MIT Infinite Dimensional Algebra Seminar (Fall 2018)

We first recall some classical inequalities and uncertainty principles on groups in Fourier analysis. Then we discuss our recent work of Fourier analysis on quantum symmetries, including subfactors, planar algebras, Kac algebras, locally compact quantum groups, modular tensor categories. Moreover, we provide a 2D picture language to study Fourier analysis. Finally, we discuss some applications and open questions.

Link to Abstract

I will start the talk by presenting a series of rather surprising (mathematical) conjectures involving various equivalences of categories; some of these conjectures provide new understanding of local geometric Langlands duality for the group GL(n) (the formulation of which will be briefly recalled) . In the main body of the talk I would like to explain how one can "invent" these conjectures while studying super-symmetric 3-dimensional quantum field theories (the relevant background will be explained in the talk, no familiarity with quantum field theory will be assumed). If time permits, I will explain some mathematical evidence for these conjectures (mostly in the case of GL(2)).

This is a joint work with Yaping Yang and Gufang Zhao. An earlier construction generalizes the construction of loop Grassmannians by replacing the data of a reductive group by a based quadratic form. One can view it as a ''homological'' construction of loop Grassmanians in comparison to the standard cohomological construction. The talk will recast this construction in terms of quivers and use this to quantize the construction.

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same structures appeared in other areas of mathematics and physics, e.g., in the study of cluster algebras, solitons, and scattering amplitudes. We discuss new ways to think about these structures, their links with polyhedral geometry, and their extensions in general Lie theoretic language. In particular, they are closely related to combinatorics of Coxeter elements in Weyl groups. The talk is partially based on a joint project with T. Lam.

In this talk, I will present a survey of recent (and ongoing) work on the geometric representation theory of moduli spaces of parabolic sheaves on surfaces. Their cohomologies, K-theories, derived categories are controlled by W-algebras, q-W-algebras, and (tentatively) yet-to-be-defined categorifications thereof. I will also mention how this relates to the AGT correspondence between gauge theory and conformal field theory.

The relation between the discrete geometry of surfaces and completely integrable discrete systems has been well established in the last few decades, through work of Bobenko, Suris and many others. The recent introduction of discrete moving frames by Liz Mansfield, Gloria Mari-Beffa and Jing-Ping Wang, and the study of the pentagram map by Richard Schwartz and many others, has produced a flurry of work connecting the discrete geometry of polygons to completely integrable PDEs (both discrete and continuous) in any dimension, including connections to Combinatorics and to bi-Hamiltonian structures that often describe Liouville integrability. In this talk I will first review definitions and background on Liouville integrable systems and discrete moving frames on polygons. I will then describe a recent proof of the integrability of a discretization of Adler-Gelfand-Dikii systems (or generalized KdV systems), that used projective geometric vector fields and their direct connection to the moduli space of projective polygons. We will describe in detail how introducing the projective group allowed us to describe the generalized KdV discretizations as completely integrable bi-Hamiltonian systems, a task that seemed out of reach through more direct classical methods. This is joint work with Jing-Ping Wang and Annalisa Calini.

Let G be any split semi-simple adjojnt group. Let S be a topological surface with punctures & marked points on the boundary, considered modulo isotopy. We assign to (G,S) a new moduli space P(G,S). When S is a surface with punctures, it is a finite cover of the moduli space of G-local systems on S. We introduce a cluster Poisson structure on the space P(G,S), equivariant under the action of the mapping class group. Using this + previous joint work with Fock, we quantize the space P(G,S) by first q-deforming the algebra of regular functions on it, and then constructing its principal series of *-representations in a Hilbert space H(G,S). By construction, the q'-deformed *-algebra of the Langlands dual group G' also acts in H(G,S). We conjecture that, given any Riemann surface C with punctures of topological type S, the space of conformal blocks on C of the principal series of unitary representations of the related W-algebra W(G) is identified with the space H(G,S). This is a joint work with Linhui Shen.

The classical limit of the global geometric Langlands correspondence is the conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. While there was a significant progress on this statement for G=GL(n), much less is known about the case of general G. In my talk, I plan to review the global setting, and then focus on the classical limit of the _local_ geometric Langlands correspondence. My goal is to explain new techniques and ideas that are available in the local case.

Monstrous moonshine motivated new methods in algebra at the end of the previous century. More recent observations relating finite groups to modular forms have suggested new approaches to enumerative and arithmetic geometry. I will present two cases of this. Specifically, in the first part of the talk I will explain a vertex algebraic construction that conjecturally relates the Conway group to the enumerative geometry of K3 surfaces. In the second part of the talk I will explain some connections between finite simple groups and the arithmetic of modular abelian varieties.