MIT Infinite Dimensional Algebra Seminar (Fall 2024)

In the geometric approach to quantum theory that I suggested several years ago we take as a starting point the set of states. This viewpoint is more general than the standard approach where the states are considered as density matrices and the algebraic approach where the states are identified with positive functionals on an associative algebra with involution. A large class of examples including conventional quantum mechanics can be constructed from classical theory where our devices can measure only a part of observables.

In a geometric approach one can derive the formulas for probabilities analyzing interaction with a random environment. If the theory is translation-invariant we can define a notion of particle and scattering of particles.

The quantum difference equations are K-theoretic analogs of Dubrovin connection in quantum cohomology. In my talk I show that over $p$-adic fields the quantum difference equations of Nakajima quiver varieties are equipped with the Frobenius automorphism $z\to z^p$.

I show that the corresponding Frobenius intertwiner is a partition function of quasimaps with special boundary conditions. I describe an explicit formula for the degree zero term of the intertwiner and explain the connection with works of Dwork, Sperber and Kedlaya.

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Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of spaces M(S,L) are finite. M.Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics.

However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani's recursions.

This part of the talk is based on the joint work with Zhe Sun.

There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, our exponential volumes are finite for any G. When P are polygons, they provide a commutative algebra of positive Whittaker functions for the group G. The tropical limits of the positive Whittaker are the (zonal) spherical functions for the group G.

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The first part of the talk will be given by Vasily Krylov and is based on the joint work with Ivan Perunov. We will discuss the geometry of Coulomb branches corresponding to finite Dynkin quivers. These Poisson varieties are known to be isomorphic to the so-called generalized slices in affine Grassmannians. Using the multiplication morphisms introduced by Braverman, Finkelberg, and Nakajima, we will construct coverings of resolutions of certain slices by affine spaces and, in particular, obtain explicit Darboux coordinates on them as well as characters of tangent spaces at torus fixed points.

The second part of the talk, given by Hunter Dinkins and based on joint work with Andrey Smirnov and Jiwu Jang, will discuss the 3d mirror dual picture. The mirror dual to the Coulomb branches of the first part are ADE type Nakajima quiver varieties. The quasimap vertex functions of such varieties are expected to encode information about the tangent spaces of the dual varieties. These vertex functions can be computed explicitly as power series using the combinatorics of minuscule posets. We will explain how to sum these series in certain cases and directly match characters of tangent spaces of the first part.

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This is based on arXiv:2408.02485, joint with Bezrukavnikov. We construct certain functors on categories of representations of rational Cherednik algebras associated with symmetric groups in zero and large positive characteristic. Our functors are indexed by pairs of a partition and a rational number, the slope. For a given slope, the functors give an action of the positive half of the Heisenberg Lie algebra, while when the slope varies and the characteristic is positive we get a categorical action of the positive half of the elliptic Hall algebra. In this talk I will explain necessary notions, sketch the construction, and provide the motivation, partly coming from the double affine representation theory.

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I will report on a new approach to interpreting the category of character sheaves as a monoidal center of the torus-equivariant version of the Hecke category. The benefits of this approach are that it at the same time stays completely within the realm of triangulated categories, works independently of sheaf-theoretic set up and goes beyond the case of unipotent monodromic sheaves. In particular, it allows to give a new proof, independent of sheaf-theoretic setting, of the fact that the Drinfeld center of the abelian Hecke category is equivalent to the abelian category of unipotent character sheaves. I will further use the construction to establish a mixed Koszul self-duality for the derived category of character sheaves. This in turn has an application to symmetry in Khovanov-Rozansky homology theories and the generalizations of these theories to links due to Gorsky-Hogancamp.

The talk is based on several projects, both published and in preparation, joint in various combinations with R. Bezrukavnikov, R. Gonin, Q.P. Ho, P. Li, K. Tolmachov and Y. Varshavsky.

In this talk we will explore how some methods and results of modular representation theory of finite groups can be extended to the representation theory of algebraic supergroups with a reductive even part over a field of characteristic zero. We begin by defining analogs of Sylow and elementary abelian subgroups, followed by "superanalogs" of Sylow theorems. We will also outline a proof of an analogue to Chouinard’s theorem for supergroups, which states that the projectivity of a representation is detected by restriction to elementary abelian subgroups. Additionally, we will discuss possible analogs of the Green correspondence, defect groups, and Broue's abelian defect groups conjecture in the context of supergroups. Finally, we will show that, unlike in the case of finite groups, the cohomological support does not coincide with the Balmer spectrum of the derived category of representations of a supergroup.The talk is based on joint work with I. Entova-Aizenbud, J. Pevtsova, A. Sherman and D. Vaintrob.

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(based on joint work with Marcelo De Martino)

For an algebraic torus defined over a local (or global) field F, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We topologize the relevant spaces of continuous homomorphisms and continuous cochains with the compact-open topology, and show that Langlands’s homomorphism is continuous and open. Moreover, we show that, in both the local and global settings, the subset of unramified characters corresponds to the identity component of the relevant space of characters when viewed in this topological framework. We discuss applications to the group of unramified characters, and to the component group of the relevant space of characters.

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In its simplest incarnation, the geometric Langlands program, as defined by Beilinson and Drinfeld in the 90’s, relates, on one side, a flat connection on a Riemann surface C, and on the other side, a more sophisticated structure known as a D-module on C. A recent generalization of the correspondence, due to Aganagic-Frenkel-Okounkov (AFO), predicts the existence of an isomorphism between q-deformed versions of conformal blocks on C (of genus 0 or 1), for a W-algebra on one side, and a Langlands dual affine Lie algebra on the other side. In the first part of the talk, I will review the current status of the program, and elucidate the meaning of tame ramification at points on C; the crucial new ingredient will be to give a definition of q-primary vertex operators on the W-algebra side, which I argue to be determined entirely from the representation theory of the dual quantum affine algebra, very much like the celebrated relation between W-algebra generators and the q-characters of Frenkel-Reshetikhin.

In the second part of the talk, I will give applications for quasimap counting in enumerative geometry, and the representation theory of quantum groups.

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We discuss ramifications of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. It turns out that many equations of mathematical physics, such as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries. We present their geodesic setting, which also allows one to describe multiphase fluids, homogenized vortex sheets and Brenier’s generalized flows. This is a joint work with Anton Izosimov.

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The goal of my talk to explain a proof of the result asserting that the trace of the Frobenius endofunctor of the category of automorphic sheaves with nilpotent singular support is isomorphic to the space of unramified automorphic functions. More generally, traces of Frobenius-Hecke functors produce shtuka cohomologies.

This is a joint work with D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin and N. Rozenblyum.

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The concept of q-deformation arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; q-deformations are important for knot invariants, combinatorial enumeration, discrete geometry... In mathematical physics, q-deformations are often understood as "quantizations".

The recently introduced notion of a q-deformed real number is based on the geometric idea of invariance by a modular group action. In this introductory and accessible talk, I will explain what is a q-rational and a q-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas.

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