Talks

Titles and abstracts.

Andrea Appel, University of Edinburgh.

Title: Monodromy of the Casimir connection and quantum Weyl groups.

Abstract: Here.

Galyna Dobrovolska, Columbia University.

Title: Wall-crossing for the rational Cherednik algebra.

Abstract: I will describe our progress (joint with Roman Bezrukavnikov and Guangyi Yue) on a question of Bezrukavnikov about wall-crossing for the rational Cherednik algebra in positive characteristic. The question is equivalent to a study of certain properties of the Mullineux involution.

Iordan Ganev, IST Austria.

Title: Beilinson-Bernstein localization via wonderful asymptotics.

Abstract: We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler.

Iain Gordon, University of Edinburgh.

Title: Parking spaces, after Armstrong, Reiner and Rhoades.

Abstract: This is joint work in progress with Martina Lanini. I will review a recent conjecture of Armstrong, Reiner and Rhoades that gives more structure to known bijections between (various complex reflection group generalisations of) non-crossing partitions and non-nesting partitions using the representation theory of rational Cherednik algebras. I will explain how this conjecture may be related to the so-called LL morphism and give some evidence, for and against. There’s much to do and help would be welcome!

Sam Gunningham, University of Texas, Austin.

Title: The Kostant catgory and its central action.

Abstract: I will describe a remarkable symmetric monoidal category associated to a reductive group G, which acts centrally on any G-category. This category has various interpretations: as bi-whittaker D-modules on G, as a quantization of the regular centralizers for G, as modules for the homology of the Langlands dual affine Grassmannian, and as modules for a certain nil-Hecke algebra. This talk is based on joint work with David Ben-Zvi and David Nadler.

Martina Lanini, Università degli Studi di Roma Tor Vergata.

Title: Sheaves on the alcoves and modular representations.

Abstract: I'll report on a joint project with Peter Fiebig. The project is meant to give a new perspective on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. Given a finite root system R and a field k we introduce an exact category C of sheaves on the partially ordered set of alcoves associated with R, and we show that the indecomposable projective objects in C encode the aforementioned characters. The category is way easier to understand than the representation theory of the group, and it can be thought of as a periodic modular generalization of the category of Soergel bimodules.

Emmanuel Letellier, Université Denis-Diderot.

Title: Exotic Fourier transforms over finite fields.

Abstract: Braverman-Kazhdan (2003) and Lafforgue (2013) developed a new approach to Langlands functoriality using Fourier transforms. Their approach makes sense over finite fields. In this talk I will explain some new results I obtained with Gérard Laumon.

Michael McBreen, MIT.

Title: Quantization and Quantum Cohomology.

Abstract: I will discuss joint work with Joel Kamnitzer and Nick Proudfoot relating the quantization of a symplectic resolution with the enumerative geometry of the symplectic dual space, providing a `quantum analogue' of Hikita's conjecture.

Kevin McGerty, University of Oxford.

Title: Kirwan surjectivity for quiver varieties.

Abstract: A classical result of Kirwan proves that cohomology ring of a quotient stack surjects onto the cohomology of an associated GIT quotient via the natural restriction map. In many cases the cohomology of the quotient stack is easy to compute so this often yields, for example, generators for the cohomology ring of the GIT quotient. In the symplectic case, it is natural to ask whether a similar result holds for (algebraic) symplectic quotients. Although this surjectivity is thought to fail in general, it is expected to hold in many cases of interest. In recent work with Tom Nevins (UIUC) we establish this surjectivity result for Nakajima's quiver varieties. An important role is played by a new compactification of quiver varieties which arises from the study of graded representations of the preprojective algebra.

Sven Meinhardt, University of Sheffield.

Title: Applications of Donaldson-Thomas Theory in Geometric Representation Theory.

Abstract: Donaldson-Thomas theory has been developed to study moduli spaces in a 3-Calabi-Yau setting strongly related to superstring theory. It does not only apply to geometry but also to topology and to representation theory. The latter case is actually best-understood and we start by giving a short introduction into DT-theory for representations of quivers with potential. In the second part we sketch applications involving intersection cohomology of moduli spaces, quantum groups and Kac-Moody algebras. This is a report of joint work with Ben Davison and Markus Reineke.

Andrei Negut, MIT.

Title: W-algebras for surfaces.

Abstract: We study the well-known K-theoretic Hall algebra of 0-dimensional sheaves on a smooth surface S. We interpret it first as a shuffle algebra, and consequently as a W∞ algebra. Somewhat surprisingly, the structure constants of the W_\infty algebra are Laurent polynomials in the two deformation parameters a and b, which one can set equal to the Chern roots of the cotangent bundle of S. Because of the general nonsense above, the W∞ algebra acts on the K-theory of moduli spaces of stable sheaves on S, but it is a nice feature that this action factors through the quotient Wr. Geometrically, this comes from a nice intersection-theoretic equality in the K-theory of the moduli space of sheaves, which I believe is special to the situation of stable sheaves.

Markus Reineke, Ruhr-Universitüt Bochum.

Title: Global topology of linear degenerations of flag varieties.

Abstract: Linear degenerations of flag varieties arise from relaxation of the containment relation between subspaces constituting a flag. We first describe the geometry of the individual degenerations qualitatively. Then we turn to global aspects, namely, how cohomology changes along the family of degenerations. We describe this change in terms of supporting perverse sheaves via quantum groups. This is a report on a joint project with G. Cerulli Irelli, X. Fang, E. Feigin and G. Fourier.

Francesco Sala, Kavli IPMU.

Title: CoHAs of Higgs sheaves on a curve.

Abstract: Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in algebraic geometry and representation theory. For example, if the quiver is the 1-loop quiver, the corresponding CoHA is the Maulik-Okounkov affine Yangian of gl(1). It acts on the equivariant cohomology of Hilbert schemes of points on the complex affine plane ("extending" previous results of Nakajima, Grojnowski, Vasserot, etc for actions of Heisenberg algebras) and of moduli spaces of framed sheaves on the complex projective plane. In the present talk, I will follow the recipe of "replacing quivers with curves" and I will end up by introducing CoHAs associated with the stack of Higgs sheaves on a smooth projective curve. (This is a joint work with Olivier Schiffmann.)

Peter Samuelson, University of Edinburgh.

Title: Hall algebras and Fukaya categories.

Abstract: The multiplication in the Hall algebra of an abelian category is defined by "counting extensions of objects," and the representation theory of this algebra tends to be quite interesting. (E.g. Ringel showed the Hall algebra of modules over a quiver is the quantum group.) Recently, Burban and Schiffmann explicitly described the Hall algebra of coherent sheaves over an elliptic curve, and various authors have connected this algebra to symmetric functions, Hilbert schemes, torus knot homology, the Heisenberg category, and the skein algebra of the torus. Motivated by this last connection and homological mirror symmetry, we discuss some computations in progress involving the Hall algebra of the Fukaya category of a (topological) surface. (joint with Ben Cooper.)

Travis Schedler, Imperial College London.

Title: Filtrations on cohomology of symplectic resolutions and Weyl group representations.

Abstract: I will explain how to dene natural ltrations on the cohomology of symplectic resolutions via berwise holomorphic forms and a conjectural link with Poisson-de Rham homology. For the Springer resolution we obtain canonical ltrations on Weyl group representations. This includes joint work with Bellamy and Etingof.

Peng Shan, Université Paris-Sud.

Title: Affine quiver Hecke algebras and Hikita’s conjecture.

Abstract: We will explain how to use some affine analogue of quiver Hecke algebras to study the conjectural relationship between the spectrum of cohomology of quiver varieties and Coulomb branch proposed by Hikita and Nakajima. This is a joint work in progress with Eric Vasserot.

Nicolo Sibilla, University of Kent.

Title: Log schemes, root stacks and parabolic bundles.

Abstract: Log schemes are an enlargement of the category of schemes that was introduced by Deligne, Faltings, Illlusie and Kato, and has applications to moduli theory and deformation problems. Log schemes play a central role in the Gross-Siebert program in mirror symmetry. In this talk I will introduce log schemes and then explain recent work joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects of their geometry. I will discuss a comparison result between two different ways of associating to a log scheme its etale homotopy type, respectively via root stacks and the Kato-Nakayama space. If time permits I will also explain a new categorified excision result for parabolic sheaves, which relies on the technology of root stacks.

Balázs Szendröi, University of Oxford.

Title: Partition functions and the McKay correspondence.

Abstract: I will explain some results on certain sheaf-theoretic partition functions defined on Calabi-Yau orbifolds, and their connection to the McKay correspondence and the representation theory of affine Lie algebras. Some of the ideas are from joint work with Gyenge and Nemethi, respectively Davison (the latter unfinished).

Support

The conference is supported by the Engineering and physical sciences reserach council (EPSRC), as part of the grant "Symplectic representation theory".