Past Semesters

Spring 2015

Move your mousepointer on the title of a talk to see an abstract (if available). The schedule is also kept up to date in a google calendar, which you can find below.

January 15 Michael Wemyss (Edinburgh) Aspects of the Homological MMP

I will outline the main ideas of arXiv:1411.7189, which uses noncommutative deformations and universal properties to jump between minimal models in the MMP in a satisfyingly algorithmic fashion. As part of this, a flop is constructed not by changing GIT, but instead by changing the algebra keeping GIT fixed, and flops are detected by whether certain contraction algebras are finite dimensional. Carrying this extra information allows us to continue to flop, and thus continue the MMP, without having to calculate everything at each stage.

Proving things in this canonical categorical manner allows us then to say things about GIT. In fact the HomMMP computes the full wall and chamber structure, and also gives a method for determining which walls produce flops and which do not. If there is time, I will explain that it also can be used to prove that flop functors braid in dimension three, however the combinatorics are not the expected one, and higher length braid relations naturally appear.

February 5 Mario García Fernández (ICMAT) Stability data, irregular connections and tropical curvesI will give an overview of recent joint work with S. Filippini and J. Stoppa, in which we construct isomonodromic families of irregular meromorphic connections on P1, with generalized monodromy in the automorphisms of a class of infinite-dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R → 0 “conformal limit” we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). In the R → ∞ “large complex structure limit” the families relate to tropical curves in the plane and tropical/GW invariants. The connections we construct are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto–Moore–Neitzke.
February 12 Alice Rizzardo (Edinburgh) An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves
February 26 Sergey Arkhipov (Aarhus) Quasi-coherent Hecke categories and affine braid group actions

We propose a geometric setting leading to categorical braid group actions. First we consider the quasi-coherent Hecke category QCHecke(G,B) for a reductive group G with a Borel subgroup B. We show that a monoidal action of QCHecke(G,B) on a triangulated category gives rise to a categorification of degenerate Hecke algebra representation known as Demazure Descent Data. Next we replace the group G by the derived group scheme LG of topological loops with values in G and consider QCHecke(LG,LB). A monoidal action of the category QCHecke(LG,LB) gives rise to a categorical action of the affine Braid group.
Trying to avoid heavy derived algebraic geometry methods, we present an example of the construction above in classical algebro-geometric terms. For a G-variety X, we construct a Braid group action on a category of equivariant matrix factorizations on the product of T∗X and the Grothendieck variety for the Lie algebra of G. The potential for matrix factorizations is provided by the moment map.

March 5 No EDGE seminar (GLEN in Glasgow)
March 12 Michael Groechenig (Imperial) Infinite-dimensional vector bundles and reciprocity
March 19 No seminar
March 26 (two talks!) Joseph Karmazyn (Edinburgh) Moduli, McKay, and Minimal Models

Moduli spaces are often used to realise derived equivalences in algebraic geometry. I will recall the examples of the derived equivalence of 3-fold flops in the minimal model program and the derived SL2 McKay correspondence.

These derived equivalences can be translated to noncommutative algebra where they have been extended to include more general settings. I will discuss how a moduli interpretation can be extended to include these derived equivalences with noncommutative algebras.

Rebecca Tramel (Edinburgh) Bridgeland stability on surfaces with curves of negative self-intersection

I consider X a smooth projective complex surface containing a curve C whose self-intersection is negative. In 2002, Bridgeland defined a notion of stability for the objects in Db(X), which generalized the notion of slope stability for vector bundles on curves. The space of such stability conditions is a complex manifold, Stab(X). If we fix a numerical class, then we can decompose Stab(X) into open chambers where the moduli space of stable objects of this class remains constant, and codimension one walls where this moduli space may change.
I consider objects in Db(X) whose numerical class is that of the skyscraper sheaf of a point. In the geometric chamber of Stab(X), the moduli space of stable objects of this class is X itself. In 2012 Toda showed that if C has self-intersection -1, there is a wall to the geometric chamber along which the points of C become semistable. I generalize this result to find such a wall when C has arbitrary negative self-intersection. I then describe the moduli space of stable objects after wall-crossing.

April 23 Tyler Kelly (Cambridge) Equivalences of (Stacky) Calabi-Yaus in Toric VarietiesGiven Calabi-Yau complete intersections in a fixed toric variety, there are possibly various constructions to compute its mirror. Sometimes these mirrors are isomorphic but sometimes not. Mirror symmetry predicts a relationship amongst these so-called double mirrors. In this talk, we will show that the stacky versions of these varieties are derived equivalent. In the proving of this theorem, we get some applications which involve polarisations of K3 surfaces, special degenerate families of CY hypersurfaces in toric varieties, and a generalization of the BHK mirror constructions to families.
May 28, JCMB 6206 Daniel Halpern-Leistner (Columbia) Equivariant Hodge theoryRecent results have revealed a mysterious foundational phenomenon: some quotient stacks for algebraic groups G acting on non-proper schemes X still behave as if they are proper schemes. I will report on one instance of this yoga: one can consider the non-commutative Hodge-to-de-Rham sequence, from Hochschild homology to periodic cyclic homology, for the derived category of coherent sheaves. This spectral sequence degenerates on the first page for smooth and proper schemes, and it turns out that this degeneration also occurs for many "cohomologically proper" quotient stacks. With a little work, this leads to a canonical weight 0 Hodge structure on the Atiyah-Segal equivariant K-theory of the complex analytification of X. The associated graded of the Hodge filtration is the space of functions on the "derived loop space" of the stack.

Autumn 2014

September 18 No Seminar
September 25 Ivan Cheltsov (Edinburgh) What are the worst singular points of plane curves of given degree?
October 2 Martin Kalck (Edinburgh) Relative singularity categories(Relative) singularity categories are triangulated categories associated with (non-commutative resolutions of) singular varieties. I will explain these notions and their mutual relations focusing on the simplest examples - the singularities of type A_1, e.g. k[x|/x^2. For these examples, everything can be understood in a rather elementary way. In particular, familiarity with triangulated categories will NOT be necessary to follow the talk. In the end, I will mention what we know for ADE-singularities in general. This is based on joint work with Dong Yang.
October 9 Brent Pym (Oxford) Quantum deformations of projective three-spaceIn noncommutative projective geometry, quantum versions of projective space are often described in terms of their homogeneous coordinate rings, which are noncommutative analogues of polynomial rings. The algebras corresponding to quantum projective planes were classified in geometric terms by Artin, Tate and Van den Bergh in a celebrated 1990 paper. The related problem for projective three-space has received considerable attention, but the full classification remains elusive. I will describe some recent progress on this problem, in which deformation quantization is combined with Cerveau and Lins Neto's classification of foliations on projective space to give a classification of the flat deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.
October 16 Balazs Szendroi (Oxford) Motivic Donaldson-Thomas series of deformed Calabi-Yau geometriesMotivic DT theory gives a refined count of objects in 3-CY categories, for example sheaves on Calabi-Yau threefolds. At least in the local quiver setting, it is easy to ask about motivic counts of objects in categories defined by deformed Calabi-Yau spaces, such as quantum three-space or the affine cone over the Jordan plane. We compute the answer in some cases, and conjecture it in some others, based on an intriguing and sofar not always precise formula involving the motivic count of simple objects only. (Joint work with Andrew Morrison and Brent Pym)
October 23 Evgeny Smirnov (HSE Moscow) Spherical double flag varieties

Classical Schubert calculus deals with orbits of a Borel subgroup in GL(V) acting on a Grassmann variety Gr(k,V) of k-planes in a finite-dimensional vector space V. These orbits (Schubert cells) and their closures (Schubert varieties) are very well studied both from the combinatorial and the geometric points of view.

One can go one step farther, considering the direct product of two Grassmannians Gr(k,V)x Gr(l,V) and the Borel subgroup in GL(V) acting diagonally on this variety. In this case, the number of orbits still remains finite, but their combinatorics and geometry of their closures become much more involved. However, something still can be said about them. I will explain how to index the closures of a Borel subgroup in Gr(k,V)xGr(l,V) combinatorially and construct their desingularizations, which are similar to Bott-Samelson desingularizations for ordinary Schubert varieties. I will also mention the analogues of these results for direct products of partial flag varieties for reductive groups of type different from A_n.

October 30 Chunyi Li (Edinburgh) Topics on the Hilbert scheme of points on projective planeThe Hilbert scheme of n points in P2 parameterizes all 0-dimensional subschemes on P2 with length n. It is a smooth version of n-th symmetric product of P2. I will introduce some of its 'neighborhoods' in the area of birational geometry and deformation theory.
November 6, JCMB 5215 Gavin Brown (Loughborough) Elliptic Gorenstein projection following Coughlan

The archetype elliptic Gorenstein singularity on a 3-fold
is the vertex of the affine cone over a K3 surface.
In favourable circumstances, within a family of 3-folds
of general type, one may be able to degenerate to
a 3-fold with an elliptic Gorenstein singularity, and then
project away from this point to a 3-fold of general type
in a smaller ambient projective space.
Used in reverse, this resembles Fano's famous strategy for
constructing Fano 3-folds whose anticanonical embedding is
in high codimension: construct the image of some projection
and then undo the projection. We try the same with canonical
3-folds (embedded by K_X = O(+1), rather than O(-1) for Fanos)
and projection from elliptic Gorenstein points.
As ever, the more subtle phenomena begin to arise
in codimension 4, when we try to write X in wP^7.
This is joint with Stephen Coughlan (Hannover).

November 13 Susan J. Sierra (Edinburgh) Moduli of point representations, the enveloping algebra of the Virasoro algebra, and noetherianityI will explain the geometric methods underlying my 2013 proof (with Walton) that the universal enveloping algebra of the Virasoro algebra is not noetherian. This was a problem that had been open for more than 20 years and had resisted many attempts at an algebraic proof.
November 20 Georg Oberdieck (Zurich) Curve counting in K3 x P1 and the Hilbert scheme of points of a K3 surfaceLet S be a K3 surface. Recent results suggest a correspondence between the Gromov-Witten and Donaldson-Thomas theory of S x P1 and the genus 0 GW theory of the Hilbert scheme of points of S. I will explain these results and how they lead to precise conjectures about the correspondence. As an application, I will discuss a conjecture for the full GW theory of the Calabi Yau threefold S x E, where E is an elliptic curve. This is joint work with R. Pandharipande.
November 27 No seminar
December 11, 3pm, JCMB 6206 Cristina Manolache (Imperial) Mirror symmetry without localizationMirror Symmetry predicts a surprising relationship between the number of curves in a target space X and certain hypergeometric functions. Existing proofs rely on complicated localisation computations. I will describe a new, more conceptual proof.

Spring 2014

January 16, JCMB 4312 Francesco Sala (Heriot-Watt) Framed sheaves on root stacks and gauge theory on ALE spacesThis talk is about a new approach to the study of supersymmetric gauge theories on ALE spaces of type $A_{k-1}$, for $k\geq 2$, by using the theory of framed sheaves on 2-dimensional root toric stacks. In particular, I will describe a ``stacky compactification" of a minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_{k}$ and moduli spaces of framed sheaves on it. These moduli spaces provide a new setting for the study of gauge theories on ALE spaces and can be used to define new geometric realizations of representations of affine/vertex algebras. In the last part of the talk I will focus on the case of rank one framed sheaves: in that case I will describe a geometric construction of a highest weight representation of $\widehat{\mathfrak{sl}}(k)$ at level one and will characterize some Carlsson-Okounkov type vertex operators on $\widehat{\mathfrak{sl}}(k)$ that have a gauge-theoretic meaning.
January 23, 4 pm, JCMB 5326 Diane Maclagan (Warwick) Tropical schemes, tropical cycles, and valuated matroidsThe tropicalization of a subvariety of a torus records the cycle of its compactification in an ambient toric variety. In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure. I will introduce these notions, and outline how to also recover the tropical cycle from this information. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon
January 30 Alastair King (Bath) Grassmannian cluster categories and dimers on a disc
February 6 Pierre Schapira (Paris VI) Microlocal Euler classes and Hochschild homology

This is a joint work with Masaki Kashiwara.
On a complex manifold $(X;O_X)$, the Hochschild homology is a powerful tool to construct characteristic classes of coherent modules and to get index theorems. Here, I will show how to adapt this formalism to a wide class of sheaves on a real manifold M by using the functor uhom of microlocalization. Hence, the analogue of the Hochschild homology lives now in T*M, the cotangent bundle. It is isomorphic to the inverse image by $\pi_M: T*M \to M$ of $\omega_M$, the topological dualizing complex on M. This construction applies in particular to constructible sheaves on real manifolds and D-modules on complex manifolds, or more generally to elliptic pairs.

February 13 Adam Boocher Closures of a linear spaceLet L be an affine linear space. Once we fix coordinates, it makes sense to discuss the closure of L inside a product of projective lines. In this talk I'll present joint work with Federico Ardila concerning the defining ideal of the closure. It turns out that the combinatorics of this ideal are completely determined by a matroid associated to L and we are able to explicitly compute its degree, universal Groebner basis, Betti numbers, and initial ideals. I'll include several examples along the way and discuss how this closure operation comes up naturally when one searches for ideals with "nice" behavior upon degeneration.
March 6 Tara Holm (Cornell) The topology of toric origami manifoldsA folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will describe how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.
March 13 Paolo Stellari (Milano) Fourier-Mukai functors: derived vs dg categoriesFourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless a basic question is still open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? We discuss the recent advances in the subject and study the same question in the context of dg categories where the problem has been settled by B. Toen. In this talk we propose a simpler approach not based on the notion of model category. This is a joint work with A. Canonaco.
March 20, 4pm, JCMB 4312 Vladimir Baranovsky (UC Irvine) Deformation quantization of smooth Lagrangian subvarietyWe consider a smooth algebraic variety X with an algebraic symplectic form and a deformation O_h (with a formal parameter h) of the structure sheaf compatible with this form. For a smooth Lagrangian subvariety Y in X and a line bundle L on Y, we describe when L also admits a deformation to a module over O_h. Joint work with V. Ginzburg, D. Kaledin and J. Pecharich.
March 21 3 pm, Faculty Room North, David Hume Tower, George Square Hodge Seminar Etienne Ghys (Lyon) Some remarks on singularities of real analytic curves in the planeThe topological nature of singularities of planar complex analytic curves has been understood for many years. Indeed, Newton-Puiseux series allow a complete understanding of the link of the singularity. Amazingly, the analogous question in the real domain leads to interesting combinatorial developments. I will start my discussion with a clever remark of M. Kontsevich.
March 27, 4 pm, JCMB 5325 Nicolò Sibilla Ribbon graphs, skeleta and homological mirror symmetryIn this talk I will review recent work of mine, partially in collaboration with H. Ruddat, D. Treumann and E. Zaslow, which centers on various aspects of Kontsevich's Homological Mirror Symmetry in the large complex limit. The one-dimensional case will be emphasized, as a convenient testing ground for more general constructions.
April 3, JCMB 5325, 3pm-5pm Klaus Altmann Update on deformations of toric varieties

We give a summary about methods and known results in the deformation theory of affine toric varieties. The convex geometric counterpart of this is the decomposition of polyhedra into Minkowski summands.
However, the presence of singularities in codimension two or the combination of different multidegrees require new tools. This leads to the concepts of so-called thickenings of morphisms - meaning deformations "without base space".
In particular, we return to one of the original tasks when dealing with toric geometry: The understanding of algebraic invariants of toric objects in terms of their combinatorics.

Wednesday, May 21, 11.30-12.30, JCMB 5205 Sam Payne (Yale) Nonarchimedean methods for multiplication maps

Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the homogeneous coordinate ring of a projective variety. I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry.

This work is parallel in many ways to the limit linear series of Eisenbud and Harris. One key difference is that we focus on degenerations in which the special fiber is not of compact type. In this context, the tropical Riemann-Roch theory of Baker and Norine and Baker’s specialization lemma are starting points for sometimes intricate calculations in component groups of Neron models and on skeletons of Berkovich curves.

Tuesday, May 27, 2pm-3pm, JCMB 5215 Jason Lo (MPI Bonn) Fourier-Mukai transforms on elliptic fibrationsIn their 2002 JAG paper, Bridgeland-Maciocia laid out a few ideas for computing Fourier-Mukai transforms for sheaves on elliptic threefolds, which allowed them to construct an isomorphism between two moduli spaces of stable sheaves. I will explain how their ideas can be developed into a toolchest for computing Fourier-Mukai transforms on elliptic fibrations. On elliptic fibrations of any dimension, this toolchest can be used to construct an open immersion from a moduli of polynomial stable complexes to a moduli of stable sheaves, as well as a 1-1 correspondence between line bundles and spectral sheaves.
Tuesday, May 27, 3pm-4pm, JCMB 5215 Ziyu Zhang (Bath) Birational geometry of singular moduli spaces of O'Grady typeWe consider 10 dimensional singular moduli spaces of semistable objects on a projective K3 surface with respect to generic Bridgeland stability conditions. We will show that they admit symplectic resolutions, which are all deformation equivalent to the exceptional 10 dimensional holomorphic symplectic manifold constructed by O'Grady. Moreover, many properties of smooth holomorphic symplectic manifolds still hold for these singular moduli spaces. By generalizing work of Bayer and Macri, we can relate wall crossing on the stability manifold of the K3 surface to birational transformations of these singular moduli spaces. This is a joint work with C.Meachan.
Thursday, June 12, 2pm, JCMB 6311 Nathan Ilten (Berkeley) Vanishing of Higher Cotangent Cohomology and Applications
Monday, June 16, 3pm, JCMB 6206 Dave Anderson (IMPA) Schubert calculus and the Satake correspondenceRecent work of Laksov, Thorup, Gatto, and Santiago has given a perspective in which Schubert calculus on Grassmannians can be realized via certain operators on exterior algebras. This point of view also applies to equivariant and quantum versions of Schubert calculus. In this mostly expository talk, I will explain how their work fits into a more general framework via the geometric Satake correspondence.
Wednesday, June 18, 3pm, JCMB 5215 Pablo Solis (Berkeley) Degenerations of the Moduli space of G bundles on a curveThis talk is about three things: infinite dimensional Lie groups, representation theory and moduli problems. The group in question is called the loop group; it is the space of maps from a circle into a topological group. The loop group has a class of representations -positive energy representations- that generalize the highest weight representations of semisimple groups over the complex numbers. These representations allow one to construct an infinite dimensional space that one can relate to the finite dimensional moduli problem of parametrizing G bundles on a compact Riemann surface. I'll explain this setup and show how it can be used to compactify this moduli space of G bundles when the Riemann surface is allowed to develop singularities. Over the last 30 years other compactifications have been presented by Caparoso, Gieseker, Pandharipande, Seshadri and Nagaraj but this is the first construction that provides a compactification not just for vector bundles but for principal G bundles for an arbitrary simple group over the complex numbers.
Tuesday, July 29, 2pm Emanuele Macri (Ohio State) Stability conditions on abelian threefolds

I will present a new proof of a result by Maciocia and Piyaratne on the existence of Bridgeland stability conditions on abelian threefolds.
As in their work, we prove a Bogomolov-type inequality involving Chern classes of certain stable objects in the derived category. Our approach uses the multiplication maps, and has the advantage of applying to abelian threefolds with arbitrary Picard number.
As a corollary, we can show the existence of Bridgeland stability conditions on all the orbifolds quotients of abelian threefolds, including Calabi-Yau threefolds of abelian-type, and those constructed by Borcea-Voisin.
This is based on joint work in progress with Arend Bayer and Paolo Stellari.

Autumn 2013

September 9 (Monday!), 4pm Hendrik Süß (Edinburgh) Equivariant vector bundles on T-VarietiesBy 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. Notes
September 9 (Monday!), 5pm Francois Petit (Edinburgh) Fourier-Mukai transform in the quantized settingAfter 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. Notes
September 19, 3pm JCMB 5215 Evgeny Shinder (Edinburgh)Exceptional collections on fake projective planesFake projective planes are surfaces of general type with the same rational cohomology as a projective plane CP^2. These surfaces appear as quotients of a complex 2-ball by an arithmetic subgroup of PU(2,1) and are completely classified into finitely many isomorphism classes. We study the derived category of coherent sheaves on fake projective planes. For fake projective planes with an automorphism group of order 21 we show that O, O(-1), O(-2) form an exceptional collection in the derived category. The orthogonal complement to this collection provides an example of a quasi-phantom (a category with vanishing Hochschild homology). Notes
September 19, 4pm JCMB 5215 Ciaran Meachan (Edinburgh) Derived autoequivalences of hyperkähler varietiesP functors are a natural generalisation of Huybrechts and Thomas' P objects and as such, they determine autoequivalences of the codomain category. We will report on the latest developments in this direction when the codomain category is the derived category of coherent sheaves on a hyperkähler variety. Notes
September 26, JCMB 5326 Costya Shramov (Steklov) Finite groups of birational automorphismsGiven a variety X over some field K, one may wonder what are the restrictions on finite groups that act on X by birational automorphisms. In a joint work with Yu. Prokhorov we have recently showed (modulo some standard conjectures of birational geometry) that for a given X such groups are always bounded if K is finitely generated over Q. I will speak about this result and relevant results over algebraically closed fields.
October 1, 3pm, joint with MAXIMALS David Andrew Jordan (Edinburgh) Quantum differential operators and the torus \( T^2 \)
October 1, 4pm, joint with MAXIMALS Natlia Iyudu (Edinburgh) A proof of the Kontsevich conjecture on noncommutative birational transformations
October 3 GLEN in Liverpool
October 4, 9:30 am - 6:00 pm Maxwell colloquium on Combinatorial Algebraic Geometry
October 10 Burt Totaro (Cambridge/UCLA) The integral Hodge conjecture for 3-folds

Hodge originally conjectured that every Hodge class
in the integral cohomology of a smooth complex projective variety
is algebraic, meaning the class of a Z-linear combination
of subvarieties. This is false in general. We discuss recent
positive and negative results on the integral Hodge conjecture
for 3-folds.

October 17 Yuri Fedorov (UPC Barcelona) Prym varieties in integrable systems: their algebraic description and separation of variablesThe most powerful method of explicit solving algebraic completely integrable systems is their Lax representation. The complex invariant tori of the systems are known to be Jacobian varieties of the corresponding spectral curves or, in most cases, Abelian subvarieties of the Jacobians, called Prym varieties. Then, if one wants to make a separation of variables for the system, it is nesesary to relate the Prym variety with an algebraic curve. I will show how to do this this in several cases by using the results of D. Mumford, A. Dalaljan, V. Enolski, P. van Moerbeke, A. Levin, and F. Koetter.
October 24 Jihun Park (Pohang) Fano threefold hypersurfacesIn 1979 Reid discovered the 95 families of K3 surfaces in three dimensional weighted projective spaces. After this, Fletcher, who was a Ph.D. student of Ried, discovered the 95 families of weighted Fano threefold hypersurfaces in his Ph.D. dissertation in 1988. These are quasi-smooth hypersurfaces of degrees d with only terminal singularities in weighted projective spaces P(1,a1,a2,a3,a4), where d =a1+a2+a3+a4. The 95 families are determined by the quadruples of non-decreasing positive integers (a1, a2, a3, a4). All Reid’s 95 families of K3 surfaces arises as anticanonical divisors in Fletcher’s 95 families of Fano threefolds.
October 31 Vasily Golyshev (IITP Moscow) \(\Gamma\) class and \(\Gamma\) conjectures for Fano varietiesWe state Gamma conjectures I and II, explain why these may be viewed as a quantum refinement of Riemann-Roch-Hirzebruch, and prove Gamma I for a particular Fano threefold, V_12, by relating it to the equation Apery had used back in 1979 in his proof of irrationality of zeta(3) (joint work with Don Zagier).
November 7 Gregory Sankaran (Bath)Stable homology of toroidal compactificationsI will describe work still in progress with J. Giansiracusa (Swansea) in which we aim to show that the homology of the matroidal partial compactification of the moduli space of abelian g-folds stabilises in small degree. Similar but not identical results have recently been obtained, independently and by entirely different methods, by Grushevsky, Hulek and Tommasi.
November 14 Johan Martens Conformal Blocks and Kummer SurfacesOriginating in statistical mechanics, bundles of conformal blocks have in the last few decades increasingly been found to be useful objects by geometers and topologists. I will sketch a brief overview of some such developments, all of which are concerned with large level asymptotics. I will then try to convince the audience that also sporadic low-level behaviour is interesting, by focusing on (WZW) conformal blocks for SU(2) at level 4, in particular on genus 2 curves. This is ongoing joint work with T. Baier and M. Bolognesi.
November 21, 3.10pm Will Donovan The Pfaffian-Grassmannian correspondence, via Landau-Ginzburg B-modelsIf two Calabi-Yau threefolds are birationally equivalent, then there is an induced Fourier-Mukai equivalence of their derived categories by a theorem of Bridgeland. The converse is not true however, as there exist pairs of Calabi-Yau threefolds which can be proved not to be birational, but are nevertheless derived equivalent. Intriguing examples were produced by Borisov and Caldararu in 2006. We give a new proof of the associated Pfaffian-Grassmannian equivalences, using the technology of Landau-Ginzburg B-models. Our proof uses ideas from a physical construction of Hori-Tong. It shows that even though these equivalences relate non-birational varieties, they are a consequence of relations between certain Landau-Ginzburg models which are birational. This is joint work with Nicolas Addington and Ed Segal.
November 21, 4.10pm Jonathan Pridham Tannaka duality for dg categories and motivesTannaka duality originally looked at reconstructing compact Lie groups or linear algebraic groups from their categories of representations, and characterised such categories. Joyal and Street generalised this to recover a coalgebra from its finite-dimensional comodules. I will explain how to extend this to dg coalgebras and dg categories, via derived Morita theory. This has consequences for associating homotopy groups to cohomology theories, and in particular for motives of algebraic varieties.
December 16, 3pm, JCMB 4312 Miles Reid Resolution of threefold singularities and quiver representations

Spring 2013

June 05, 2013 Alastair Craw Geometric Reids recipe for dimer modles. mp4
April 04, 2013 Michael Thaddeus A tale of two compactifications. mp4
March 15, 2013 Daniel Greb Compact moduli spaces for slope semistable sheaves. mp4
March 14, 2013 Ugo Bruzzo Noether lefschetz theory for hypersurfaces in toric 3 folds. mp4
March 12, 2013 Emma Praviato The cavalcade of poncelet s theorem. mp4
March 07, 2013 John Ottem Ample subschemes and partially ample line bundles
February 28, 2013 Tom Bridgeland Cluster varieties and stability conditions. mp4
February 07, 2013 Victor Lozovanu An extension of kodaira vanishing in arbitrary codimension mp4

Autumn 2012

November 29, 2012 Timothy Logvinenko Spherical dg-functors mp4 D
November 27, 2012 Yuji Odaka Towards algebro-geometric understanding of k-stability of fano varieties mp4 D
November 08, 2012 Dave Anderson Pfaffian formulas for symplectic degeneracy loci
November 01, 2012 Alastair Craw Mori dream spaces as fine moduli of quiver representations mp4 D
October 25, 2012 Charles Vial Non-commutative resolutions and grothendieck groups mp4 D
October 18, 2012 Andrei Trepalin Rationality of quotients of projective plane over non-closed fields
October 11, 2012 Jonathan Pridham Semiregularity and reduced obstruction theories
October 04, 2012 Jason Lotay Uniqueness of lagrangian self-expanders
September 21, 2012 Arend Bayer Wall crossings and birational geometry D
September 20, 2012 Milena Hering Cox rings of toric bundles D

Spring 2012

May 31, 2012 Hugues Auvray Uniqueness and obstructions to existence of constant scalar curvature kahler metrics: the quasi-projective case
May 25, 2012 Julius Ross Local moment maps associated to okounkov bodies
March 29, 2012 Michael Singer Partial bergman kernels and toric k-stability
March 08, 2012 Kuzma Khrabrov Graded rings and q-fano threefolds with exotic weil class group
March 01, 2012 Jose Miguel Figueroa-O'Farrill Why we like homogeneous manifolds
February 16, 2012 Jesus Martinez Garcia Degenerations of del pezzo surfaces after hacking and prokhorov
February 09, 2012 Antony Maciocia Computing the walls for bridgeland stability
February 02, 2012 Damiano Testa The surface of cuboids and siegel modular threefolds mp4 D
January 26, 2012 Sergey Grigorian Deformations of g2-structures with torsion mp4 D
January 19, 2012 Rafael Torres Constructions of generalized complex structures in dimension four mp4 D

Autumn 2011

December 08, 2011 Alexander Kasprzyk Small polygons and toric codes
December 01, 2011 Joan Simon Topology from cosmology mp4 D
November 24, 2011 Dmitrijs Sakovics Weakly-exceptional quotient singularities mp4 D
November 17, 2011 Will Donovan Derived autoequivalences of hyper-kahler manifolds from heisenberg categorification
November 10, 2011 Andriy Haydys Higher dimensional gauge theory and fueter maps mp4 D
November 03, 2011 Ed Segal Window-shifts and grassmannian twists mp4 D
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