Speaker: Mark Walker
Title: The Hodge conjecture for matrix factorizations
Abstract: Over 20 years ago, Eric Friedlander and I developed semi-topological K-theory, an invariant of complex varieties that lies part-way between algebraic K-theory and topological K-theory. Since then, Anthony Blanc extended the notion of semi-topological K-theory to dg categories over the complex numbers. By inverting the Bott element, this leads to a topological K-theory of such dg categories, also developed by Blanc. Among other things, this allows one to place a "non-commutative Hodge structure" on suitably nice dg categories --- the role of singular cohomology with complex coefficients is played by periodic cyclic homology, the Hodge filtration comes from negative cyclic homology, and the rational structure is given by the Blanc's Chern character map from topological K-theory. In particular, one can formulate an "nc Hodge conjecture" for certain dg categories. In this talk, I will describe in more detail the ideas outlined above. Then I'll talk about recent joint work with Michael Brown in which we relate the nc Hodge conjecture for the dg category of matrix factorization of homogenous polynomials with the classical Hodge conjecture for smooth projective hyper-surfaces.
Speaker: Jeremiah Heller
Title: Some computations of equivariant motivic cohomology
Abstract: For a finite group G, Bredon motivic cohomology is an invariant for smooth G-schemes, over a base field, which mixes motivic cohomology and topological Bredon cohomology. I'll talk about a computation of the coefficient ring and ongoing work computing the Steenrod algebra and operations in this theory, mostly for k=C, and for G = C_2. This is joint work with M. Voineagu and P. A. Ostvaer.
Speaker: Roman Bezrukavnikov
Title: Frobenius kernel cohomology, p-cells and loop spaces
Abstract: Humphreys conjectures relate cohomological support of a tilting module over a quantum group at a root of unity to Kazhdan-Lusztig cells in the affine Weyl group. A proof was obtained by the speaker 20 years ago, along with an explicit geometric description of cohomology of such a module. Loop spaces appear in that story in two different ways: via the realization of the module category in terms of perverse sheaves on the affine Grassmannian and via the localization theorem identifying them with coherent sheaves on the space of small loops into G/B in the sense of derived algebraic geometry. I will review results and conjectures by Achar and Riche and by Boixeda Alvarez and the speaker aiming at a parallel theory for tilting modules over an algebraic group in positive characteristic. A central role there is played by the classical result of Friedlander and Suslin describing the spectrum of cohomology of a nilpotent group scheme.
Speaker: Daniel Nakano
Title: On Donkin's conjectures
Abstract: I will begin by presenting background material on two famous conjectures formulated by Donkin at MSRI in 1990. The first conjecture is Donkin's Tilting Module Conjecture (DTilt), and the second conjecture is Donkin's p-Filtration Conjecture (DFilt). For over 30 years, these conjectures have eluded verification for primes smaller than 2h-2 where h is the Coxeter number of the underlying algebraic group. Recent progress by Kildetoft-Nakano and Sobaje has shown that there are important connections between these conjectures. In particular, Jantzen's Question posed in 1980 on the existence of Weyl p-filtrations for Weyl modules for a reductive algebraic group constitutes a central part of the new developments. I will later describe how we produced infinite families of counterexamples to Jantzen's Question and Donkin's Tilting Module Conjecture. Counterexamples can be produced via our methods for all groups other than when the root system is of type A_n or B_2. Furthermore, I will also present a complete answer to Donkin's Tilting Module Conjecture for rank 2 groups. At the end of the talk, I will show a few results that indicate that there might be a positive answer to the Tilting Module Conjecture for groups whose underlying root system is of type A_n. These results represent joint work with Christopher Bendel, Cornelius Pillen and Paul Sobaje.
Speaker: Alexander Merkurjev
Title: Massey products in Galois cohomology
Abstract: A fundamental question in Galois theory is the profinite inverse Galois problem: Which profinite groups are realizable as absolute Galois groups of fields? A historically fruitful approach to the profinite inverse Galois problem has been to give constraints on the cohomology of absolute Galois groups. The most spectacular example of this is the Norm-Residue Theorem (the Bloch–Kato Conjecture), proved by Rost and Voevodsky. Another tool to establish constraints on the Galois cohomology of absolute Galois groups is the theory of Massey products. The Massey Vanishing Conjecture of Minac and Tan predicts that all Massey products in the Galois cohomology of a field vanish as soon as they are defined. I will give an introduction to Massey products and describe recent progress on the Massey Vanishing Conjecture.
Speaker: Sarah Witherspoon
Title: Cohomology of Nichols algebras and quantum groups
Abstract: Friedlander's breakthrough work with Suslin in the 90s - on cohomology of finite group schemes - together with other work on finite groups, restricted enveloping algebras, quantum groups and more, inspired a conjecture: The cohomology of finite dimensional Hopf algebras, and more generally finite tensor categories, is always finitely generated. Some current work on this conjecture focuses on known classes of finite dimensional Hopf algebras, for example, the pointed Hopf algebras that include the small quantum groups. Underlying such Hopf algebras and largely governing their cohomology are the Nichols algebras, defined via braid group actions on vector space tensor powers. In this talk, we will introduce all these topics and describe some recent results for Hopf algebras arising from Nichols algebras.
Speaker: Vera Serganova
Title: Commonality between finite groups and supergroups (towards super Green correspondence)
Abstract: In this talk I will discuss similarities and differences between modular representation theory of finite groups and algebraic supergroups over C with reductive underlying groups. We try to generalize defect theory and Green correspondence to supergroups. At the moment we have precise results in defect 1 case and some conjectures and open questions in general. This is a joint work with I. Entova-Aizenbud and A. Sherman.
Speaker: Burt Totaro
Title: Actions of the group Z/p in characteristic p
Abstract: For an action of a finite group on a smooth variety in characteristic zero, the action can be linearized near a fixed point. This is false for actions of G=Z/p in characteristic p, essentially because G is not linearly reductive. And in general, the singularities of X/G can be bad, for example not Cohen-Macaulay. The good news is: we give a sufficient condition for a G-quotient singularity to be a mu_p-quotient singularity. (And we know everything about mu_p-quotient singularities.)
Speaker: Kirsten Wickelgren
Title: A1-enumerative geometry and a wall-crossing formula for A1 Gromov-Witten invariants
Abstract: A1-homotopy theory was introduced in the late 90s by Morel and Voevodsky and gives a method for transporting methods from algebraic topology into algebraic geometry. It is a feature of A1-homotopy theory that analogous results over the complex and real numbers may indicate the presence of a common generalization valid over a general field k. We will discuss consequences for enumerative geometry. For example, in joint work with Jesse Kass, we show there are 15<1>+12<-1> lines on a cubic surface in an appropriate sense and Marc Levine has done much work in this field. The counts now take values in the Grothendieck--Witt group of stable isomorphism classes of non-degenerate, symmetric bilinear forms. <a> represents the class of the bilinear form k x k -> k mapping (x,y) to axy. Classical results over R and C can be recovered by taking the rank and the signature. This talk will introduce A1-enumerative geometry and present a wall-crossing formula for A1 Gromov-Witten invariants joint with Erwan Brugallé.
Speaker: Julia Plavnik
Title: Semisimplification of contragredient Lie algebras
Abstract: TBD
Speaker: Raphael Rouquier
Title: Higher representation theory
Abstract: I will discuss recent work towards constructing braided monoidal 2-categories associated with simple (super)Lie algebras and connections with 3-dimensional topology.
Speaker: Cris Negron
Title: Quantum Frobenius kernels at arbitrary roots of 1
Abstract: I will discuss constructions of quantum Frobenius kernels at (completely) arbitrary roots of 1. These investigations are motivated by recent developments in quantum field theory, where quantum groups at "bad" roots of unity appear naturally.