Abstract:  

Abstract:  Gauged Linear Sigma Models (GLSM) are curve-counting theories of a critical locus of a function in a GIT quotient variety. The central part is played by a special matrix factorization on the moduli space of GLSM maps. GLSM invariants satisfy many remarkable properties, many of which are captured by the central charges, that are particular generating series of GLSM invariants that depend on matrix factorizations. In the talk, I will explain what these objects are and how to use central charges to prove symmetry and wall-crossing results for GLSM.

Abstract:  This is a report on work of my graduate student Cristian Rodriguez. A Q-Fano 3-fold is a complex projective variety with mild singularities such that its 1st Chern class is positive. Q-Fano 3-folds with b_2=1 arise as end products of Mori's minimal model program. Thousands of families are expected, whereas there are only 17 in the smooth case. We will describe mirror symmetry for Q-Fano 3-folds in terms of the Strominger-Yau-Zaslow conjecture and Kontsevich's homological mirror symmetry conjecture, building on work of Auroux. The mirror of a Q-Fano 3-fold is a K3 fibration over the affine line such that the total space is log Calabi--Yau and some power of the monodromy at infinity is maximally unipotent. In 95 cases the Q-Fano is realized as a hypersurface in weighted projective space and we describe the mirror K3 fibration explicitly.  

Abstract:  

Abstract:  Quantum groups are related to 3-dimensional topological quantum field theories. Downsizing from three dimensions to one but adding defects, I will explain a surprising relation between topological theories for one-dimensional manifolds with defects and values in the Boolean semiring and finite-state automata and their generalizations. I will then discuss the easier, linear case of these theories where symmetric Frobenius algebras appear, with connections to thin surface cobordisms. This is joint with Mikhail Khovanov. 

Abstract:  The Steinberg variety and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in zero and positive characteristics. Based on arXiv:2302.05782 and work in progress. 

Abstract:  The moduli spaces of one-dimensional sheaves on P^2 are first studied by Simpson and Le Potier, and they admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one expects to obtain certain BPS invariants from the perverse filtration on cohomology induced by this morphism, which motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present some recent progress on this cohomology ring, including a minimal set of tautological generators, and a “Perverse = Chern” conjecture which specializes to an asymptotic product formula for refined BPS invariants of local P^2. This can be viewed as an analogue of the recently proved P=W conjecture for Hitchin systems. Based on joint work with Junliang Shen, and with Yakov Kononov and Junliang Shen. 

Abstract:  

Abstract:  In this talk, I discuss how an infinite dimensional convex geometry of interest to physicists exhibits "flattening," which manifests as emergent equalities among naively independent coordinates. This flattening behavior is intrinsically tied to the infinite dimensional nature of the convex geometry, as these emergent equalities only appear in the infinite dimensional limit. In more detail, the space of causal and unitary theories, called the EFT-Hedron, is identified as the intersection of a convex region given by the Minkowski sum of two moment curves and a hyperplane in an infinite dimensional projective space. I use linear programming to provide strong numeric evidence that the EFT-hedron "flattens out." For example, restricting a finite fraction of the coordinates to be even-zeta values, the remaining coordinates are (conjecturally) fixed to take odd-zeta values. I will conclude by briefly sketching how this conjecture relates to Type-I superstring theory, which corresponds to a particular point in the EFThedron. 

Abstract: We study D. Kazhdan and G. Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U and explore unexpected connections to other interesting objects in representation theory. We first define an analogue of Category O in the context of Kazhdan-Laumon categories and explicitly classify its simple objects, and then use this combinatorial data to discuss its connections to Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety. Finally, we study the action of the braid group appearing in the definition of Kazhdan-Laumon categories and give a categorification of the "algebra of braids and ties" occuring in the context of knot theory.  

Abstract:  In analogy with the (generalized) Springer correspondence relating perverse sheaves on a nilpotent cone to representations of the Weyl group, we consider perverse sheaves on the symmetric product of n copies of the plane C2, constructible with respect to the natural stratification by collision of points. This category is semisimple when the coefficients have characteristic zero, but with positive characteristic coefficients it can be very complicated. We show that this category is equivalent to modules over a convolution algebra given by K-theory of sheaves on the symmetric group, equivariant for the action of Young subgroups on the left and right.  Up to Morita equivalence, this algebra has a Schur algebra as a quotient.  I will also explain how this algebra arises using the K-theory of Hilbert schemes and a theorem of Bridgeland, King, and Reid. Joint work with Carl Mautner. 

Abstract:  

Abstract:  The irreducible characters of a finite reductive group are partitioned into sets called Harish–Chandra series, describing their behavior under induction from and restriction to Levi subgroups. Broué, Malle, and Michel found a ‘cyclotomic’ generalization of this story, depending on geometric induction and restriction functors introduced by Deligne–Lusztig and Lusztig. I present some conjectures, joint with Ting Xue, that relate these cyclotomic Harish–Chandra series to the representation theory of certain Hecke algebras at roots of unity. These conjectures seem to be new even in type A. They suggest studying a certain virtual Hecke-algebra bimodule depending on two parameters. When these parameters are the same prime power, this virtual bimodule can be constructed from the cohomology of Deligne–Lusztig varieties. When they are certain roots of unity, I conjecture that it can be constructed from the cohomology of homogeneous affine Springer fibers. The motivation for this conjecture comes from work of Oblomkov–Yun on affine Springer fibers, my own work on braid varieties, and nonabelian Hodge theory.