroids that gives further insight into the infinitesimal theory and also provides methods for identifying special bar lengths for which a generically rigid framework is flexible. We use circuit polynomials to identify stresses, or dependence relations among the linearized distance equations and to find bar lengths which give rise to motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

Abstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542).

Abstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542).

Abstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo.

Abstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo.

Abstract: I will explain how the ideas from Part 1 connect to difficult parts of analysis and/or topology(Nevanlinna Theory, Morse Theory and Shapiro’s Conjecture from 1950 about common zeros of exponential polynomials). Serious work from Diophantine geometry is involved, due to Bombieri, Masser and Zannier.

Abstract: Exponential algebra extends commutative algebra by taking account of rings (particularly fields) with an exponential function. Classically there are few examples, but those, the real and the complex exponentials, are of great importance for all of mathematics. It is not even clear what should be the definition of an exponential ring, and it is certainly not at all clear what exponentially algebraic should mean. Historically Hardy (around 1911) used some tricks of the trade to get good information about zeros of one variable exponential polynomials, and Ritt, in the late 1920’s, established a quite subtle factorization theorem for one variable exponential polynomials. These in turn linked to questions about the distribution of zeros of systems of exponential polynomials. Some of these problems have remained open, and turn out to be connected both to transcendental number theory and to mathematical logic (decidability and definability issues). In the first part I will explain some basic definitions and constructions (e.g of free exponential rings),and sketch the connection to Schanuel’s Conjecture from transcendental number theory. I will also explain how logicians came to these problems, and what difference their ideas made in establishing a quite elaborate subject of exponential algebra.

Abstract: I will give some details about the proof of our results on almost commuting matrices that includes effective algebraic geometry and commutative algebra as well as the algebraic Ornstein Weiss principle.

Abstract: I will talk about a classical problem of Halmost on almost commuting matrices and our recent result with Lukasz Grabowski.

Abstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case.

Abstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case.

Abstract: A major current goal for noncommutative projective geometers is the classification of so-called “noncommutative surfaces”. Let S denote the 3-dimensional Sklyanin algebra, then S can be thought of as the generic noncommutative surface. In recent work Rogalski, Sierra and Stafford have begun a project to classify all algebras birational to S. They successfully classify the maximal orders of the 3-Veronese subring T of S. These maximal orders can be considered as blowups at (possibly non-effective) divisors on the elliptic curve E associated to S. We are able to obtain similar results in the whole of S.

Abstract: I will introduce a few key concepts in the theory of noncommutative projective geometry. In particular, I aim to give an idea of how one should think of a noncommutative curve/ surface. I will also describe a key example called a twisted homogeneous coordinate ring: a ring built out of geometry which plays a vital role in the theory.

Abstract: We answer a question of Ufnarovskii whether an automaton algebra must possess a generating set and an order on monomials with respect to which the reduced Groebner basis in the ideal of relations is finite. Namely, we present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Groebner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. Note that extending the ground field does not help. The proof is partially based on sensitivity of the growth of an algebra to characteristic of the ground field, which is restricted in case of finite Groebner basis.

Abstract: We introduce the concepts of a Groebner basis, Hilbert series and growth of an algebra. These notions will be demonstrated on the following result. We present a simple example (4 generators and 7 relations) of a quadratic semigroup algebra of intermediate growth. The proof is obtained by computing the (infinite) reduced Groebner basis in the ideal of relations. Although the basis follows a clear and simple pattern, the corresponding set of normal words fails to form a regular language. The latter is noteworthy in its own right. The only previously known example of a quadratic algebra of intermediate growth due to Kocak is non-semigroup and is given by 14 generators and 96 quadratic relations.

Abstract: Bertini Theorems for F-signature Abstract: In characteristic zero, it is well known that multiplier ideals and log terminal singularities satisfy Bertini-type theorems for hyperplane sections. The analogous situation in characteristic p > 0 is more complicated. While F-regular singularities satisfy Bertini, the test ideal does not. In this talk, I will describe joint work with Karl Schwede and Javier Carvajal-Rojas showing that the F-signature -- a numerical invariant of singularities that detects F-regularity -- satisfies the relevant Bertini statements for hyperplane sections. In particular, one can view this as a generalization of the corresponding results for F-regularity.

Abstract: We will give a gentle introduction to some of the basic invariants of singularities of rings in positive characteristic defined via the Frobenius endomorphism. In particular, we will pay close attention to F-signature and Hilbert-Kunz multiplicity, highlighting the known examples for each.

Abstract: It is well known that the universal enveloping algebra of a finite dimensional Lie algebra admits no non-trivial deformations as an algebra. In particular, there exists a (non-explicit) isomorphism between the trivial deformation of the enveloping algebra and the corresponding quantum group. An explicit description of such isomorphism was known only for sl(2). In this talk, we introduce a new realisation of the evaluation homomorphism of the Yangian of sl(n) and we use it to obtain an explicit isomorphism between classical and quantum sl(n). This is a work in progress with S. Gautam.

Abstract: In this talk, we recall some basic facts about the Yangian of gl(n) and, in particular, the role played by the Capelli identity in the definition of the Yangian and in the construction of the evaluation homomorphism. We then describe a similar result for sl(n), which leads to an apparently new presentation of the evaluation homomorphism in this case.

Abstract: By a result of Ingalls and Thomas, one can think of the bounded derived category of finite dimensional representations of an ADE quiver as a categorification of non-crossing partitions of the corresponding type. The non-crossing partitions are precisely the lattice of exact localizations of the bounded derived category. I'll discuss various directions in which one can generalise this, such as the extension to doubly infinite Dynkin type A, representations over more general rings, and (time permitting) the situation for tame quivers. This is based on joint work with Gratz, with Antieau, and with Krause.

Abstract: Abstract TBA

Abstract: Conformal field theories, and the fusion categories derived from them, provide classes in group cohomology that generalize characteristic classes. These classes are called "anomalies", and obstruct the existence of constructing orbifold models. I will discuss two of the most charismatic groups --- the Conway group Co_0 and the Fischer--Griess Monster group M --- and explain my calculation that in both cases the anomaly has order exactly 24. The Monster calculation relies on a version of "T-duality" for finite groups which in turn relies on fundamental results about fusion categories. I will try to explain everything from the beginning, and assume no knowledge of the Monster or its cousins.

Abstract: Abstract: I will explain how to study the Poisson (co)homology of a Poisson variety locally via D-modules. When there are finitely many symplectic leaves, the zeroth Poisson homology is finite-dimensional, and as an application, one has finitely many irreducible representations of every quantization. In the case that the variety is conical and admits a symplectic resolution, this conjecturally recovers the cohomology of the resolution and equips it with filtrations recording the order of vanishing of fiberwise closed differential forms on smoothings. In the case of nilpotent cones, this recovers a conjectural formula of Lusztig in terms of Kostka polynomials. In the case of smooth Poisson varieties, work in progress with Brent Pym shows that the entire Poisson cohomology is finite-dimensional when the modular filtration is finite and defines a perverse sheaf. This has applications to Feigin-Odesski Poisson structures on even-dimensional projective spaces.

Abstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.

Abstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.

Abstract: A notion of prime ideal in a commutative ring can be expressed in many equivalent statements that become distinct when the ring is assumed to be noncommutative. This leads to: completely prime ideals, strictly prime ideals, strongly prime ideals, s-prime ideals, l-prime ideals, etc. Each of the aforementioned “prime” has a module analogue; and these analogues collapse to just one notion when a module is defined over a commutative ring. In this talk, I discuss the completely prime (sub)modules, their properties and torsion theories induced by the completely prime radical. Secondly, by comparing completely prime (sub)modules with prime (sub)modules, I talk about a class of 2-primal modules which is a module analogue of 2-primal rings.

Abstract: The integrals appearing in Kontsevich's deformation quantization formula are notoriously difficult to compute. As a result, direct calculations with the formula have so far been intractable, even in very simple examples. In forthcoming work with Peter Banks and Erik Panzer, we give an algorithm for the exact evaluation of the integrals in terms of special transcendental constants: the multiple zeta values. It allows us to calculate the formula on a computer for the first time. I will give an overview of our approach, which recasts the integration problem in purely algebraic terms, using Francis Brown's theory of single-valued multiple polylogarithms.

Abstract: Deformation quantization is a process that assigns to any classical mechanical system its quantum mechanical analogue. The problem can be phrased in purely algebraic terms: we would like to start with a commutative ring equipped with a Poisson bracket, and produce a noncommutative deformation of its product. A priori, the Poisson bracket only specifies the deformation to first order in the deformation parameter, but a deep theorem of Maxim Kontsevich extends the deformation to all orders in a canonical way. While the problem is algebraic, his solution is transcendental: it involves integrals over high-dimensional configuration spaces. I will give an elementary introduction to his formula, and talk about the (very few) examples in which it can actually be computed by hand.

Abstract: The Hall algebra associated to a category is related to the Waldhausen S-construction in the work of Kapranov and Dyckerhoff. We explain how the higher associativity data can be extracted from this construction in a natural way, thus allowing for various higher categorical versions of Hall algebra. We then discuss a natural and systematic extension of this construction providing a bi-algebraic structure. We show how it can be used to provide a more transparent proof for the Green's theorem for the Hall algebras of hereditary categories and discuss possible extension to the higher categorical setting.

Abstract: We discuss the notion of a “mate” of a square in a 2-category. We will explain how it is related to base change in algebraic geometry, and that it can be understood as a homotopic condition. We then explain how this can be used to categorify the notion of Hopf algebra, and the Heisenberg double construction.

Abstract: The Yang-Baxter equation is an important tool in theoretical physics and pure mathematics, with many applications in different domains going from condensed matter to topology. The importance of this equation led Drinfeld to ask for studying the simplest family of solutions: combinatorial or set-theoretical solutions. In this talk we review the basic theory of set-theoretical solutions, we discuss some problems and solutions and we give some application.

Abstract: The numerical classification of "noncommutative surfaces" of rank 4 suggests the existence of a family previously not considered in the literature. By generalising Orlov's blowup formula to blowups of sheaves of maximal orders outside the ramification locus, we construct these starting from Artin--Schelter regular algebras which are finite over their center for all cases in the classification. Previously the first non-trivial case in the classification was constructed by de Thanhoffer--Presotto using noncommutative P^1-bundles. We can compare this to the blowup construction using some very classical geometry of linear systems. This comparison can be seen as a noncommutative instance of the classical isomorphism between the first Hirzebruch surface and the blowup of P^2 in a point. This is joint work with Dennis Presotto and Michel Van den Bergh.

Abstract: I will explain how one can describe derived categories of smooth projective varieties using full and strong exceptional collections, i.e. via an explicit finite-dimensional algebra. I will also explain the mutation of exceptional collections, and the role of the braid group. Another important ingredient in the description of a derived category is the Serre functor. It is given by Serre duality in algebraic geometry, and the Auslander-Reiten translation in the representation theory of algebras. The Serre functor induces an automorphism of the Grothendieck group, and in the case of a smooth projective surface this automorphism satisfies additional strong properties. I will review these results, and explain how it leads to the numerical classification of "noncommutative surfaces" of rank 4 due to de Thanhoffer--Van den Bergh.

Abstract: Spherical varieties form a remarkable class of complex algebraic varieties equipped with an action of a reductive group G, which includes toric, flag and symmetric varieties. Smooth affine spherical varieties are the local models of multiplicity free (real) Hamiltonian and quasi-Hamiltonian manifolds. A natural invariant of an affine spherical variety X is its weight monoid, which is the set of irreducible representations (or dominant weights) of G that occur in the coordinate ring of X. In the 1990s F. Knop conjectured that the weight monoid is a complete invariant for smooth affine spherical varieties, and in 2006 I. Loseu proved this conjecture. I will present joint work with G. Pezzini in which we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties. I will also discuss some applications obtained with Pezzini and K. Paulus.

Abstract: I will illustrate with many examples how one can use the representation theory of complex reductive groups to obtain combinatorial invariants of algebraic varieties equipped with an action of such a group. The main focus will be on invariants of (affine) spherical varieties.

Abstract: The representation theory of Lie algebras of type D is studied by Ehrig and Stroppel in an approach generalizing Arakawa and Suzuki’s work in type A. More specifically, they use affine Nazarov-Wenzl algebras, and their cyclotomic quotients, instead of the degenerate affine Hecke algebra to describe the endomorphism ring of certain representations in type D. We will discuss a signed version, or odd VW algebras living in an extension of the odd Brauer category, in a similar approach towards describing endomorphism rings of the periplectic Lie superalgebra p(n). The key ingredient will be a certain sneaky quadratic Casimir element and the corresponding Jucys-Murphy components.

Abstract: As motivation for the work to be described in the second hour, I will discuss the classical Schur-Weyl duality for gl(n), describing the endomorphism ring of tensor powers of the vector representation in terms of the symmetric group, as well as higher Schur-Weyl duality involving more general representations and the degenerate affine Hecke algebra in place of the symmetric group. This construction has been further extended in using diagrammatic algebras to describe the representation theory of other Lie algebras and Lie superalgebras. We will also discuss some Lie supertheory towards looking at the periplectic Lie superalgebra.

Abstract: I'll explain the sense in which the map from the stack of finite-dimensional representations of an algebra to the coarse moduli space behaves as though it were a proper map. This turns out to be a vital piece in proving the following statement: the cohomological Hall algebra associated to the preprojective algebra of a quiver is a quantum enveloping algebra, for the strictly positive part of a new type of Kac-Moody algebra, which carries a cohomological grading. This gives a mathematical formulation for the physicists' Lie algebra of BPS states. The cohomological degree zero piece of this algebra is the positive part of the usual Kac-Moody algebra of the largest subquiver without imaginary simple roots, but even if this is the entire quiver, there's much more to this algebra than the usual Kac-Moody algebra of the quiver.

Abstract: The main talk concerns the search for the "Lie algebra of BPS states" associated to a preprojective algebra. In the pretalk I'll explain why, as a mathematician, one would go looking for such a thing. The evidence pointing to the existence of this algebra involves many nice results from combinatorics of representations of quivers, Donaldson-Thomas theory, and Nakajima quiver varieties. I'll try to give a geographical sketch of these other results in order to motivate the main talk.

Abstract: The Kleinian singularities make up a family of well-understood (commutative) surface singularities. In 1998, Crawley-Boevey and Holland introduced a family of algebras which may be viewed as noncommutative deformations of Kleinian singularities. Using singularity categories, I will make comparisons between the types of singularity arising in the commutative and noncommutative settings. I will also show that the "most singular" of these noncommutative deformations has a noncommutative resolution for which an analogue of the geometric McKay correspondence holds.

Abstract: Let G be a reductive algebraic group over a field k. The notion of a $G$-complete reducible subgroup of G was introduced by Serre; in the special case G= GLn(k), a subgroup H of G is G-completely reducible if and only if the inclusion of H in G is completely reducible in the sense of representation theory. G-complete reducibility has turned out to be an important tool for investigating the subgroup structure of simple algebraic groups. In this talk I will discuss the interplay between geometric invariant theory and the theory of G-complete reducibility.

Abstract: One of the main tools for understanding automorphisms of free groups is via the action on Culler Vogtmann Space. More recently, the geometry of this space has been the subject of intense study. We will provide an introduction to these objects as well as presenting a report on some recent joint work with Stefano Francaviglia showing that the set of "minimally displaced points" for a given automorphism is connected, and that this is enough to solve the conjugacy problem in some limited cases.

Abstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,…

Abstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,…

Abstract: In recent years there has been renewed interest in the construction of finitely generated algebraic division algebras that are not finite-dimensional. This is division ring version of Kurosh Problem. There are results as local versions of Kurosh problem, for example a theorem due to Jacobson asserts: every division ring whose elements are algebraic of bounded degree over its center, is centrally finite. Recently, this result has been generalized by Jason Bell et al to left algebraic division rings over not necessarily central subfields. Using combinatorics of words, in this seminar we show the statement holds for division rings whose commutators are left algebraic over not necessarily central subfields.

Abstract: In this talk I am going to give a brief historic overview about the origin of commutators in group theory. Then I will pass to division rings and will show how one can obtain essential information about the structure of a division ring in terms of commutators and structures generated by commutators. Also, you will find generalization of some classic results due to Jacobson, Kaplansky and Noether about division rings.

Abstract: Consider a positively graded, connected algebra A, finitely generated in degree 1, for example the polynomial ring of global dimension n. Assume that there exists some reductive group G acting on A as gradation preserving algebra automorphisms, then each degree k part decomposes as a finite sum of simple G-modules. Then a natural question is: do there exist other graded algebras B such that 1) G acts on B, with the action preserving the gradation and 2) the degree k parts of A and B are isomorphic as G-modules for each natural number k ? As one may suspect, this depends greatly on G and A itself. Some constructions and the motivating example of the 3-dimensional Sklyanin algebras will be discussed. For this example, if time permits, I will show that some additional information about these algebras can be gained by this construction.

Abstract: A quick introduction to noncommutative projective geometry in the style of Artin-Tate-Van den Bergh. In this field, one studies noncommutative graded algebras with 'similar' properties to the commutative polynomial ring. Such properties can be either of homological or algebraic nature. I will talk about the classification of AS-regular algebras and define the problem I've been working on in the context of this field.

Abstract: The starting point for this talk is Chouinard’s theorem, which states that a module for a finite group is projective if and only if its restriction to every elementary abelian p-subgroup is projective; and Dade’s lemma, which gives an easy test for whether a module for an elementary abelian group is projective. I shall talk about analogous results for finite group schemes and finite supergroup schemes, and their consequences for the structure of the stable module category. Parts of this talk represent joint work with Iyengar, Krause and Pevtsova.

Abstract: Permutations subject to equivalences can be used to classify invariants of unitary group actions on polynomial functions of matrices and tensors. These equivalence classes are related to permutation centralizer algebras. One sequence of these algebras is closely related to Littlewood-Richardson coefficients. Structural properties of these algebras as well as Fourier transforms on them have applications in dualities between gauge theories and string theories. They yield results on the counting and correlators of multi-matrix invariants, relevant to the physics of super-symmetric states in the AdS/CFT correspondence. Combinatoric questions on the structure of these algebras are related to the complexity of spaces of super-symmetric states.

Abstract: In the pre-seminar, I will give an overview of some aspects of conformal field theories and representation theory which play an important role in understanding the duality (the AdS/CFT correspondence) between strings in 10 dimensions and conformal field theories in 4 dimensions. This will include BPS states, matrix correlators, large N expansions and Schur-Weyl duality.

Abstract: A character variety of a manifold X is a moduli space of representations of pi_1(X). It was shown by Atiyah-Bott and Goldman that character varieties of surfaces are naturally symplectic, and that character varieties of 3-manifolds define Lagrangian subvarieties in the character varieties of their boundary surfaces. In this talk, I'll explain that all this structure can be ``quantized", giving rise to a TFT which we call the quantum character TFT. We obtain in this way manifestly topological constructions of many gadgets traditionally thought of as living in the world of representation theory: quantum coordinate algebras, q-difference operator algebras, double affine Hecke algebras, etc. Quantizing the Lagrangians of different 3-manifolds gives a new approach to studying the representation theory of these objects: the quantization of a Lagrangian should be (roughly) a simple module for the quantization of the symplectic variety it lives on. The main technical ingredient in the construction and any computations with it is the Morita theory of tensor categories, as developed in the preseminar. This is joint work with Ben-Zvi, Brochier and Snyder.

Abstract: "Categorification" means replacing vector spaces with categories, in an artful way. When we categorify the notion of a ring, and a module over it, we get the notions of a tensor category, and a module category over it. Examples of these are ubiquitous throughout representation theory and algebraic geometry. If you hand a representation theorist a ring, she will ask "what are its modules?". In this talk, I'll develop some tools which you can use in case someone ever hands you a tensor category, and asks what are its module categories? It turns out that the resulting "Morita theory" for tensor categories plays a crucial role in the next talk.

Abstract: Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley-Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

Abstract: This will be a short reminder on the basic properties of quiver varieties. As well as giving the basic definitions, I’ll explain how one computes their dimensions, when they are smooth etc. No prior knowledge will be assumed.

Abstract: The wonderful compactification of a group encodes the asymptotics of matrix coefficients for the group and captures the rational degenerations of the group. In this talk, we will explain a construction of the wonderful compactification via the Vinberg semigroup which makes these properties explicit. We will then introduce quantum group versions of the Vinberg semigroup, the wonderful compactification, and the latter's stratification by G x G orbits. Our approach relies on a theory of noncommutative projective schemes, which we will review briefly.

Abstract: The wonderful compactification of a group plays a crucial role in several areas of geometric representation theory and related fields. The aim of this talk is to give a construction of the wonderful compactification and explain how its rich structure links the geometry of the group to the geometry of its partial flag varieties. We will describe several examples in detail. The first part of the talk will be an overview of necessary background from the representation theory of complex reductive groups.

Abstract: I will discuss how questions on Sklyanin algebras can be solved using combinatorial techniques, namely, Groebner bases theory. Elements of homological algebra also feature in our proofs. We calculate Hilbert series, prove Koszulity, PBW, Calabi-Yau etc., depending on parameters of Sklyanin algebras. Similar methods are used for generalized Sklyanin algebras, and for other potential algebras.

Abstract: TBA

Abstract: Infinite-dimensional representation theory of finite dimensional Lie algebras is a rich topic with many interesting results. One of the most beautiful pieces of this subject is a description of primitive and prime ideals of universal enveloping algebras of finite-dimensional Lie algebras, and this involves quite advanced algebraic, geometric, and combinatorial techniques. It is natural to generalize the classification of primitive and prime ideals to the setting of infinite-dimensional Lie algebras, and in my talk I will provide a description of primitive ideals of the universal enveloping algebra of sl(infinity). I hope that I will be able to explain in an understandable way algebraic and combinatorial aspects of this result.

Abstract: A quantum cluster (or quantum torus) is an algebra over C(q) with q-commuting generators. Various embeddings of quantum groups into quantum tori have been studied over the past twenty years in relation with modular doubles, quantum Gelfand-Kirillov conjecture, and construction of braided monoidal categories. In a recent paper by K. Hikami and R. Inoue, such an embedding of the quantum group U_q(sl_2) was used to relate the corresponding R-matrix with quantum cluster mutations and half-Dehn twists. I will discuss how to generalize the results of Hikami and Inoue to U_q(sl_n). The quantum group is embedded into the tensor square of the quantized categorification space of 3 flags and 3 lines in C^n, which were studied in detail in the works of V. Fock and A. Goncharov. I also plan to show how the conjugation by the R-matrix can be expressed via a sequence of cluster mutations. If time permits, I will outline a way to generalize the above construction to quantum groups of arbitrary finite type and discuss its applications to the representation theory.

Abstract: Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding planar algebras fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These planar algebras are the ones attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don't fit into any of the classical families: G_2, F_4, E_6, E_7, and E_8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-paramter family of planar algebras which introduces a variable q, and yields a new exceptional knotpolynomial. In joint work with Scott Morrison and Dylan Thurston, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.

Abstract: In geometric representation theory, we typically study certain categories associated to a reductive group G (e.g. certain representations of G or g=Lie(G), D-modules on Bun_G(Curve), Character sheaves on G, ...). Often there are functors of parabolic induction and restriction going between the categories associated to G and to Levi subgroups L of G. I will explain how these functors allow us to break up our category into pieces, indexed by classes of cuspidal objects on Levi subgroup (an object is called cuspidal if it is not seen by induction from a proper Levi). We will see how things like Hecke algebras and (relative) Weyl groups naturally appear. Later, I will focus on the case of adjoint equivariant D-modules on the Lie algebra g, and indicate how this case may be generalized to other settings - mirabolic, quantum, elliptic...

Abstract: Quantum groups play a prominent role in many branches of mathematics, from gauge theory and enumerative geometry, to knot theory and quantum computing. In many cases, this is due to their tight relation with the braid groups. More specifically, to the fact that a quantum group is in the first place a Hopf algebra whose representations carry a natural action of the braid group. In the first part of the talk, I will explain how the quantum groups are in fact analytic objects, describing the monodromy of certain systems of difeerential equations arising in Lie theory. I will first review the renowned Drinfeld-Kohno theorem, describing the monodromy of the Knizhnik-Zamolodchikov equations associated to a simple Lie algebra in terms of the universal R-matrix of the corresponding quantum group. I will then present an extension of this result, providing a descrip- tion of the monodromy of the Casimir equations associated to a simple Lie algebra (in fact, to any symmetrisable Kac-Moody algebra) in terms of the quantum Weyl group operators of the corresponding quantum group. The proof relies on the notion of generalised braided category (or quasi-Coxeter category), which is to a generalised braid group what a braided monoidal category is to the standard braid group on n strands. In the second part of the talk, I will explain how Tannaka{Krein dual- ity, quantization of Lie bialgebras, dynamical KZ equations, and Hochschild cohomology in the framework of appropriate PROP categories play a funda- mental role in the proof of the monodromy theorem.

Abstract: One of the earliest successes in geometric representation theory was Springer's construction of the irreducible representations of the symmetric group (or any Weyl group) in the top cohomology of fibers of a resolution of singularities of the nilpotent cone of GL(n) (or a reductive group). More recently there has been considerable progress extending these ideas to representations and sheaves with positive characteristic coefficients. Life is more complicated in positive characteristic: the category of representations is no longer semisimple, and on the geometric side this is reflected in the failure of some important geometric tools from characteristic 0 such as the decomposition theorem. In this talk I will describe work with Carl Mautner giving a picture similar to Springer theory where the role of the nilpotent cone is played by hypertoric varieties. We obtain representations of a new class of algebras which we call "Matroidal Schur algebras", which share many features with their classical cousins. In particular the categories are highest weight, and the categories for Gale dual hypertoric varieties are related by Ringel duality. In the second part of the talk I will explain some of the ideas in the proof and a conjectural geometric approach to proving that certain categories of perverse sheaves are highest weight.

Abstract: We explain two ways in which geometry can be used to produce solutions of the Yang-Baxter equation. First, we introduce Maulik and Okounkov's "stable envelope" construction of R-matrices acting in the cohomology of a symplectic variety, and describe some of the geometric properties these R-matrices enjoy. Second, we produce an R-matrix from the Hilbert scheme of points on a general surface from an intertwiner of certain highest weight Virasoro modules; for the surface C^2, this construction is due to Maulik and Okounkov. We also explain how to modify this construction to produce formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme.

Abstract: This is joint work with S. Sierra. We explore the (noncommutative) geometry of representations of locally finite Lie algebras. Let L be one of these Lie algebras, and let I ⊆ U(L) be the annihilator of a locally simple L-module. We show that for each such I, there is a quiver Q so that locally simple L-modules with annihilator I are parameterized by “points” in the “noncommutative space” corresponding to the path algebra of Q. We classify the quivers that occur and along the way discover a beautiful connection to characters of the symmetric groups S_n.

Abstract: I will describe the geometric R-matrix formalism, developed in the work of Maulik and Okounkov, that leads to the construction of a Hopf algebra $Y_Q$ acting on the equivariant cohomology of the Nakajima varieties associated to a quiver $Q$. In the first hour, I will give an introductory overview of the basic machinery of Nakajima varieties and their equivariant cohomology, which underpins the Maulik-Okounkov construction. In particular, we’ll illustrate the general definitions by focusing on the concrete example of cotangent bundles to Grassmannians. In the second hour, I’ll describe the stable basis construction in the equivariant cohomology of a symplectic variety, which is the key technical tool used to construct the geometric R-matrices. In our example of cotangent bundles to Grassmannians, we’ll see that this yields a geometric construction of the Yangian of $gl_2$. Finally, we will conclude by discussing some work in progress extending the Maulik-Okounkov construction to encompass Yangian coideal subalgebras, such as reflection equation algebras.

Abstract: First hour: The Witten-Reshetikhin-Turaev knot invariants are polynomials associated to a knot in S^3 constructed using representation theory of quantum groups. A concrete combinatorial construction of these invariants is given by the Kauffman bracket skein relations. In this talk we first discuss some of the background for these constructions. We then discuss how the skein relations are related to representation varieties, and to the Poisson structure on representation varieties of topological surfaces. Second hour: The double affine Hecke algebra is a noncommutative algebra depending on parameters q and t which is associated to a Lie algebra. The DAHA has been connected to various parts of math, including symmetric polynomials, integrable systems, Hilbert schemes, and more. Frohman and Gelca showed that the skein algebra of the torus is the t=q specialization of the sl_2 DAHA. We discuss this result (and some background), and then discuss a conjecture involving the DAHA and modules coming from knot complements.

Abstract: We study the representation theory of finite-dimensional graded algebras which admit a triangular decomposition similar to universal enveloping algebras of Lie algebras. Such a decomposition implies a rich combinatorial structure in the representation theory (this was discovered in this generality by Holmes and Nakano) and there are many examples like restricted quantized enveloping algebras at roots of unity, restricted rational Cherednik algebras, etc. We show that even though our algebras have in general infinite global dimension, the graded module category is in fact a highest weight category (with infinitely many simple objects, however). Under certain conditions we are able to establish a proper tilting theory in this category and use this to show that the degree zero part of the algebra is a cellular algebra. This is joint work with Gwyn Bellamy (Glasgow).

Abstract: The noncommutative and nonassociative algebra which arises in the description of the target space of non-geometric string theory fits naturally as a commutative and associative algebra object in a certain closed braided monoidal category, the representation category of a triangular quasi-Hopf algebra. In this talk I will show how exploring the syntax of category theory enables one to express notions of geometry on an algebra object in terms of universal constructions internal to the representation category of any triangular quasi-Hopf algebra.

Abstract: By quantum matrix algebras I mean algebras related to quantum groups and close in a sense to that Mat(m). These algebras have numerous applications. In particular, by using them (more precisely, the so-called reflection equation algebras) we succeeded in defining partial derivatives on the enveloping algebras U(gl(m)). This enabled us to develop a new approach to Noncommutative Geometry: all objects of this type geometry are deformations of their classical counterparts. Also, with the help of the reflection equation algebras we introduced the notion of braided Yangians, which are natural generalizations of the usual ones and have many beautiful properties.

Abstract: A hovel is degraded building. It can be used to associate to a Kac-Moody group over a local field a bunch of Hecke algebras. First, I will explain the definition of the hovel which generalises the construction of the Bruhat-Tits building associated to a reductive group. Then I will present three kind of algebras that one can associate to the hovel: the spherical Hecke algebra, the Iwahori-Hecke algebra and the Bernstein-Lusztig-Hecke algebra.

Abstract: Set-theoretic solutions of the Yang--Baxter equation form a meeting ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. Associated to each set-theoretic solution are several algebraic constructions: the monoid $S(X, r)$, the group $G(X, r)$, the semigroup algebra $kS(X, r)$ over a field k, generated by X and with quadratic relations $xy = .r(x, y)$, a special permutation group $\mathcal{G}$ and a left brace $(G, +,.)$. In this talk I shall discuss some of the remarkable algebraic properties of these object.

Abstract: Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, or all of $M_n(K)$. We prove the conjecture for $K=\mathbb{R}$ or for quadratically closed field and $n=2$, and give a partial solution for an arbitrary field $K$. We also consider homogeneous and Lie polynomials and provide the classifications for the image sets in these cases.

Abstract: By a result of J. Scott, the homogeneous coordinate ring of the Grassmannian Gr(k,n) can be realised as a cluster algebra. The Pluecker coordinates of the Grassmannian are all cluster variables. I will talk about joint work with J. Scott. We introduce a twist map on the Grassmannian and show that it is related to a twist of Berenstein-Fomin-Zelevinsky and can be implemented by a maximal green sequence, up to frozen variables. We give Laurent expansions for twists of Pluecker coordinates as scaled dimer partition functions (matching polynomials) on weighted versions the plabic (planar bicoloured) graphs arising in the cluster structure.

Abstract: Motivic Donaldson-Thomas invariants of quivers were defined by M. Kontsevich and Y. Soibelman as a mathematical definition of string-theoretic BPS state counts. We discuss several results relating these invariants to the geometry of moduli spaces of quiver representations.

Abstract: The prime spectrum of a quantum algebra has a finite stratification in terms of a set of distinguished primes called H-primes, and we can study these strata by passing to certain nice localizations of the algebra. H-primes are now starting to show up in some surprising new areas, including combinatorics (totally nonnegative matrices) and physics, and we can borrow techniques from these areas to answer questions about quantum algebras and their localizations. In particular, we can use Grassmann necklaces -- a purely combinatorial construction -- to study the topological structure of the prime spectrum of quantum matrices.

Abstract: Factorisation algebras and equivalently chiral algebras are geometric versions of vertex algebras, introduced by Beilinson and Drinfeld. There is also a non-linear analogue of a factorisation algebra, called a factorisation space. I will define these objects, and furthermore show how we can use the Hilbert scheme of points of a smooth d dimensional variety X to construct examples.

Abstract: I will begin presenting (part of) the history of the Krull-Schmidt-Remak Theorem. Then I will show a number of results concerning uniqueness of direct-sum decompositions of right modules over a ring R and uniqueness of direct-product decompositions of a group G. I will conclude giving some results about the category of G-groups, which is a category rather similar to the category Mod-R of right R-modules. Here a G-group is a group H on which G acts as a group of automorphisms.

Abstract: In the late 1970s and early 1980s, Dixmier and Moeglin gave algebraic and topological conditions for recognising the primitive ideals (namely the kernels of the irreducible representations) of the enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, such algebras are said to satisfy the "Dixmier-Moeglin equivalence". Many interesting families of algebras, including many families of quantum algebras, have since been shown to satisfy this equivalence. I will outline work of Jason Bell, Stephane Launois, and myself, showing that in several families of quantum algebras, an arbitrary prime ideal is equally close (in a manner which I will make precise) to being primitive, rational, and locally closed. The family on which I shall focus is that of the quantum Schubert cells U_q [w]. For a simple complex Lie algebra g, a scalar q which is not a root of unity, and an element w of the Weyl group of g, U_q [w] is a subalgebra of U_q^+(g) constructed by De Concini, Kac, and Procesi; familiar examples include the algebras of quantum matrices.

Abstract: Let G be a complex reductive group and B a Borel subgroup of G, a subgroup H of G is called spherical if it acts with finitely many orbits on the flag variety G/B. For example, if H coincides with B, then the orbits are parametrized by the Weyl group of G and the orbits are the Schubert cells. Even though spherical subgroups are classified combinatorially, the corresponding orbit decompositions of G/B are not yet understood in general. In this seminar I will consider two special cases, namely that of a solvable spherical subgroup and that of a symmetric subgroup of G corresponding to an involution of Hermitian type. In these cases, I will explain how to attach a root system to every H-orbit in G/B, and how these root systems allow to parametrize the H-orbits in G/B. These parametrizations are compatible with a general action of the Weyl group of G that Knop defined on the set of H-orbits in G/B, and I will explain how to recover the Weyl group action from the parametrization of the orbits. The talk is based on two joint works, respectively with Andrea Maffei and with Guido Pezzini.

Abstract: Roughly speaking, an element s in a commutative ring A is said to be weakly proregular if every module over A can be reconstructed from its localisation at s considered along with its local cohomology at the ideal generated by s. This notion extends naturally to finite sequences of elements: a precise definition will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. In particular, every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A. In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.

Abstract: Spherical subgroups of finite type of a Kac-Moody group have been recently introduced, in the framework of a research project aimed at bringing the classical theory of spherical varieties to an infinite-dimensional setting. In the talk we discuss their definition and some of their properties. We introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, which satisfies the same axioms as in the finite-dimensional case.

Abstract: We describe the finite dimensional representation category of gl(m|n) and of its quantized enveloping algebra using variations of Howe duality, and we review the Reshetikhin-Turaev construction of the corresponding link invariants of type A. We discuss then some results (and some open questions) on their categorification, in particular using the BGG category O.

Abstract: In pioneering work, Khovanov introduced the so-called arc algebra. His arc algebra is a certain algebra built from cobordisms turning up in TQFT’s and is known to be related to Khovanov homologies, categorification of tensor products of sl(2) and to a certain 2-block parabolic of category O for gl(n) - as follows from work of many researchers in the last 15 years. Sadly, in Khovanov’s original construction, the functoriality of Khovanov homologies cannot be encoded directly nor is it clear how to generalize his construction to obtain relations to e.g. tensor products of sl(N) or N-block parabolics of category O for gl(n). For this purpose, one needs to modify his arc algebra by using singular TQFT’s instead of “usual” TQFT’s. In this talk I will explain Khovanov’s topologically and elementary, yet powerful, construction in details as well as its relations to categorification and category O. I will then sketch how to use singular TQFT’s to generalize the construction.

Abstract: If X is a determinantal variety then there are a number of objects that may be regarded as "resolutions of singularities" of X: (1) the classical Springer resolution by a vector bundle over a Grassmannian, (2) a suitable quotient stack, (3) various non-commutative resolutions. In the talk we will discuss how these different resolutions are related. For ordinary determinal varieties this is joint work with Buchweitz and Leuschke. For determinantal varieties of symmetric and skew-symmetric matrices new phenomena occur due to the fact that the Springer resolution is no longer crepant. This is joint work with Špela Špenko.

Abstract: Abstract: Flag varieties are fertile soil where it grows geometry, algebra and combinatorics. Motivated by the PBW filtration of Lie algebras, E. Feigin defined the degenerate flag varieties, which are flat degenerations of the corresponding flag varieties. The purpose of this talk is to introduce a new family of (flat) degenerations of flag varieties of type A, called linear degenerate flag varieties, by classifying flat degenerations of a particular quiver Grassmannian. The geometry of these degenerations will also be presented. This talk is based on a joint work (in progress) with G. Cerulli Irelli (Rome), E. Feigin (Moscow), G. Fourier (Glasgow) and M. Reineke (Wuppertal).

Abstract: In this talk I will discuss how categorification methods can be used to obtain a graded version of the Brauer algebra, especially in non semi-simple cases. This will involve the category O for certain orthogonal Lie algebras and their parabolic and graded analogues. The categorifications involved in this process are a generalised version of the ones introduced by Rouquier and Khovanov-Lauda in the sense that the categorified object is not a Kac-Moody algebra or quantum group, but a so-called quantum symmetric pair. This is joint work with Catharina Stroppel.

Abstract: The Birch—Swinnerton-Dyer conjecture is one of the most mysterious open problems in number theory, predicting a relation between arithmetic objects, such as the points on an elliptic curve, and certain complex-analytic functions. A powerful approach to the conjecture is via a tool called an ‘Euler system’. I will explain the idea behing this approach, and some recent new results in this direction.

Abstract: The natural problem of determining characters of simple objects in suitable representation categories can be rephrased in terms of multiplicities of irreducible modules in standard objects. In this talk, I will focus on the case in which these multiplicities are governed (or expected to be governed) by certain combinatorial families of polynomials, and explain how moment graph techniques can be used to approach such a problem.

Abstract: The mapping class group Mod(S) of a surface S appears in a variety of contexts, for example, as a natural analogue both of arithmetic groups and of automorphism groups of free groups, and as the (orbifold) fundamental group of the moduli space of Riemann surfaces. However, its subgroup structure is not at all well understood. In this talk we will discuss a certain rigidity displayed by a wide class of subgroups of Mod(S): any normal subgroup of Mod(S) that contains a "small" element has Mod(S) as its group of automorphisms. This result is proved using a resolution of a metaconjecture posed by N. Ivanov stating that every stating that every "sufficiently rich" complex associated to S has Mod(S) as its group of automorphisms. (This is joint work with Dan Margalit.)

Abstract: We review the definition of Hattori-Stallings traces of projective modules and their relation to Morita equivalence. As an application, we will discuss Berest-Etingof-Ginzburg's work on Morita equivalence of rational Cherednik algebras of type A.

Abstract: Open Richardson varieties are the intersections of opposite Schubert cells in full flag varieties. They play a key role in Schubert calculus, total positivity and cluster algebras. We will show how to realize the quantized coordinate ring of each open Richardson variety as a normal localization of a prime factor of a quantum Schubert cell algebra. Using a combination of ring theoretic and representation theoretic methods, we will produce large families of toric frames for all quantum Richardson varieties. This has applications to cluster algebras and to the construction of a Dixmier type map from the symplectic foliation of each Schubert cell to the primitive spectrum of the corresponding quantum Schubert cell algebra. This is a joint work with Tom Lenagan (University of Edinburgh).

Abstract: I will discuss what a graded version of a category is and why graded versions of categories of representations are interesting to study. I want to discuss how recent advances in the theory of motives help to better understand these graded versions.

Abstract: In this talk I will explain a joint work with Y. Grimeland. Surface algebras are algebras of global dimension 2 associated to an unpunctured surface $S$ with an admissible cut. It is possible to associate to each such algebra an invariant in an affine space of $H^1(S,\mathbb Z)$ up to an action of the mapping class group which determines the derived equivalence class of the algebra. The proof uses strongly the graded mutation introduced in a joint work with S. Oppermann. This invariant is closely linked with the Avella-Alaminos-Geiss invariant for gentle algebras, and gives some information on the AR structure of the corresponding derived category.

Abstract: Abstract: In this talk I will present basic results of cluster-tilting theory developed in [Iyama Yoshino 2008: Mutation in triangulated categories and Rigid CM modules] and [Buan-Iyama-Reiten-Scott 2009: Cluster structures for 2-Calabi-Yau categories]. I will explain how these results were a motivation for generalising the construction of cluster categories. I will first recall the motivation and definition of the acyclic cluster category due to Buan Marsh Reineke Reiten Todorov in 2006, and then focus to the construction of the generalised cluster category associated with algebras of global dimension 2 [Amiot 09]. Then I will explain how cluster-tilting theory can apply in classical tilting theory via graded mutation in a joint work with Oppermann.

Abstract: The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. One can enlarge these algebras by a commutative subalgebra $X$ to anaffine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right $X$-module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of $X$-semisimple, or calibrated, representations, tha t don't necessarily factor through the affine Hecke algebra. I will discuss Walker's TQFT-motivated 2-handle construction of the $X$-semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description. This is joint work with Kevin Walker.

Abstract: The 45th ARTIN meeting will take place at the University of Edinburgh on the 11th and 12th of September 2015. All talks will be in the James Clerk Maxwell Building, room 6206. The theme of the meeting is noncommutative ring theory, with an emphasis on noncommutative algebraic geometry. See http://hodge.maths.ed.ac.uk/tiki/ARTIN-45 for more details

Abstract: Abstract: In search of L-infinity mappings between homotopy Poisson algebras of functions, we discovered a construction of a certain nonlinear analog of pullbacks. Underlying such "nonlinear pullbacks" there are formal categories that are "formal thickening" of the usual category of smooth maps of manifolds (or supermanifolds). Morphisms in these categories are called ''thick morphisms'' (or microformal morphisms). They are defined by formal canonical relations between the (anti)cotangent bundles of the manifolds. Thick morphisms have beautiful properties. For example, we can define the adjoint of a nonlinear morphism of vector bundles, as a thick morphism of the dual bundles, which reduces to the ordinary adjoint in the linear case. In the talk, I will explain the construction of microformal morphisms and nonlinear pullbacks, and their applications to homotopy Poisson structures, vector bundles and L-infinity algebroids. (See preprints: http://arxiv.org/abs/1409.6475 and http://arxiv.org/abs/1411.6720.)

Abstract: Abstract: A nonnoetherian singularity may be viewed geometrically as an algebraic variety with positive dimensional `smeared-out' points. I will describe how isolated nonnoetherian singularities admit noncommutative blowups which are smooth, in a suitable geometric sense. Furthermore, I will describe how a class of isolated nonnoetherian noncommutative singularities, namely non-cancellative dimer algebras, also admit noncommutative blowups which are smooth.

Abstract: Abstract: A rack is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode two of the three Reidemeister moves. We will begin by generalizing Whitehead's notion of a crossed module of groups to that of a crossed module of racks. Motivated by the relationship between crossed modules of groups and strict 2-groups, we then will investigate connections between our rack crossed modules and categorified structures including strict 2-racks and trunk-like objects in the category of racks. We will conclude by considering topological applications, such as fundamental racks. This is joint work with Friedrich Wagemann.

Abstract: Abstract: I will discuss a cohomological field theory associated to a quasihomogeneous isolated singularity W with a group G of its diagonal symmetries (a Landau-Ginzburg A-model, in physical parlance). The state space of this theory is the equivariant Milnor ring of W and the corresponding invariants can be viewed as analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). In the case of simple singularities of type A they control the intersection theory on the moduli space of higher spin curves. The construction is based on derived categories of (equivariant) matrix factorizations of singularities with the role of the virtual fundamental class from the Gromov-Witten theory played by a "fundamental matrix factorization" over a certain moduli space.

Abstract: Abstract: The idea behind this talk is to describe two-sided ideals of U(g) for an infinite dimensional Lie algebra g using known classification of prime two-sided ideals of U(g') for finite dimensional Lie subalgebras g' of g. In particular, if g is semisimple, by analogy with a finite dimensional case, one may expect that all primitive two-sided ideals are annihilators of highest weight modules. To start with we focus on infinite-dimensional Lie algebra sl(\infty). We will see that the annihilators in U(sl(\infty)) of most highest weight sl(\infty)-modules equal (0) and explicitly describe all highest weights for which this annihilator is not (0). We also prove that in the latter case the annihilator is an integrable ideal and provide a classification of such ideals. The proof will use the classification of two-sided ideals of U(sl(n)) (and thus a little bit of Young diagrams and Robinson–Schensted algorithm). Title/Abstract for the second part: Title: On ideals in the enveloping algebra of a locally simple Lie algebra Abstract: Let g be a Lie algebra with universal enveloping algebra U(g). To a two-sided ideal I of U(g) one can canonically assign a Poisson ideal gr I in S(g). It turns out that very frequently S(g) has no non-trivial Poisson ideals (and I hope I give some idea why it is so). As a consequence very frequently U(g) has no non-trivial two-sided ideals. As a final result I will provide some quite explicit description of countable dimensional locally simple Lie algebras g such that U(g) affords a non-trivial two-sided ideal.

Abstract: I will discuss work (in progress) with Ben Webster on homological mirror symmetry for hypertoric varieties. Hypertoric varieties are a family of noncompact algebraic symplectic spaces associated to hyperplane arrangements; we show how the quantization of such spaces in finite characteristic has a natural description on the mirror side.

Abstract: Abstract: I am going to discuss two subalgebras in the rational Cherednik algebra associated with a Coxeter group. These subalgebras satisfy Poincare-Birkhoff-Witt property and they are given by quadratic relations. They deform semidirect product of quotients of universal enveloping algebras of so(n) and gl(n) with the Coxeter group algebra, and they are related to quantisation of functions on the Grassmanian of two-planes and on the space of matrices of rank at most 1 respectively. The centres of these subalgebras give quantum Hamiltonians related to Calogero-Moser integrable systems which I plan to discuss as well. This is based on joint work with T. Hakobyan.

Abstract: Starting from solutions of the Yang-Baxter equation we construct a noncommutative bi-algebra which can be described in purely combinatorial terms using non-intersecting lattice paths. Inside this noncommutative algebra we identify a commutative subalgebra, called the Bethe algebra, which we identify with the direct sum of the equivariant quantum cohomology rings of the Grassmannian. We relate our construction to results of Peterson which describe the quantum cohomology rings in terms of Kostant and Kumar's nil Hecke ring and the homology of the affine Grassmannian. This is joint work with Vassily Gorbounov, Aberdeen.

Abstract: I will explain how (quasi-)Hamiltonian reduction fits into the framework of derived symplectic geometry. (Quasi-) Hamiltonian spaces are interpreted as Lagrangians in shifted symplectic stacks and the reduction corresponds to Lagrangian intersection. This gives a new perspective on deformation quantization of Hamiltonian spaces. This could also be used to make sense of deformation quantization of quasi-Hamiltonian spaces.

Abstract: Abstract: The study of separating invariants is a relatively recent trend in invariant theory. For a finite group acting linearly ona vector space, a separating set is a set of invariants whose elements separate the orbits of the action. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the leastpossible cardinality of a separating set for a given action. Our main result is a lower bound which generalizes the classical result of Serre that if the ring of invariants is polynomial, then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples. This is based on joint work with Emilie Dufresne.

Abstract: We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Poisson Gelfand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as a Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients. For instance this class includes Poisson determinantals varieties.

Abstract: Abstract: A uniform categorical description for both the Drinfeld center and a Heisenberg analogue called the Hopf center of a monoidal category (relative to a braided monoidal category) is presented using morphism categories of bimodules. From this categorical definition, one obtains a categorical action as well as a definition of braided Drinfeld and Heisenberg doubles via braided reconstruction theory. In examples, this categorical picture can be used to obtain a categorical action of modules over quantum enveloping algebras on modules over quantum Weyl algebras. Moreover, certain braided Drinfeld doubles give such an action on modules over rational Cherednik algebras using embeddings of Bazlov and Berenstein of these algebras into certain braided Heisenberg doubles which can be thought of as versions of the Dunkl embeddings. We argue that the corresponding braided Drinfeld doubles can serve an quantum group analogues in the setting of complex reflection groups. Finally, the categorical description can be extended naturally to give TQFTs with defects using recent work of Fuchs-Schaumann-Schweigert.

Abstract: Tom Leinster recently introduced an invariant of graphs called the magnitude. In this talk I will define a homology theory for graphs that categorifies the magnitude, in the sense that the magnitude of a graph can be recovered from its homology by taking the Euler characteristic. (Thus this is categorification in the same sense that Khovanov homology categorifies the Jones polynomial.) Important properties of the magnitude can then be seen as shadows of properties of homology. For example, magnitude satisfies an inclusion exclusion formula that can be recovered from a Mayer-Vietoris theorem in homology. The talk will (hopefully) be accessible for anybody who knows what graphs and chain complexes are, and I will try to illustrate it with lots of pictures and examples.

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://www.ma.hw.ac.uk/~ndg/nbsan.html

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract:

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: See the maximals webpage for detalis.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: For title and abstract, click here

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS

Abstract: Click here for title and abstract.

Abstract: For title and abstract, click here

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: Click here for title and abstract.

Abstract: We describe the recent research started by Nekrasov Shatashvilly Braverman Maulik Okounkov on correspondence between the quantum cohomology of the quiver varieties and the quantum integrable systems. Our main example will be the cotangent spaces to partial flag varieties.

Abstract: We explain how one can relate the Hochschild (co)homologies of Hopf algebras having equivalent tensor categories of comodules, in case the trivial module over one of the Hopf algebras admits a resolution by free Yetter-Drinfeld modules. This general procedure is applied to the quantum group of a bilinear form, for which generalizations of results by Collins, Hartel and Thom in the orthogonal case are obtained. It also will be shown that the Gerstenhaber-Schack cohomology of a cosemisimple Hopf algebra completely determines its Hochschild cohomology. Basic facts and definitions about Hopf algebras will be recalled first.

Abstract: I will discuss properties of resolutions of powers of ideals. This will be done in the framework of Betti diagrams and their polyhedral structure. At the end a conjecture regarding monomial ideals will be stated.

Abstract: I will talk about symmetries of derived categories of symmetric algebras, known as spherical twists, and explain when they satisfy braid relations. Using a more general collection of symmetries, known as periodic twists, I will explain how lifts of longest elements of symmetric groups to braid groups act on the derived category. This was first described in a special case by Rouquier and Zimmermann and, if time permits, I hope to present a new proof of their result.

Abstract: The multiple CSP is the following: Given two conjugate lists of elements A=[a_1,a_2,...,a_n] and B=[b_1,b_2,...,b_n], can we find a conjugator x such that x^{-1}a_{i}x=b_i ? We show that one can put a linear upper bound on the geodesic length of a shortest conjugator.

Abstract: I will explain how some nice Lie structures appear when one is trying to compute Ext and Tor of a closed subvariety. I will use it as an excuse to introduce some nice concepts of derived geometry. I'll end the talk with a striking analogy between two great results: one in Lie theory and the other one in algebraic geometry.

Abstract: Title: Morita equivalences from Higgsing toric superpotential algebras Abstract: Let A and A' be superpotential algebras of brane tiling quivers, with A' cancellative and A non-cancellative, and suppose A' is obtained from A by contracting, or 'Higgsing', a set of arrows to vertices while preserving a certain associated commutative ring. A' is then a Calabi-Yau algebra and a noncommutative crepant resolution of its prime noetherian center, whereas A is not a finitely generated module over its center, often not even PI, and its center is not noetherian and often not prime. I will present certain Morita equivalences that relate the representation theory of A with that of A'. I will also describe the Azumaya locus of A, and relate it to the Azumaya locus of A'. Along the way, I will introduce the notion of a non-local algebraic variety, and show how this notion is intimately related to these algebras.

Abstract: Title: Beilinson-Drinfeld factorization algebras and QFT. Abstract: I will explain what are factorization algebras and how they can be defined in very general settings of geometries with a good notion of D-modules. I will then talk about applications of this theory. In particular I will discuss the differentiable case and its relations to quantum field theory.

Abstract: Dirac Operators on Quantised Hermitian Symmetric Spaces In this joint work with Matthew Tucker-Simmons (U Berkeley) the \bar\partial-complex of the quantised compact Hermitian symmetric spaces is identified with the Koszul complexes of the quantised symmetric algebras of Berenstein and Zwicknagl. This leads for example to an explicit construction of the relevant quantised Clifford algebras. The talk will be fairly self-contained and begin with three micro courses covering the necessary classical background (one on Dirac operators, one on symmetric spaces, one on Koszul algebras), and then I'll explain how noncommutative geometry and quantum group theory lead to the problems that we are dealing with in this project.

Abstract: Wendy Lowen (Antwerp) On compact generation of deformed schemes. We discuss a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's Result on compact generation of quasi-compact separated schemes. We give an application of our theorem to non-commutative deformations of such schemes.

Abstract: Michele D'Adderio (University Libre de Bruxelles) A geometric theory of algebras. I will introduce some classical notions of geometric group theory (like growth and amenability) in the setting of associative algebras, and I will show how they interact with other classical invariants (like the Gelfand-Kirillov dimension and the lower transcendence degree).

Abstract: Oleg Chalykh (Leeds) Calogero-Moser spaces for algebraic curves. I will discuss two existing definitions of Calogero-Moser spaces for curves: one in terms of Cherednik algebras, another - in terms of deformed preprojective algebras, the link between them, and explain how one can compute geometric invariants of these spaces, such as the Euler characteristic and Deligne-Hodge polynomial.

Abstract: Adrien Brochier (Edinburgh) On finite type invariants for knots in the solid torus. Finite type knot invariants are those invariants vanishing on the nth piece of some natural filtration on the space of knots. This notion was introduced by Vassiliev and it turns out that most of known numerical invariants are of finite type. Kontsevich proved the existence of a "universal" invariant, taking its values in some combinatorial space, of which every finite type invariant is a specialization. This result involves some complicated integrals, but can be made combinatorial using the theory of Drinfeld associators. We will review this construction and explain why the naive generalization of this theory for knot in thickened surfaces fails. We will suggest a general way of overcoming this obstruction, and prove an analog of Kontsevich theorem in this framework for the case M=C^*, i.e. for knots in a solid torus. Time permitting, we will give an explicit construction of specializations of our invariant using quantum groups.

Abstract: Stephane Launois (Kent) Efficient recognition of totally nonnegative cells. In this talk, I will explain how one can use tools develop to study the prime spectrum of quantum matrices in order to study totally nonnegative matrices.

Abstract: Title: Characterizations of left orders in left Artinian rings. Abstract: Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In my talk, three more new criteria are given.

Abstract: Path algebras with relations constructed from a superpotential were studied by Bocklandt, Schedler and Wemyss in 'Superpotentials and Higher Order Derivations'. I consider when deformations of these algebras have relations given by an inhomogenous superpotential. This encompasses many interesting examples, such as deformed preprojective algebras and symplectic reflection algebras.

Abstract: An endomorphism T of an object can be viewed as a discrete-time dynamical system: perform one iteration of T with every tick of the clock. This dynamical viewpoint suggests questions about the long-term destiny of the points of our object. (For example, does every point eventually settle into a periodic cycle?) A fundamental concept here is the "eventual image". Under suitable hypotheses, it can be defined as the intersection of the images of all the iterates T^n of T. I will explain its behaviour in three settings: one set-theoretic, one algebraic, and one geometric. I will then present a unifying categorical framework, using it to explain how the concept of eventual image is a cousin of the concepts of spectrum and trace.

Abstract: Title: Chevalley restriction theorem for vector-valued functions on quantum groups Abstract: For a simple finite dimensional Lie algebra g, its Cartan subalgebra h and its Weyl group W, the classical Chevalley theorem states that, by restricting ad-invariant polynomials on g to its Cartan subalgebra, one obtains all W-invariant polynomials on h, and the resulting restriction map is an isomorphism. I will explain how to generalize this statement to the case when a Lie algebra is replaced by a quantum group, and the target space of the polynomial maps is replaced by a finite dimensional representation of this quantum group. I will describe all prerequisites for stating the theorem and sketch the idea of the proof, most notably the notion of dynamical Weyl group introduced by Etingof and Varchenko.

Abstract: Lie superhomomorphisms of superalgebras. The relationship between the associative and Lie structure of an associative algebra was studied by Herstein and some of his students in the 1950's and 1960's. After some initial partial results the complete classification of Lie isomorphisms was obtained by Matej Bre\vsar in 1993. It now seems natural to continue the investigation of Lie homomorphisms in the setting of superalgebras. Let $A=A_0\oplus A_1$ be an associative superalgebra over a field $F$ of characteristic not $2$. By replacing the product in $A$ by the superbracket $[\cdot,\cdot]_s$, $A$ becomes a Lie superalgebra. Recall that $[\cdot,\cdot]_s$ is defined for homogeneous elements $a,b\in A$ as $[a,b]_s=ab-(-1)^{|a||b|}ba$. A bijective linear map $\phi:A\to A$ is a Lie superautomorphism of $A$ if $\phi(A_i)=A_i$, $i\in \mathbb{Z}_2$, and $\phi([a,b]_s)=[\phi(a),\phi(b)]_s$ for all $a,b\in A$. We will present a characterization of Lie superautomorphisms of simple associative superalgebras, obtained in a joint work with Yuri Bahturin and Matej Bre\vsar.

Abstract: Title: Rational Cherednik algebras and Schubert cells Abstract: I will recall the connection between rational Cherednik algebras, the Calogero-Moser space and the adelic Grassmannian. Then I will try to explain how one can interoperate Schubert cells, and conjecturally their intersection, in the Grassmannian in terms of the representation theory of the rational Cherednik algebra.

Abstract: We will tell some of the story around a classical elementary congruence due to Wolstenholme (1862) that deal with prime numbers. Like for the classical Fermat and Wilson congruences various generalizations were soon discovered. Surprisingly, yet another generalization was discovered only very recently. It involves Lucas sequences, which are a generalization of the Fibonacci numbers.

Abstract: Title: $\tau$-tilting theory Abstract: Mutation is a basic operation in tilting theory, which covers reflection for quiver representations, APR tilting modules and Okuyama-Rickard construction of tilting complexes. In this talk we introduce the notion of (support) $\tau$-tilting modules, which `completes' tilting modules from viewpoint of mutation in the sense that any indecomposable summand of a support $\tau$-tilting module can be replaced in a unique way to get a new support $\tau$-tilting module. Moreover, for any finite dimensional algebra we show that there exist bijections between (1) support $\tau$-tilting modules, (2) functorially finite torsion classes, and (3) two-term silting complexes. Moreover if the algebra comes from a 2-Calabi-Yau triangulated category, (4) cluster-tilting objects also correspond bijectively. This is a joint work with Takahide Adachi and Idun Reiten.

Abstract: Title: 'The mixed Littlewood Conjecture'.

Abstract: Title : Quadratic differentials as stability conditions Abstract : This talk is about how spaces of quadratic differentials on Riemann surfaces arise as stability conditions on certain CY3 categories. These categories are defined by quivers with potential but can also be viewed (heuristically?) as Fukaya categories of symplectic manifolds. I will try to explain what all this means, and give the main idea of the construction. This is joint work with Ivan Smith, inspired by a paper of physicists Gaiotto, Moore and Neitzke.

Abstract: Title: n-representation infinite algebras Abstract: We introduce a distinguished class of finite dimensional algebras of global dimension n which we call n-representation infinite. For the case n=1, they are path algebras of non-Dynkin quivers. Taking (n+1)-preprojective algebras, they correspond bijectively with (n+1)-Calabi-Yau algebras of Gorenstein parameter 1. I will discuss 3 important classes of modules, preprojective, preinjective and regular as an analogue of the classical case n=1. This is a joint work with Martin Herschend and Steffen Oppermann.

Abstract: Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set

Abstract: Speaker: Christian Korff Title: Quantum cohomology via vicious and osculating walkers Abstract: We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder. These lattice paths can be described in terms of the combinatorial R-matrix of Kirillov-Reshetikhin crystal graphs. Crystal graphs are special directed, coloured graphs which are combinatorial objects encoding the representation theory of quantum algebras. Using the path description we identify the quantum Kostka numbers of Bertram, Ciocan-Fontanine and Fulton with the cardinality of a special subset of vertices in these graphs. Speaker: James Mitchell Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set Abstract: In this talk I will review some recent results relating to the lattice of subgroups of the symmetric group and its semigroup theoretic counterpart the lattice of subsemigroups of the full transformation semigroup on an infinite set. As might be expected, these lattices are extremely complicated. I will discuss several results that make this comment more precise, and shed light on the maximal proper sub(semi)groups in the lattice. I will also discuss a natural related partial order, introduced by Bergman and Shelah, which is obtained by restricting the type of sub(semi)groups and considering classes of, rather than individual, (semi)groups. In the case of the symmetric group, this order is very simple but in the case of the full transformation semigroup it is again very complex.

Abstract: Title: "Spectral Curves ans Number Theory" Abstract: The modern approach to integrable systems typically proceeds via a curve, the parameters of the curve encoding the actions and its Jacobian (or possibly some related Prym) encoding the angles. Physically relevant families of curves are often described by fixed relations amongst differentials on the curve. We shall look at number theoretic properties of these curves. For many integrable systems the curves are transcendental. I shall review W\"ustholz's Analytic subgroup theorem giving simple examples before applying this in the spectral curve context.

Abstract: Title: "Classifying Noncommutative surfaces: Subalgebras of the Sklyanin algebra" Abstract: Noncommutative projective algebraic geometry aims to use the techniques and intuition of (commutative) algebraic geometry to study noncommutative algebras and related categories. A very useful intuition here is that (the category of coherent sheaves over) a noncommutative projective scheme is simply the category of finitely generated graded modules modulo those of finite length over a graded algebra R. One of the major open problems here is to classify the noncommutative projective irreducible surfaces aka noncommutative graded domains of Gelfand-Kirillov dimension three. After surveying some of the known result on this question I will describe some very recent work of Rogalksi, Sierra and myself describing the subalgebras of the Sklyanin algebra.

Abstract: Title: Generation of Finite Groups

Abstract: Title: On the categorification of Reid's recipe Abstract: For a finite abeilan subgroup G of SL(3,C), Reid's recipe is a combinatorial cookery that describes very simply the relations between tautological line bundles on the G-Hilbert scheme. Building on results of Cautis-Logvinenko, I'll describe joint work that reveals the importance of this cookery for the derived category of the G-Hilbert scheme.

Abstract: Title: A family of 4-dimensional algebras Abstract: We construct an interesting family of algebras of global dimension and GK-dimension 4, and show that the general member of this family is noetherian and birational to P2 (in the appropriate sense). Such algebras were conjectured not to exist by Rogalski and Stafford. We show also that these algebras have counterintuitive homological properties: in particular, the Auslander-Buchsbaum formula fails for them. This is joint work with Rogalski.

Abstract: Title: "Geometry of random groups"

Abstract: Title: Height zero characters of covering groups. Abstract: The characters of height 0 of finite groups are the object of numerous theorems and conjectures. If G is a finite group, and p a prime, we write Irr_0(G) for the set of characters of p-height 0 of G. The Alperin-Mckay Conjecture states that, if B is a p-block of G with defect group D, and with Brauer correspondent b in N_G(D), then |Irr_0(B)|=|Irr_0(b)|. In 2002, Isaacs and Navarro formulated a refinement of this conjecture. For any integer 0 < k < p, we denote by M_k(B) the set of height 0 characters of B whose degree has a p'-part congruent to ± k modulo p. The Isaacs-Navarro Conjecture then states that |M_{ck}(B)|=|M_k(b)|, where c is the p'-part of the index of N_G(D) in G. In this talk, I want to present (an idea of) the proof of this result in the Schur extensions of the symmetric and alternating groups. As in the symmetric groups, it is in this case possible to exhibit an explicit bijection, by using the combinatorics that describes the characters and blocks. I also show how these groups fit within the frame of a recent conjecture by Malle and Navarro on nilpotent blocks. Finally, I want to conclude with some related results about the combinatorics we use, in particular about hooks in partitions and bars in bar-partitions.

Abstract: Title: "Automorphisms of generalized R. Thompson groups via dynamics"

Abstract: Title: "Conjugacy of algebraic numbers with rational parameters" Abstract: "We consider algebraic numbers having either rational real part, rational imaginary part or rational modulus, and discuss the question of whether such numbers can share their minimal polynomial. To answer this question, we apply some Galois theory and group theory." This work is joint with Karl Dilcher and Rob Noble.

Abstract: Title: "Harish-Chandra bimodules of rational Cherednik algebras" Abstract: "Harish-Chandra bimodules are a class of bimodules defined for rational Cherednik algebras that have attracted much recent research interest. In this talk, I will attempt to explain some of the motivation behind this interest and then move on to present some results regarding Harish-Chandra bimodules of rational Cherednik algebras, with particular emphasis on the case of cyclic groups."

Abstract: Title: Symmetric Auslander and Bass categories Abstract: We define the symmetric Auslander category $\sA^{\s}(R)$ to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains $\sA(R)$, the ordinary Auslander category. It is well known that $\sA(R)$ is intimately related to Gorenstein projective modules, and our main result is that $\sA^{\s}(R)$ is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories \[ \underline{\GMor}(R) \stackrel{\simeq}{\rightarrow} \sA^{\s}(R) / \sK^{\bounded}(\Prj\,R) \] where $\underline{\GMor}(R)$ is the stable category of Gorenstein projective objects in the abelian category $\Morph(R)$ of homomorphisms of $R$-modules. This result is set in the wider context of a theory for $\sA^{\s}(R)$ and $\sB^{\s}(R)$, the symmetric Bass category which is defined dually. This is joint work with Peter Jorgensen.

Abstract: Title: On 1-dimensional representations of finite W-algebras. Abstract: 1-dimensional representations of finite W-algebras enable one to construct completely prime primitive ideals with a prescribed associated variety and quantise coadjoint nilpotent orbits. A few years ago I conjectured that all finite W-algebras admits such representations. In my talk I am going to discuss the current status of this conjecture.

Abstract: The abstract notion of recognition: algebra, logic and topology (Joint work with M. Gehrke and S. Grigorieff) We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topology is the key ingredient. We prove the existence of a minimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a compact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular languages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered.

Abstract: Title: Higgs algebra of weighted projective lines and loop crystals. Abstract: In this talk we contruct enveloping algebras of loop Lie algebras via geometry, considering constructible functions on the space of Higgs bundles on a weighted projective line. The geometry of this space then leads to nice elements in the algebra, which forms a basis called the semicanonical basis. Another interested feature coming from geometry is the construction of a loop crystal, which is an analog of a crystal in the loop case.

Abstract: Title: Modular representation theory of the Ariki-Koike algebra in characteristic 0. Abstract: The Ariki-Koike algebra is a natural generalisation of the Iwahori-Hecke algebras of types A and B. Much of its representation theory is controlled by the Schur elements, which are Laurent polynomials attached to its irreducible representations. We will give a new, pretty formula for these elements, and study the applications of our result to the representation theory of the Ariki-Koike algebra in characteristic 0

Abstract: Title: Rational Cherednik algebras in positive characteristic. Abstract: In this talk I will describe some of the basic features of rational Cherednik algebras in positive characteristic. There is a close relationship between the representation theory of these algebras and the geometry of their centres. I will show how their representation theory can be used to determine when the centre of the algebra is a regular ring. This is based on joint work with M. Martino.

Abstract: Title: Affine W-algebras Abstract: Affine W-algebras may be considered as a generalization of infinite dimensional Lie algebras such as Kac-Moody algebras and the Virasoro algebra. They may be also regarded as an affinization of finite W-algebras, but affine W-algebras are introduced earlier than finite W-algebras in physics literature. Affine W-algebras are related with conformal field theories, integrable systems, quantum groups, the geometric Langlands program, and 4 dimensional gauge theories. In my talk I will discuss about their structure and describe their representation theory by focusing on type A cases.

Abstract: Title: Finite Subgroups of the Classical Groups Abstract: A theorem of Jordan (1878) states that there is a function f on the natural numbers such that if G is a finite subgroup of GL(n,C), then G has an abelian normal subgroup of index at most f(n). Several years ago, I determined the optimal value for f(n), and I will talk about this and recent work that extends the result to the finite subgroups of all classical groups, both real and complex.

Abstract: Title: "J. Sally's question and a conjecture by Y. Shimoda" Abstract: In 2007, Y. Shimoda, in connection with an open question of J. Sally from 1978, conjectured that a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. This talk surveys the results that have been obtained to date concerning this conjecture. First we indicate that we can reduce to the case of dimension three, and that the conjecture has a positive answer if the ring is either regular, or is complete with infinite residue field and multiplicity at most three. Finally we consider the case of the appropriate analogue of the conjecture for standard graded rings, and indicate how a mix of algebraic and geometrical methods yields a definite answer in this setting. (Joint work with S. Goto and F. Planas-Vilanova)

Abstract: Title: Totally nonnegative matrices Abstract: A real matrix is totally nonnegative if each of its minors is nonnegative (and is totally positive if each minor is positive). The talk will survey elementary properties of these matrices and present new results which have surprising links with the theory of quantum matrices.

Abstract: Title: Flops and mutations of polyhedral singularities Abstract: Let G be a finite subgroup of SO(3) and consider the so called polyhedral singularity C^3/G. It is well known that the G-Hilb is a distinguished crepant resolution which plays a central role in the so called McKay correspondence. I will explain in the talk how every crepant resolution of C^3/G is a moduli space of quiver representations showing that there exists a 1-to-1 correspondence between between flops of G-Hilb and mutations of the McKay quiver. This is a joint work with Y. Sekiya.

Abstract: Title: Fourier--Mukai partners of elliptic surfaces. Abstract: For smooth projective varieties X and Y, when their derived categories are equivalent we say that X is a Fourier--Mukai partners of Y. We study the set of isomorphism classes of Fourier--Mukai partners of elliptic surfaces with negative Kodaira dimensions.

Abstract: Title: "Parahoric torsors and Parabolic bundles on compact Riemann surfaces and representations of Fuchsian groups." Abstract: Let X be an irreducible smooth projective algebraic curve of genus g ≥ 2 over the ground field of complex numbers and let G be an arbitrary semisimple simply connected algebraic group. The aim of the talk is to introduce the notion of a semistable and stable parahoric torsor under certain Bruhat-Tits group schemes and construct the moduli space of semistable parahoric G –torsors and identify the underlying topological space of this moduli space with spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles. The talk is on a joint work with C.S. Seshadri.

Abstract: Title: "Slow but not too Slow: Nil Algebras and Growth"

Abstract: Title: Some quantum analogues of properties of Grassmannians Abstract: The classical coordinate ring of the Grassmannian has many nice structural properties and one expects these to carry over to its quantum analogue. We will discuss two properties for which this does indeed happen, namely a cluster algebra structure (recent work with Launois, quantizing work of Scott) and an action of the dihedral group (work with Allman, extending a recent construction of Launois and Lenagan). We will also mention an extension in a different direction, namely to infinite Grassmannians (work with Gratz).

Abstract: Title: Canonical birationally commutative factors of noetherian graded algebras Abstract: It is known that if a graded k- algebra R is strongly noetherian (that is, it remains noetherian upon commutative base-change), then there is a canonical map from R to a twisted homogeneous coordinate ring on some projective scheme. We show this can be generalized to algebras that are merely noetherian, and the resulting factor satisfies a universal property. Further, we show that under suitable conditions on the geometry of the Hilbert schemes of point modules over R, this canonical factor is a naive blowup algebra, in the sense of Keeler-Rogalski-Stafford.

Abstract: Title: Mutation of rigid objects and partial triangulations. Abstract: By several results (due to Amiot, Fomin--Shapiro--Thurston, Labardini-Fragoso and Keller--Yang) a cluster category can be associated with any compact Riemann surface with boundaries and marked points. The triangulations of the marked Riemann surface correspond to the so-called cluster-tilting objects of the cluster category. These objects are of particular interest since they categorify the clusters of Fomin--Zelevinsky's cluster algebras. In particular, they have a nice theory of mutation. This mutation turns out to be the categorical analogue of the flip of triangulations. Brustle--Zhang proved that some more general objects, the rigid objects, categorify the partial triangulations of the surface. In this talk, based on a joint paper with Robert Marsh, I will explain how both flips and mutations can be generalised to this situation. Our main tool is a result showing that Iyama--Yoshino reduction for cluster categories correspond to cutting along an arc the associated Riemann surface. All statements and results will be illustrated with some (small) geometric examples.

Abstract: Title: Additive polylogarithms Abstract: In this talk we will define additive polylogarithms and describe how they are related to motivic cohomology over the dual numbers of a field of characteristic zero. In the characteristic p case, and in weight 2, we will also describe how the additive dilogarithm is related to Kontsevich's logarithm.

Abstract: Title: Braid group actions on quantum symmetric pair coideal subalgebras Abstract: It was noted recently by Molev and Ragoucy, and idependently by Chekhov, that the nonstandard quantum enveloping algebra of so(N) allows an action of the Artin braid group. We interpret and generalize this action within the theory of quantum symmetric spaces.

Abstract: Title: "Quadratic algebras: the Anick conjecture on Hilbert series, Koszulity, NCCI and RCI" Abstract: We present several results on the Anick conjecture which asserts that the lower bound for the Hilbert series, known as the Golod-Shafarevich estimate is attained on generic quadratic algebra. The technique (due to Anick), allowing to write down precisely the formula for the Hilbert series will be demonstrated. We will discuss also related questions of Koszulity and being noncommutative complete intersection (NCCI). Connections to the latter property on the level of finite dimensional representations, namely, introduced by Ginsburg and Etingof notion of representational complete intersection (RCI) will be considered and some examples given.

Abstract: Title: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators''

Abstract: Some aspects of the modular representation theory of a finite group can be described by a tree. Such trees have been determined for almost all finite simple groups, but some cases remain unknown. Starting from the example of the group SL2(q) I will explain how geometric methods can be used to solve this problem for finite reductive groups.

Abstract: Jorgensen is speaking on "A Caldero-Chapoton map for infinite clusters" Minasyan is speaking on ""Fixed subgroups of automorphisms of non-positively curved surfaces."

Abstract: Cluster Algebras come equipped with a natural Poisson bracket , or potentially a family of Poisson brackets which are "related" via the actions of an algebraic torus. In this talk I will show how to classify the torus invariant Poisson prime ideals of a cluster algebra using the combinatorial data obtained from the exchange matrix. Finally I will apply this method to the case of the cluster algebra of functions on matrices. Of course, I will also introduce all the relevant notions.

Abstract: Joint with Geometry

Abstract: I will introduce some of the important models of computation, give some examples of important techniques, highlight some central results and address some open questions.

Abstract: Speaker: Christopher Phan (Glasgow) Title: Generalised Koszul properties for noncommutative graded algebras Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.

Abstract: Speaker: Tom Leinster (Glasgow) Title: Rethinking set theory Abstract: At the heart of mathematical culture is a niggling worry. We use basic set-theoretic language all the time, and we are informed that ZFC is the "foundation of mathematics". Yet most of us sail through life neither knowing nor much caring what the axioms of ZFC are; and if we do stop to look at the axioms, they seem curiously remote from what we actually do as mathematicians. I will present a solution to this problem, due to Lawvere. It is a radical reshaping of set theory. The axioms boil down to 10 totally mundane properties of sets, used every day by ordinary mathematicians. In this way, I hope to persuade you that set theory is not to be sniffed at.

Abstract: Schubert calculus answers enumerative questions such as, how many lines meet four given (sufficiently general) lines in 3-dimensional space? The answer to such a question is a non-negative integer, and can be arrived at by a calculation in the cohomology ring of the variety of lines in 3-dimensional projective space. Analogous calculations can be carried out for more general cohomology theories and more general homogeneous spaces. In these more general settings it may not be clear what the answers are enumerating, that they should be positive, or even what "positive" should mean. We'll explain what type of positivity holds for equivariant K-theory of compact homogeneous spaces, and how to prove it using souped-up versions of Kleiman transversality and Kodaira vanishing.

Abstract: In the theory of quantum groups, Lie subalgebras of semisimple Lie algebras should be realised as coideal subalgebras of quantised enveloping algebras. While many classes of such coideal subalgebras of are known, there is so far no general classification. In this talk, a classification of coideal subalgebras of the positive Borel part of a quantised enveloping algebra is presented. The result is expressed in terms of characters of quantisations of nilpotent Lie subalgebras, which were introduced by de Concini, Kac, and Procesi for any element in the Weyl group. The study of such characters naturally leads to fun Weyl group combinatorics. The talk is based on joint work with I.~Heckenberger.

Abstract: Speaker: Gwyn Bellamy Title: Cuspidal representations for rational Cherednik algebras

Abstract: Speaker: Cornelius Reinfeldt (HWU) Title: The structure of homomorphisms into hyperbolic groups

Abstract: Speaker: Martin Bridson (Oxford) Ttile: Curvature, dimension, and representations of mapping class groups Abstract: In this talk I'll discuss constraints on the way in which mapping class groups of surfaces can act on spaces of non-positive curvature and explain how these constraints lead to conclusions about homomorphisms between mapping class groups, and how they inform us about the (linear) representation theory of such groups.

Abstract: In general, local models are schemes defined over DVRs. In all examples known to me, one of the most important techniques to study a given local model is to embed its special fiber in an appropriate affine flag variety; in this way, the special fiber becomes stratified into Schubert cells. In this talk I will discuss some of the combinatorial and algebro-geometric problems in the affine flag variety that arise from these considerations. As in my other talk, the emphasis will be placed heavily on understanding concrete examples.

Abstract: Speaker: Michael Heusener (Clermont-Ferrand) Title: Infinitesimal inflexibility under Dehn filling (joint work with Joan Porti) Abstract: A closed hyperbolic 3-manifold inherits a natural projective structure. Though the hyperbolic structure is rigid (Weil,Mostow), the projective one may be rigid or not. Johnson and Millson provide nonrigid examples, by means of bending along totally geodesic surfaces. Rigid examples are constructed by Cooper, Long and Thistlethwaite and are called \emph{inflexible}. The aim of this talk is to present a method to construct infinite families of closed inflexible manifolds. In particular we shall show that for $n$ sufficienly large, the homology sphere obtained by $1/n$-Dehn filling on the figure eigth knot is infinitesimally inflexible.

Abstract: TBA

Abstract: Speaker: David Jones (HWU) Title: Strong representations of the polycyclic monoids: cycles and atoms.

Abstract: Speaker: Mark Lawson (HWU) Title: A non-commutative generalization of Stone duality

Abstract: Speaker: Tom Lenagan (UoE) Ttile: From totally nonnegative matrices to quantum matrices and back, via Poisson geometry

Abstract: Speaker: Owen Cotton-Barratt (Oxford) Title: When good groups go bad Abstract: Much of group theory is concerned with whether one property entails another. When such a question is answered in the negative it is often via a pathological example. The Rips construction is an important tool for producing such pathologies. We will consider the construction and a recent refinement which makes the output group conjugacy separable. The motivation for this was an application in profinite group theory; the context for this theorem will be described.

Abstract: TBA

Abstract: http://www.maths.abdn.ac.uk/artin/meeting.php?id=21

Abstract: http://www.maths.abdn.ac.uk/artin/meeting.php?id=21

Abstract: Speaker: Alina Vdovina Title: Cayley graph expanders, pro-p-groups and buildings

Abstract: Mirror symmetry, Langlands duality and the Hitchin system

Abstract: Multiplicity-Free Actions of Classical Groups

Abstract: Speaker: Gerald Williams (Essex) Ttile: Finiteness of some cyclically presented and Fibonacci-like groups

Abstract: Rouquier families for Hecke algebras

Abstract: Speaker: Tara Brendle (Glasgow) Title: The symmetric Torelli group

Abstract: Speaker: Brendan Owens (Glasgow) Ttile: Knot surgeries bounding definite 4-manifolds

Abstract: Speaker: Vic Reiner (University of Minnesota). Title: Extending the coinvariant theorems of Chevalley, Shephard-Todd and Springer. Abstract: (This is joint work with B. Broer, L. Smith, and P. Webb.) The theorems in the title are classical results in the invariant theory of finite subgroups of GL_n(C) generated by reflections. After reviewing these results, we show how to extend them in several directions, removing many of their hypotheses. In particular, our results work over an arbitrary field k rather than the complex numbers. Also our version of the Chevalley-Shephard-Todd theorem applies to any finite subgroup of GL_n(k), not just reflection groups. If time permits, we will mention the combinatorial applications in characteristic p that motivated us.

Abstract: Speaker: Jochen Heinloth (University of Amsterdam) Title: Twisted groups on curves and some related moduli spaces Abstract: To study the spaces of bundles on a Riemann surface (or algebraic curves), the so called uniformization theorem has been a very useful tool. This result says that these spaces can be viewed as a quotient of the space of all maps of the circle into a Lie group. A similar result has been conjectured by Pappas and Rapoport for spaces of bundles equipped with different types of extra structure. I would like to explain, what these are, how they are related to twisted loop groups and why the general setup allows to give a short proof of the conjecture. At the end of the talk I will try to indicate an application of this to point counting arguments for these moduli spaces.

Abstract: Speaker: Sue Sierra (University of Washington) Title: Primitivity of twisted homogeneous coordinate rings Abstract: Let B = B(X, L, f) be the twisted homogeneous coordinate ring associated to a complex projective variety X, an automorphism f of X, and an appropriately ample invertible sheaf L. We study the primitive spectrum of B, and show that there is an intriguing relationship between primitivity of B and the dynamics of the automorphism f. In many cases Dixmier and Moeglin's characterization of primitive ideals in enveloping algebras generalizes to B; in particular, this holds if X is a surface. This is joint work with J. Bell and D. Rogalski.

Abstract: Speaker: Sinead Lyle (University of East Anglia) Title: Jucys-Murphy elements and homomorphisms between Specht modules. Abstract: In the representation theory of the symmetric group, an important open problem is to determine the structure of certain objects known as Specht modules. I will talk about a method of constructing homomorphisms between pairs of Specht modules using the Jucys-Murphy elements. This is joint work with Andrew Mathas.

Abstract: Speaker: Kenny Brown (University of Glasgow) Title: Connections between generic q and roots of unity: q-modular systems

Abstract: Speaker: Toby Stafford (University of Manchester) Title: Equidimensionality for Cherednik algebras

Abstract: Speaker: Natalia Iyudu (University of Belfast) Title: Quadratic algebras: the Anick conjecture, representation spaces and Novikov structures

Abstract: Speaker: Nick Gilbert Title: Diagram groups and rewriting for words and trees.

Abstract: Speaker: Vladimir Dotsenko (University of Dublin Trinity College) Title: Parking functions and vertex operators Abstract: The goal of this talk is to discuss several series of graded vectors spaces whose series of dimensions include the series of Catalan numbers (and their generalisations), and the sequence (n+1)^{n-1} of "parking functions numbers". First of all, we show how these vector spaces arise from representation-theoretical constructions for some associative algebras. Another way to construct vector spaces with same dimensions and graded characters is to consider spaces of global sections of certain vector bundles on (zero fibres of) Hilbert schemes (for the latter the dimension and character formulae were obtained by Haiman in his works on diagonal harmonics and the "n! conjecture"). I shall formulate a conjecture relating these two constructions and try to explain some reasons for this conjecture to be true.

Abstract: Speaker: Emmanuel Letellier (Université de Caen) Title: Topology of character varieties and Macdonald polynomials Abstract: We conjecture a formula for the mixed Hodge polynomials of representations varieties of the fundamental group of punctured Riemann surfaces in terms of Macdonald polynomials. In this talk we will bring evidences for this conjecture and see some applications in the representation theory of quivers.

Abstract: Speaker: Charudatta Hajarnavis (University of Warwick) Title: Polynomial Identity Rings of Finite Global Dimension Abstract: A commutative Noetherian ring of finite global dimension is a direct sum of integral domains (including fields). In the dimension 1 case (i.e. hereditary rings) these are Dedekind domains. In the non-commutative case there is an extensive theory of hereditary rings showing a much more complex situation. In this survey talk we look at the situation for dimension 2 and higher and also mention some recent work.

Abstract: Speaker: Susan Cooper (University of Nebraska - Lincoln) Title: In Search of Exactness Abstract: Certain data about a finite set of distinct, reduced points in projective space can be obtained from its Hilbert function. It is well known what these Hilbert functions look like, and it is natural to try to generalize this characterization to non-reduced schemes. In particular, we consider a fat point scheme determined by a set of distinct points (called the support) and non-negative integers (called the multiplicities). In general, it is not yet known what the Hilbert functions are for fat points with fixed multiplicities as the support points vary. However, if the points are in projective 2-space and the number of support points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities (due to Guardo-Harbourne and Geramita-Harbourne-Migliore). In this talk we focus on what can be said, in projective 2-space, given information about what collinearities occur among the support points. Using this information we measure how far related sequences can be from being exact on global sections. Doing so, we obtain upper and lower bounds for the Hilbert function of the fat point scheme. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case. This is joint work with B. Harbourne and Z. Teitler.

Abstract: Speaker: Max Neunhoeffer (University of St Andrews) Title: Finding normal subgroups Abstract: In the context of the Matrix Group Recognition Project the following is an important task: Given G=< g_1, ..., g_k >, find a non-trivial element x in G that is contained in a proper normal subgroup or fail if G is simple. In this talk I explain why this problem is important and present some ideas how to tackle it.

Abstract: Speaker: Rick Thomas (University of Leicester) Title: FA-presentable structures Abstract: We are interested in the notion of computing in structures. One approach would be to take a general model of computation such as a Turing machine. A structure would then be said to be computable if its domain can represented by a set which is accepted by a Turing machine and if there are decision-making Turing machines for each of its relations. However, there have been various ideas put forward to restrict the model of computation used; whilst the range of possible structures decreases, certain properties of the structures may become decidable. One interesting approach was introduced by Khoussainov and Nerode who considered structures whose domain and relations can be checked by finite automata; such a structure is said to be FA-presentable. This was inspired, in part, by the theory of automatic groups; however, the definitions are somewhat different. We will survey some of what is known about FA-presentable structures, contrasting it with the theory of automatic groups and posing some open questions. The talk is intended to be self-contained, in that no prior knowledge of these topics is assumed. Reply

Abstract: Speaker: Pramod Achar (Louisiana State University) Title: Introduction to staggered sheaves Abstract: Perverse sheaves, introduced around 1980, have many remarkable properties, involving such notions as Poincare-Verdier duality, weight filtrations and "purity," and the celebrated Decomposition Theorem. These properties have made perverse sheaves into an incredibly powerful tool, especially for applications in representation theory. "Staggered sheaves" are a new attempt to duplicate some of these properties in the setting of vector bundles and coherent sheaves. I will discuss the ingredients that go into defining staggered sheaves, state the main results that are known so far, and perhaps speculate on potential applications. This will be an introductory talk: I will not assume any familiarity with perverse or staggered sheaves, and I will try to focus on examples on A^1 or P^1.Some of the results on staggered sheaves are joint work with D. Sage and D. Treumann.

Abstract: Speaker: Arun Ram (University of Melbourne) Title: Two boundary Braid groups, Hecke algebras and tantalizer algebras Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will a medley of vignettes around double affine type C braid groups and quotient algebras.

Abstract: Speaker: Pepe Burillo (Barcelona) Title: Higher-dimensional Thompson's groups Abstract: Higher-dimensional analogs of Thompson's group V have been introduced recently by Brin. We will recall their definition and find the analog of the standard interpretation of Thompson's groups by tree pair diagrams. We will use this interpretation to give presentations for them (both finite and infinite), and to find estimates for the word metric of these groups in terms of the number of carets in the tree pair diagram. Finally, we will show that the inclusion of F, T and V in the higher-dimensional groups is exponentially distorted

Abstract: Speaker: Saul Schleimer (University of Warwick) Title: Polynomial-time word problems Abstract: We will sketch a proof that Aut(G) has polynomial-time word problem when G is a word hyperbolic group. The heart of the argument is the idea from computer science; straight-line programs are widely studied in the field of data compression. As it so happens, they are also well suited for analyzing group automorphisms.

Abstract: Speaker: Rudolf Tange (University of York) Title: The centre of the universal enveloping algebra in characteristic p. Abstract: Let U be the universal enveloping algebra of the Lie algebra g of a reductive group G over an algebraically closed field of characteristic p and let Z be the centre of U. The algebraic variety corresponding to Z is called the Zassenhaus variety of g. Unlike in characteristic 0, Z is not a polynomial ring, in fact the Zassenhaus variety is not smooth. I will show (under certain mild assumptions) that Z is a unique factorisation domain and that its field of fractions is purely transcendental over k (i.e. the Zassenhaus variety is rational). If time allows I will indicate the relevance of the Zassenhaus variety for the representation theory of g and a relation with the Gelfand-Kirillov conjecture.

Abstract: Speaker: Nicolas Guay (Maxwell Institute) Title: Representations of double affine Lie algebras

Abstract: Speaker: Paul Martin (University of Leeds) Title: The complex representation theory of the Brauer algebra

Abstract: Speaker: Paul Turner (Heriot-Watt) Title: Homology of algebras with coefficients in a graph Abstract: Starting with a directed graph, I will describe the construction of a homology theory for algebras, related to the Khovanov homology of graphs. When the graph is the n-gon, this homology agrees with Hochschild homology up to degree n.

Abstract: Speaker: Andrew Duncan (Newcastle) Title: From pregroups to groups: decision problems and universal theory

Abstract: Speaker: Will Turner (University of Aberdeen) Title: Representation theory and four dimensional topology

Abstract: Speaker: Mathieu Carette (Brussels) Title: The automorphism group of accessible groups

Abstract: Speaker: Chris Smyth (Maxwell Institute) Title: Integer symmetric matrices and Coxeter graphs

Abstract: Speaker: Jaimal Thind (Stony Brook) Title: Coxeter Elements and Periodic Auslander--Reiten Quiver Abstract: Traditionally, to study a root system $R$ one starts by choosing a set of simple roots $\Pi\subset R$ (or equivalently, polarization of the root system into positive and negative parts) which is then used in all constructions and proofs. We discuss a different approach, starting not with a set of simple roots but with a choice of a Coxeter element $C$ in the Weyl group. We show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of $R$ with a certain subset $\Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $\Ihat$ has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram $I$: roots are vertices of $\Ihat$, and the root lattice and the inner product admit an explicit description in terms of $\Ihat$. Time permitting we will discuss how this picture can be used to obtain a description of the corresponding Lie algebra. (This is joint work with A. Kirillov Jr)

Abstract: Speaker: Jasper Stokman Title: Double affine Hecke algebras and bispectrality Abstract: One of the Macdonald conjectures is the duality -by now- theorem. The duality theorem points out the bispectral nature of the Macdonald polynomials. It was proven by Cherednik using the double affine Hecke algebra. In this talk I will establish the interplay between the double affine Hecke algebra and bispectrality on a more fundamental level. It leads to a bispectral version of the quantum Knizhnik-Zamolodchikov equations and to an integrable bispectral problem associated to the Macdonald operators. This is joint work with Michel van Meer.

Abstract: Title: The grammar of the word problem Speaker: Sarah Rees, University of Newcastle. Abstract: I shall discuss the word problem for groups, introduced a century ago by Dehn, and solved by him for surface groups. I am interested in how the structure of the set of words representing the identity element of a group (this set is called the `word problem' for the group) is related to the structure of the whole group. In the late 1980's Muller and Schupp proved that the word problem for a group can be recognised using a pushdown automaton (a machine using a simple stack memory) precisely when the group is virtually free. In this case it is Dehn's algorithm that can be programmed on the pushdown automaton. A set of words that can be recognised using a pushdown automaton can be constructed using a context-free grammar, and conversely. The proof of Muller and Schupp's result is based on the realisation that the underlying context-free grammar of the word problem puts a restriction on the geometry of the group. There's a correspondence between the types of machines that recognise sets of strings and the grammars that build them, but this result suggests that the grammar constructing the word problem of a group is more clearly related to the structure of the group than is the machine that recognises it. I shall report on my recent joint work with Holt and Shapiro. This examines the grammar associated with word problems that can be solved using a generalisation of Dehn's algorithms developed by Goodman and Shapiro; we see that in this case the grammar is always `growing context-sensitive'. We extend Goodman and Shapiro's work, and find a host of examples of groups with word problems that are context-sensitive but not growing context-sensitive. Hence we can answer questions of Kambites and Otto, who found the first example of a word problem in that category.

Abstract: Speaker: Simon Goodwin (Birmingham) Title: Representations of finite W-algebras Abstract: To each nilpotent element e in a complex semisimple Lie algebra \g, one can associate a finite W-algebra denoted by U(\g,e). This algebra can be viewed as the enveloping of the Slodowy slice through the adjoint orbit of e, and has many connections to other areas of Lie theory. After presenting some history and motivation we will present an approach, due to Brundan, Kleshchev and the author, to highest weight representation theory of finite W-algebras. There is not a natural comultiplication on finite W-algebras; however, it is possible to give the tensor product of a U(\g,e)-module with a finite dimensional U(\g)-module the structure of a U(\g,e)-module. We will discuss properties of these tensor products, which are expected to be of importance in understanding the representation theory of U(\g,e).