• Sergey ARKHIPOV
• Braid group action on equivariant matrix factorizations
• Abstract. Given an algebraic variety X acted on by a simple algebraic group G, we define a certain category of matrix factorizations on the product of the cotangent bundle to X and of the Borel subalgebra in Lie(G). The potential is provided by the moment map. We construct a categorical braid group action on the corresponding derived category. The well known construction of Bezrukavnikov and Riche appears to be a specialization of ours for X being a point.


• David BEN-ZVI
• Elliptic Character Sheaves
• Abstract. I will describe two dual approaches to character sheaves for loop groups via Geometric Langlands of the two-torus. The spectral approach (appearing in preprints 1312.7164, 1312.7164) involves the B-model of the Hitchin system of a two-torus. The automorphic approach (initial steps appearing in 1302.7053) involves the A-model of the dual Hitchin system. I will emphasize relations to topological field theory and trace formulae. The results intertwine various strands joint with D. Nadler, A. Preygel and P. Li (Berkeley).


• Pavel ETINGOF
  • D-modules on Poisson varieties, Poisson homology, and symplectic resolutions
  • Abstract. Let X be an affine Poisson algebraic variety over complex numbers with finitely many symplectic leaves. Then the zeroth Poisson homology O(X)/{O(X),O(X)} is finite dimensional. This has applications to representation theory (if A is a filtered quantization of O(X) then A has finitely many irreducible finite dimensional representations). The proof is based on attaching to X a right D-module M(X) on X which is the quotient of the canonical D-module D(X) on X by the action of hamiltonian vector fields, and showing that it is holonomic, and that the zeroth Poisson homology of X is just the underived direct image of M(X) to the point. This motivates considering the full direct image of M(X) in the more general case when X is not necessarily affine. This direct image is called the Poisson-de Rham homology of X. If X is symplectic, then M(X) is the canonical sheaf, and thus the Poisson-de Rham homology coincides with the usual De Rham cohomology of X, but in general the Poisson-de Rham homology is hard to compute. However, if X has a symplectic resolution of singularities, we conjectured with T. Schedler that the Poisson-de Rham homology of X is isomorphic to the de Rham cohomology of the resolution. I will explain the motivation for this conjecture, and describe a few cases when it is proved (symmetric powers of simple singularities, Springer fibers). I will also describe some other cases (such as surfaces in C^3 with isolated singularities) when there is no symplectic resolution but there is a smooth symplectic deformation, and the Poisson-de Rham homology equals the cohomology of the deformation. Finally, I'll describe the structure of M(X) for isolated quasihomogeneous surface singularities. This is joint work with T. Schedler.


• Victor GINZBURG
  • Counting indecomposables over a finite field
  • Abstract. We develop an alternative approach to the theorem of Hausel, Letellier, and Rodriguez-Villages on counting inddecomposable quiver representations. Our approach is based on character sheaves and it does not involve combinatorial tools. A similar approach applies for counting indecomposable vector bundles on an algebraic curve in terms of the geometry of Higgs bundles. There seems to be a connection with a recent work of Deligne on counting irreducible local systems.


• Sam GUNNINGHAM
  • A conceptual approach to the generalized Springer correspondence.
  • Abstract. I will sketch a proof of the generalized Springer correspondence using a Mackey theorem on functors of parabolic induction and restriction for perverse sheaves on the nilpotent cone. This approach allows for substantial (further) generalizations; in particular, a description of the whole derived category of adjoint equivariant D-modules on a reductive Lie algebra (or algebraic group).


• David JORDAN
  • Quantizing character varieties via 4D TFT
  • Abstract. The character variety $Ch_G(S)$ is a moduli space of representations of $\pi_1(S)$ into G. When $G$ is reductive, $Ch_G(S)$ carries a canonical Poisson bracket constructed by Goldman and Atiyah-Bott. The assignment $S \mapsto Ch_G(S)$ is natural and fully local in $S$, and hence defines a topological field theory $Ch_G$. In this talk, I'll explain joint work with D. Ben-Zvi and A. Brochier quantizing each $Ch_G(S)$ compatibly with its TFT structure, leading to a (3+1)-dimensional TFT $Ch^q_G$, which can be viewed as an extension to generic $q$ of the Reshetikhin-Turaev invariants.
    Using tools of factorization homology and representation theory of tensor categories, it is possible to give fairly complete computations of $Ch^q_G(S)$. Familiar algebras, such as reflection equation algebras, quantum differential operators, and double affine Hecke algebras appear naturally in the computations.


• Daniel JUTEAU
  • Finite dimensional representations of rational Cherednik algebras: a necessary condition
  • Abstract. In joint work with Stephen Griffeth, Armin Gusenbauer and Martina Lanini, we give a necessary condition for a simple module of a rational Cherednik algebra (associated to an arbitrary complex reflection group, and for any choice of the $c$ parameter) to be finite dimensional.


• Kobi KREMNITZER
  • Analytic geometry and representation theory
  • Abstract. I will describe a new approach to analytic geometry (Archimedean and non-Archimedean). In this approach analytic geometry is viewed as algebraic geometry relative to an appropriate category of topological vector spaces. Using this one can develop analytic derived geometry and analytic Tannakian formalism. These tools can then be used to develop analytic geometric representation theory. This is joint work with Oren Ben-Bassat.


• Ivan LOSEV
  • Derived equivalences for Cherednik algebras
  • Abstract. A Rational Cherednik algebra is constructed for every complex reflection group and depend on a parameter that is a collection of complex numbers. For these algebras, one can define categories O. Rouquier conjectured that two categories O corresponding to parameters with integral difference are derived equivalent. In my talk, I'll speak about the proof.


• Kevin MCGERTY
  • Morse theory decomposition for categories of D-modules on stacks
  • Abstract. Motivated by the study of localisation theory for quantizations of Hamiltonian reductions, we will describe an algebro-geometric version of Morse stratifications yields a recollement of the derived category of D-modules on algebraic stacks. Time permitting we will discuss work in progress on the relation between this decomposition and the question of when the hyper-Kahler Kirwan map is surjective. This is joint work with Prof. T. Nevins (UIUC).


• Simon RICHE
  • Geometric Koszul duality for modular perverse sheaves on flag varieties
  • Abstract. I will report on a joint work with Pramod Achar which aims at describing some structural properties of the category of Bruhat-constructible perverse sheaves on the flag variety of a connected complex reductive group, with coefficients in a field of positive characteristic. In particular we construct a modular ``mixed derived category'', which allow us to construct a Koszul duality relating sheaves on our given flag variety with sheaves on the flag variety of the Langlands dual group. This Koszul duality exchanges tilting perverse sheaves and the parity sheaves of Juteau-Mautner-Williamson.


• Laura RIDER
  • Modular representation theory and the geometric Satake equivalence
  • Abstract. The geometric Satake equivalence proven by Mirkovic--Vilonen (building on work of Lusztig and Ginzburg) allows one to study the representation theory of an algebraic group via the topology of the affine Grassmannian for the Langlands dual group. It is hoped that the modular representation theory of an algebraic group may be understood geometrically, with a key role played by a class of objects known as parity sheaves, defined by Juteau--Mautner--Williamson. In my talk, I will describe this dictionary between representation theory and geometry, with an emphasis on the Mirkovic--Vilonen conjecture, recently proven in joint work with Pramod Achar.


• Travis SCHEDLER
  • Poisson-de Rham homology of cones and special polynomials
  • Abstract. This is a sequel to Etingof's talk and includes joint work with Proudfoot. Given a Poisson cone, its Poisson-de Rham homology has an additional weight grading, coming from dilations of the cone. The resulting two-variable Hilbert series recovers special polynomials: for hypertoric varieties, this recovers the Tutte polynomial and a polynomial defined by Denham using the combinatorial Laplacian. By a conjecture of Lusztig, for Slodowy slices of the nilpotent cone, this recovers Kostka polynomials. Along the way we will prove my conjecture with Etingof in the hypertoric case (and, in general, reduce it to a combinatorial statement involving slices of symplectic leaves). If time permits, I will explain a related conjecture with Etingof on the two-variable Hilbert series of symplectic resolutions, and a conjecture of Proudfoot relating it to symplectic duality.


• Peng SHAN
  • Koszulity for category O of affine Lie algebras and rational Cherednik algebras
  • Abstract. I will explain the standard Koszul property of a parabolic O of affine Lie algebras, and an equivalence of highest weight categories between affine parabolic category O for gl_n and the category O of cyclotomic rational Cherednik algebras. As an application, this gives the character formulae and standard Koszul property of the category O of cyclotomic rational Cherednik algebras. (Joint work with Raphael Rouquier, Michela Varagnolo and Eric Vasserot.)


• Wolfgang SOERGEL
  • Unicity of the grading of category O
  • Abstract. This is joint work with Michael Rottmaier. I want to explain why the usual grading of category O is uniquely caracterized by its compatibility with the action of the center of the enveloping algebra, and what in fact all these words mean.


• Geordie WILLIAMSON
  • Global and local Hodge theory of Soergel bimodules
  • Abstract. I will give an overview of the global and local Hodge theory (i.e. hard Lefschetz and Hogde-Riemann bilinear relations) of Soergel bimodules. I will focus on interesting new aspects including the difficulty of determining the "hard Lefschetz locus" and the local intersection forms, which are a version for intersection cohomology version of the equivariant multiplicity. I will also try to explain the role played by the nil Hecke ring.