Welcome to the homepage of the MAXIMALS algebra seminar, at the University of Edinburgh. The seminar represents the interests of all Edinburgh faculty working in algebra and number theory, and is currently being coordinated by Andrea Appel and Alexander Shapiro. We meet regularly on Tuesdays in the Bayes Centre (5th floor), from 2:00-3:00pm.

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Upcoming MAXIMALS events at University of Edinburgh

Upcoming MAXIMALS events at Heriot-Watt

Past MAXIMALS events at University of Edinburgh

### Talks in past semesters

Apologies, the calendar for Spring 2015 is temporarily down.

 Tuesday 20 January 3pm Robert Laugwitz (Oxford) Braided Drinfeld and Heisenberg doubles and TQFTs with defects 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. Tuesday 13 January 3pm Richard Hepworth (Aberdeen) A homology theory for graphs 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.

### Talks from Fall 2014:

 Tuesday 2 December 3pm Bin Shu (East China Normal) Generic property and conjugacies of Borel subalgebras for restricted Lie algebras For a finite-dimensional restriicted Lie algebra $g$ over an algebraically closed field of prime characteristic, we introduce the notion "generic property", saying that a restricted Lie algebra satisfies such a property if it admits generic tori introduced in BFS. A Borel subalgebra of $g$ is defined as a maximal solvable subalgebra containing a maximal torus of $g$, which is further called generic if additionally containing a generic torus. In this talk, we first verify that a conjecture of Premet on regular Cartan subalgebras for $g$ is valid when $g$ satisfies the generic property, and show that his conjecture does not hold whenever $g$ does not satisfy the generic property. We finally classify the conjugacy classes of Borel subalgebras of the restricted simple Lie algebras $g=W(n)$ under $Aut(g)$-conjugation when $p>3$, and present the representatives of these classes. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras. Tuesday 25 November 3pm Martin Kalck (Edinburgh) Spherical subcategories and new invariants for triangulated categories Motivated by examples arising in algebraic geometry, we study objects of k-linear triangulated categories with two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical, i.e. a Calabi-Yau object. In many examples, both from representation theory and geometry these spherical subcategories admit explicit descriptions. Furthermore, the collection of all spherical subcategories ordered by inclusion yields a new invariant for triangulated categories. We derive coarser invariants like height, width and cardinality of this poset. This talk is based on joint work with A. Hochenegger & D. Ploog. Tuesday 18 November 3pm David Pauksztello (University of Manchester) An introduction to co-t-structures and co-stability conditions In this talk we introduce the ideas of co-t-structures and co-stability conditions and compare and contrast with t-structures and stability conditions. We show that the space of co-stability conditions on a triangulated category forms a complex manifold, and give some examples. Part of this talk is joint work with Peter Jorgensen (Newcastle-upon-Tyne). Tuesday 18 November 4pm Rupert Yu (University of Reims) Jet schemes of nilpotent orbit closures In this talk, we investigate jet schemes of nilpotent orbit closures in semisimple Lie algebras. For the regular nilpotent orbit, its closure is the nilpotent cone, and their jet schemes are always irreducible. This was conjectured by Eisenbud and Frenkel, and was proved as a special case of a result of Mustata in a more general setting. We shall see that for a non regular and non zero nilpotent orbit, the jet schemes of its closure are not irreducible in general, and we may obtain from this information on certain geometric properties of nilpotent orbit closures. This is a joint work in progress with Anne Moreau. Tuesday 11 November 4pm Arend Bayer (Edinburgh) The space of stability conditions on abelian threefolds (and a few CY 3folds) I will explain recent joint work with Emanuele Macri and Paolo Stellari, in which we describe the space of Bridgeland stability conditions on three-dimensional abelian varieties, and on (crepant resolutions of) their quotients by finite group actions. As I will explain, a lot of the structure is described by some elementary real algebraic geometry (configurations of points with respect to quadratic forms and components of the space of self-maps of the real projective line). Understanding this geometry allows us to strengthen and generalize the results by Antony Maciocia and Dulip Piyaratne (who were first to construct stability conditions on abelian varieties of Picard rank one with principal polarization). Tuesday 4 November 3pm Karel Casteels (University of Kent) Combinatorial Models of Quantum Matrix Algebras Some so-called "quantum matrix algebras" that are often defined by generators and relations (e.g, quantum matrices, the quantum special linear group, the quantum grassmannian, quantum symmetric and skew-symmetric matrices) can be embedded into a quantum torus by way of a certain directed graph. We are then able to "see" the generators and relations quite naturally, and, perhaps more importantly, we can use these models to study the prime and primitive spectra. Some of the work to be discussed is based on discussions with Stephane Launois and Tom Lenagan. Tuesday 28 October 3pm Christian Lomp (University of Porto) Semisimple Hopf algebra actions We review Hopf algebra actions on rings with particular emphasis on a question raised by Miriam Cohen in 1985 whether the smash product A # H of a semisimple Hopf algebra H acting on a semiprime algebra A is itself a semiprime ring. This question is open until now. In my talk I give a survey on known results concerning Cohen's question. Tuesday 28 October 4pm Paula Carvalho (University of Porto) On the injective hulls of simple modules over Noetherian Rings The Jacobson's conjecture is an open problem in ring theory and asks whether the intersection of the powers of the Jacobson radical of a two-sided Noetherian ring is zero. Jategaonkar answered the conjecture in the affirmative for a Noetherian ring R under an additional assumption (called FBN) which in particular implies that any finitely generated essential extension of a simple left R-module is Artinian. The latter condition, denoted by $( \diamond )$, is a sufficient condition for a positive answer to the Jacobson's conjecture. In this talk we will consider some Noetherian algebras and study if or when they satisfy $(\diamond)$. In particular, we will be interested in the case of differential operator rings $R\theta; d$ with $R$ a commutative Noetherian ring and $d$ a derivation. Tuesday 30 September 3pm Evgeny Feigin (National Research University, Russia) PBW filtration and nonsymmetric Macdonald polynomials We discuss the recently conjectured connection between the nonsymmetric Macdonald polynomials and the Poincaré–Birkhoff–Witt filtration on Demazure modules of affine Kac-Moody Lie algebras. The conjecture has a surprising consequence relating PBW degrees of the extremal vectors in finite dimensional representations of simple Lie algebras with the extremal part of Macdonald polynomials. We also describe the connection with the Kostant q-partition function. The talk is based on joint work with I.Cherednik and I.Makedonskyi.

Talks take place in the James Clerk Maxwell Building, room 6311.

Not part of MAXIMALS but related:

From 15-19 December 2014, there will be a workshop on Homological Interactions between Representation Theory and Singularity Theory at the University of Edinburgh.

### Talks 2013-2014:

WARNING: Any error in the notes of the talks is probably due to the note-taker.

 Tuesday May 27th 2:30pm Andre Leroy, (University of Artois) Euclidean pairs , quasi Euclidean rings and continuant polynomials Starting with problem of decomposition of 2 by 2 singular matrices we will introduce Euclidean pairs and quasi Euclidean rings. We will characterise these rings in different ways and show that unit-regular rings are quasi-Euclidean. We will present relations with constant polynomials and other families of polynomials defined by linear recurrence relations. Tuesday May 27th 1:00pm Maria Chlouveraki (Versailles) Yokonuma-Hecke and Yokonuma-Temperley-Lieb algebras Yokonuma-Hecke algebras were introduced by Yokonuma in the 60's as generalisations of Iwahori-Hecke algebras. They have recently attracted the interest of topologists, because they naturally give rise to invariants for framed and classical knots. In this talk we will introduce and study the Yokonuma-Hecke algebras of type A and the Yokonuma-Temperley-Lieb algebras, which are the generalisations of classical Temperley-Lieb algebras in this case, mainly from the algebraic point of view. Friday May 16th 1:00pm JCMB 6206 Noah Snyder (IU Bloomington) Local Topological Field Theory and Fusion Categories Topological field theories give a close relationship between topology and algebra. Traditionally the main application has been from algebra to topology: using algebraic constructions like quantum groups to produce topological invariants. However, you can also run the applications the other way, using topology to arrange and clarify your knowledge about algebra. The goal of this talk is to explain one such application. More specifically, a fusion category is a category that looks like the category of representations of a finite group: it has a tensor product, duals, is semisimple, and has finitely many simple objects. A somewhat mysterious fact about fusion categories (generalizing a theorem of Radford's about Hopf algebras) is that the quadruple dual functor is canonically isomorphic to the identity functor. I will explain this mystery by showing that it follows directly from the Dirac belt trick. The main technique in this proof is the construction of a local topological field theory attached to any fusion category. Topological field theories are invariants of manifolds which can be computed by cutting along codimension 1 boundaries. Local topological field theories allow cutting along lower codimension boundaries. Since manifolds with corners can be glued together in many different ways, this can be formalized using the language of n-categories. Using Lurie's version of the Baez-Dolan cobordism hypothesis, we describe local field theories with values in the 3-category of tensor categories. This is joint work with Chris Douglas and Chris Schommer-Pries. I will not assume prior familiarity with fusion categories or n-categories. Tuesday, May 13th 3:00pm Carl Mautner (MPIM Bonn) Modular Representation Theory and Parity Sheaves In the 1920's, Weyl proved a formula for the characters of the irreducible representations of reductive groups (e.g., the general linear group) over the complex numbers. In 1979, George Lusztig announced a conjectural character formula for representations of reductive groups over fields of positive characteristic. I will briefly discuss a history of the problem and explain the role of geometric objects called parity sheaves in recent advances. Tuesday, April 22nd 3:00pm Dorette Pronk (Dalhousie) Weakly globular double categories - a new model for weak 2-categories. In this talk I will discuss a new model for weak 2-categories. We will first show that there is more than one way to make 2-categories weak. Bicategories are obtained from 2-categories by relaxing the associativity and unit conditions, and requiring that they only hold up to coherent isomorphisms. But they are strict in the sense that we require them to have a set (or, class) X_0 of objects. To introduce the notion of weakly globular double category, we we will relax this requirement: rather, than requiring that we have a discrete set of objects (we call that the globularity condition), we will require that X_0 be a posetal groupoid (i.e., a set with equivalence relation). This is the weak globularity condition. The arrows in X_0 are distinct from the other arrows in the category, so we have now two classes of arrows, and this leads us to the notion of a double category. Weakly globular double categories are strict double categories that satisfy the weak globularity condition and an additional lifting condition on its double cells. I will discuss these conditions in detail and the show that this is sufficient to model weak 2-categories in the sense that there is a biequivalence between the 2-category of bicategories, homomorphisms and icons, and the 2-category of weakly globular doublce categories, pseudo functors and vertical transformations. This biequivalence factors through the 2-category of Tamsamani weak 2-categories. If there is enough time I will discuss the construction of a weakly globular double category of fractions and its universal properties. Tuesday, Apr 8th 3pm Alexey Sevastyanov (Aberdeen) A proof of De Concini-Kac-Procesi conjecture In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irredicible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by $$m^{1/2 dim O}$$. In this talk I shall prove an improved version of this conjecture and derive some important consequences of it related to q-W algebras. Tuesday, Apr 1st 3pm Elisa Gorla (Basel) Universal Groebner bases for ideals of maximal minors In 1993, Bernstein, Sturmfels, and Zelevinsky proved that the maximal minors of a matrix of variables form a universal Groebner basis. We present a very short proof of this result, along with a broad generalization to matrices with multi homogeneous structures. Our main tool is a rigidity statement for radical Borel fixed ideals in multigraded polynomial rings. This is joint work with A. Conca ed E. De Negri (University of Genoa). Tuesday, Mar 25th 4pm Claudio Sibilia (ETH Zürich) Chen homological connection for G- spaces. Chen homological connections are flat connections on a trivial bundle that can be build on any topological smooth manifold. The goal of the talk is to explain the relation between the Chen homological connection and the KZ equations in genus 0, and extends this formalism to compute some formal connections in genus 1 that are very close to the KZB connection. Tuesday, Mar 25th 3pm Iordan Ganev (Univ. Texas) Quantizations of multiplicative hypertoric varieties Multiplicative hypertoric varieties are symplectic analogues of toric varieties related to symplectic resolutions, hyperplane arrangements, and geometric representation theory. We construct quantizations, depending on a parameter $q$, of multiplicative hypertoric varieties using an algebra of difference operators on affine space. Furthermore, when $q$ is a root of unity, we show that the quantization acquires a large center and defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety. This is joint work with David Jordan. Tuesday, Mar 18th 3pm Ben Webster (U Virginia/Paris 6) Hodge seminar Tuesday, Mar 4th 3pm Felipe Rincon (Warwick) Positroids and the totally nonnegative Grassmannian Positroids are combinatorial objects that can be used to index the different strata in certain well-behaved decompositions of the Grassmannian and its positive part. In this talk I will present joint work with Federico Ardila and Lauren Williams, in which we study some of the combinatorial properties of positroids. As an application, we prove da Silva's 1987 conjecture that any positively oriented matroid is representable over the field of rational numbers. In particular, this implies that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. Wendesday, Feb 26th 1:10pm Adam Chapman (Belgium) Clifford Algebras of Binary Cubic Forms in Characteristic 3 his talk is based on a joint work with Jung-Miao Kuo. The structure of Clifford algebras of binary cubic forms in characteristic different from 3 has been described in full detail by Haile. Kuo has generalized his results for the algebra associated to a ternary cubic curve in characteristic different from 3. Here we shall talk about the analogous results in case of characteristic 3. Tuesday, Feb 25th 3pm Johanna Hennig (UCSD) Locally finite Lie algebras in positive characteristic For finite dimensional Lie algebras, there is the well-known Ado’s theorem: Every finite dimensional Lie algebra embeds into a finite dimensional associative algebra. Bahturin, Baranov, and Zalesski proved an infinite dimensional version of Ado’s theorem for a simple, locally finite Lie algebra L over a field of characteristic zero: L embeds into a locally finite associative algebra if and only if L is isomorphic to the commutator of skew-symmetric elements of a locally finite, associative algebra with involution. We extend this result to fields of positive characteristic—we provide two structure theorems which reduce to Bahturin, Baranov, Zalesski’s result in characteristic zero and also generalize classical structure theorems for finite dimensional Lie algebras in characteristic p. Friday, Feb 14th 3pm Erez Sheiner (Bar Ilan) Exploded Layered Tropical Algebra ELT algebra is an extension of the max-plus algebra, which is a lesser degeneration of algebraic varieties. Using the layered structure we define roots of polynomials and singularity of matrices, despite the lack of an additive inverse. I will not assume familiarity with tropical geometry. Tuesday, Feb 11th 3pm Greg Ginot (Paris 6) Factorization algebras and applications to centralizers and Bar construction. in the first part of the talk, we will discuss the notion of centralizers of En-algebras maps which shall can be thought as "higher" generalizations of Hochschild cohomology and we will explain how one can see describe them in terms of factorization algebras. Factorization algebras will be detailled in the second part of the talk. They can be thought of as a "non-abelian" version of cosheaves. We will also give an analogue of the Bar constructions for Factorization algebras. Tuesday, Feb 4th 3pm Sian Fryer (Manchester) The q-Division Ring and its Fixed Rings The q-division ring (denoted here by D) is one of the easiest examples of noncommutative infinite dimensional division rings to define, but answering even fairly basic questions concerning the structure of its automorphism group or its sub-division rings of finite index is still quite difficult. The second question in particular is of interest due to its connections with Artin's conjectured classification of surfaces in non-commutative algebraic geometry. I will describe the structure of the fixed rings of D for a certain class of finite groups and use this to construct some rather unexpected examples of homomorphisms on D, including a conjugation automorphism which is not inner and a conjugation homomorphism which is not even bijective. Tuesday, Jan 28th 4pm Michael Wemyss (Edinburgh) Noncommutative deformations and applications I will mainly talk on my joint work with Will Donovan (1309.0698) which gives new invariants to any contractible curve using noncommutative deformation theory. I will explain a little about our motivation, and why studying noncommutative deformations is strictly necessary. In the setting of 3-fold flops, we obtain many new examples of finite dimensional self-injective algebras, and I will give some of their properties. If I have time, I will say a bit about how this helps to run the minimal model program in dimension three. Tuesday, Jan 21st 3pm Jean-Marie Bois (Kiel) Weyl groups for restricted lie algebras, and the Chevalley restriction theorem Tuesday, Jan 7th 3pm David Gepner (Regensburg) Brauer groups of commutative ring spectra We define and study Azumaya algebras over a commutative (or E_\infty) ring spectrum R, and study the homotopy groups of the resulting Brauer space (of Azumaya R-algebras up to Morita equivalence). Our main technical tool is an etale-local triviality result for Azumaya algebras over connective derived schemes (in the sense of J. Lurie, and inspired by a similar result of B. Toen) and a local-to-global result for the compact generation of R-linear \infty-categories with descent. The result is a spectral sequence for the homotopy groups of the Brauer space whose E_2 term consists entirely of ordinary etale cohomology groups, and which is often entirely computable. This is joint work with Ben Antieau. December 6th, 3pm @ICMS Joint w/ Heriot-Watt Anne Thomas (Glasgow) Quasi-isometry of right-angled Coxeter groups A group G with a finite generating set S can be considered as a metric space by endowing it with the word metric with respect to S. Up to quasi-isometry, this metric on G does not depend on S. A major theme in geometric group theory is to classify all finitely generated groups up to quasi-isometry. We investigate the quasi-isometric classification of right-angled Coxeter groups using divergence of geodesics and topological features of their boundaries at infinity. This is joint work with Pallavi Dani. December 6th, 4pm @ICMS Joint w/ Heriot-Watt Ghislain Fourier (Glasgow) Fusion products and symmetric functions Fusion products for current algebras have been introduced fifteen years ago. Roughly speaking, they make use of the natural grading of the polynomial ring to form graded tensor products of simple modules for a simple complex Lie algebra. These fusion products play a crucial role in the study of finite-dimensional modules for current or loop algebras, for instance they recover Weyl modules and Demazure modules to name but a few. Although intensively studied, various fundamental questions are not answered yet, for example about defining relations or graded character formulas. The current state of art and as well as the strong connection to conjectures about Schur positivity of symmetric functions (and recent results here) will be presented. November 26, 3pm Emily Dufresne (Basel) Separating invariants and local cohomology The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits of the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. (joint with Jack Jeffries) November 26, 4pm Ines Henriques (Sheffield) F-thresholds and Test ideals for determinantal ideals of maximal minors. Test ideals first appeared in the theory of tight closure, and reflect the singularities a ring of positive characteristic. Motivated by multiplier ideals in characteristic zero, N. Hara and K. Yoshida defined (generalized) test ideals as their characteristic p analogue. Whereas multiplier ideals are defined geometrically, using log resolutions, test ideals are defined algebraically using the Frobenius morphism. The test ideals of an ideal I form a non-increasing, right continuous family, {τ(c . I)}, parametrized by a positive real parameter c. The points of discontinuity in this parametrization, are called F-thresholds of I and form a discrete subset of the rational numbers (Blickle-Mustaţă-Smith, Hara, Takagi-Takahashi, Schwede-Takagi, Katzman-Lyubeznik-Zhang). We consider ideals generated by maximal minors of a matrix of indeterminates, in its polynomial ring over a field of positive characteristic. Using an algebraic approach, we identify their F-thresholds and test ideals. November 19, 3pm Tom Lenagan (Edinburgh) Totally nonnegative matrices A real matrix is totally nonnegative if each of its minors is nonnegative, and is totally positive if each minor is greater than zero. We will outline connections between the theory of total nonnegativity and the torus invariant prime spectrum of the algebra of quantum matrices, and will discuss some new and old results about total nonnegativity which may be obtained using methods derived from quantum matrix methods. Most of the material is joint work with Stephane Launois and Ken Goodearl. NOTE: much of the material to be presented has been included in earlier seminars in Edinburgh. November 13 (Wednesday!), 4pm David Alan Jordan (Sheffield) Connected quantized Weyl algebras Connected quantized Weyl algebras are algebras with a PBW basis in which any two generators either q-commute or satisfy a quantum Weyl relation xy-qyx=1-q. The connectedness condition ensures, in some sense, enough quantum Weyl relations. The talk will begin with some discussion of how the motivating examples arose from work of Fordy and Marsh on periodic quiver mutation and Poisson algebras and then proceed to a classification of the connected quantized Weyl algebras and the determination of their prime ideals. October 25 (Friday!), 4-5pm Chris Dodd (Toronto) Cycles of Algebraic D-modules in positive characteristic I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle. October 28 (Monday!), 3pm David Evans (Cardiff) The search for the exotic - subfactors and conformal field theory Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories. I will describe work with Terry Gannon on the search for exotic theories beyond those from symmetries based on loop groups, Wess-Zumino-Witten models and finite groups. October 23 2-3pm, 4-5pm, 24th 4-5pm, (WTh!) Chris Dodd (Toronto) Quantizations of Conic Symplectic Varieties and Representation theory I will describe some recent progress- joint with G. Bellamy, K. McGerty, and T. Nevins- in understanding modules over algebraic quantizations of certain nice classes of symplectic varieties. In particular, I will explain in a leisurely fashion how quantizations come up in representation theory, how one can use the presence of a torus action to study modules over these quantizations, and the types of results (geometric and representation-theoretic) that come from this way of thinking. October 22, 3pm Nick Gurski (Sheffield) Categorical operads Operads are a convenient tool for encoding certain kinds of algebraic structures, and they are in heavy use in algebraic topology and homological algebra. There are some special features of operads in Cat, the (2-)category of categories, as well as a number of features shared with operads in other categories. I will review the basics of the general theory, and then talk about a couple of things special to the case of Cat. In the second hour, I plan to focus on how equivariance plays an important role in this story. October 15, 3pm Joe Chuang (City University London) Algebra with surfaces Frobenius algebras give rise to topological invariants of surfaces. I will review this idea (two-dimensional topological ﬁeld theory) and describe joint work with Andrey Lazarev on a similar construction. October 8, 3pm Uzi Vishne (Bar Ilan) p-central elements and subspaces in central simple algebras The major open question on central simple algebras is the cyclicity problem: are all algebras of prime degree cyclic? Any cyclic algebra of degree p has p-central elements: non-central elements whose p-power is central. The cyclicity problem can thus be studied using subspaces of p-central elements, which will be the main topic of the lecture. We will discuss such subspaces from various points of views: chain lemmas, the symbol length problem, connections with elementary number theory, and a new construction in nonassociative algebra. October 8, 4pm Be'eri Greenfeld (Bar Ilan) Unions over chains of prime ideals In a commutative ring, the union over a chain of prime ideals is prime. This is not true in the general case: there exist counterexamples from many classes, including nil rings, locally finite rings, affine algebras with polynomial growth and rings satisfying a polynomial identity (PI). In the latter case, though, the maximal number of non-prime unions of subchains of a chain of prime ideals is (tightly) bounded by the PI-class ,hence finite; this is far from being true in general. In this talk we discuss several examples with additional properties (e.g. primitivity), positive results and suggestions for further research directions. This is based on joint work with Louis Rowen and Uzi Vishne. October 1, 3pm David Andrew Jordan (Edinburgh) Quantum differential operators and the torus $$T^2$$ Abstract: The algebra $$D_q(G)$$ is a $$q$$-deformation of the algebra $$D(G)$$ of differential operators on a semi-simple algebraic group. In this talk, I will explain an intimate relationship between $$D_q(G)$$ and the torus $$T^2$$: namely, $$D_q(G)$$ carries an action by algebra automorphisms of the torus mapping class group $$SL_2(\mathbb{Z})$$, and also yields representations of the torus braid group extending the well-known action of the planar braid group on tensor powers of quantum group representations. Finally, the so-called Hamiltonian reduction of $$D_q(G)$$ quantizes the moduli space $$Loc_G(T^2)$$ of $$G$$-local systems on $$T^2$$, or equivalently, homomorphisms $$\pi_1(T^2)\to G$$,and this observation allows us to generalize the construction of $$D_q(G)$$ to quantize $$Loc_G(\Sigma_{g,r})$$, for an arbitrary surface with genus g and r punctures. Time permitting, I will outline work in progress with David Ben-Zvi and Adrien Brochier putting all of the above into the context of topological field theories. October 1, 4pm Natalia Iyudu (Edinburgh) A proof of the Kontsevich conjecture on noncommutative birational transformations I will talk about our recent proof (arXiv1305.1965) of the Kontsevich conjecture dated back at 1996, and mentioned at the 2011 Arbeitstagung talk on 'Noncommutative identities' (arXiv1109.2469). This conjecture says that certain transformations given by matrices over free noncommutative algebra with inverses ('free field' due to P.Cohn) are periodic, on the level of orbits of the left/right diagonal action. Namely, let $$(M_{ij})_{1 \leq i,j \leq 3}$$ be a matrix, whose entries are independent noncommutative variables. Let us consider three 'birational involutions' $$I_1: \,\, M \to M^{-1}$$ $$I_2: \,\, M_{ij} \to (M_{ij})^{-1}, \,\, \forall i,j$$ $$I_3: \,\, M \to M^t$$ Then the composition $$\Phi = I_1 \circ I_2 \circ I_3$$ has order three. September 24, 3pm Gwyn Bellamy (Glasgow) Generalizing Kashiwara's equivalence to conic quantized symplectic manifolds Abstract: Kashiwara's equivalence, saying that the category of D-modules on a variety X support on a smooth, closed subvariety Y is equivalent to the category of D-modules on Y, is a key result in the theory of D-modules. In this talk I will explain how one can generalize Kashiwara's result to modules for deformation-quantization algebras on a conic symplectic manifold. As an illustrative application, one can use this result to calculate the additive invariance such as the K-theory and Hochschild homology of these module categories. This is based on joint work C. Dodd, K. McGerty and T. Nevins. September 17, 3pm Stefan Kolb (Newcastle) Radial part calculations for affine sl2 Abstract: In their seminal work in the 1970s Olshanetsky and Perelomov used radial part calculations for symmetric spaces to prove integrability of the Calogero-Moser Hamiltonian for special parameters. In this talk, restricting to affine sl2, we will explore what happens if one extends their argument to Kac-Moody algebras. I will try to explain how this leads to a blend of the KZB-heat equation and Inozemtsev's extension of the elliptic Calogero-Moser Hamiltonian. September 9 (Monday!), 5pm Francois Petit (Edinburgh) Fourier-Mukai transform in the quantized setting Abstract: After reviewing some elements of the theory of Deformation Quantization modules (DQ-modules), I will show that a coherent DQ-kernel induces an equivalence between the derived categories of coherent DQ-modules if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent O-modules. September 9 (Monday!), 4pm Hendrik Suess (Edinburgh) Equivariant vector bundles on T-Varieties Abstract: By Klyachko's work there is an equivalence of categories between equivariant vector bundles on toric varieties and families of vector space filtrations. In this talk I will discuss an generalization of this equivalence to bundles on varieties with smaller torus actions. Now, vector space filtrations are replaced by filtrations of vector bundles on some quotient space. This description comes with a nice splitting criterion and allows to prove that vector bundles of low rank on projective space, which are equivariant with respect to special subtori of the maximal acting torus must split. September 4th (Wednesday!), 3pm Qendrim Gashi (Pristina) Mazur's inequality, its converse and generalizations Abstract: We will study the classical version of Mazur's inequality, comparing Newton and Hodge polygons, and a converse thereof (due to Kottwitz and Rapoport). We will then discuss group-theoretic generalizations of these results and implications for affine Deligne-Lusztig varieties. We will conclude with some results from root theory and toric geometry.