Abstracts

Week of February 16, 2025

February 17, 2025
Geometry, Symmetry and Physics Some Computations with Representations of Finite Chevalley Groups in Equal Characteristic 4:30pm -
KT 801

I will discuss two computations involving representations of finite Chevalley groups in equal characteristic. The first one reduces by an argument of Bao Le Hung and Tony Feng to description of a class in top homology of a certain affine Springer fiber (performed jointly with Bao Le Hung, Tony Feng and Pablo Boixeda Alvarez). The second one (joint with Michael Finkelberg, David Kazhdan and Calder Morton-Ferguson) involves a new basis for the ring O(T) of regular functions on the maximal torus T as a module over the ring O(T)^W of invariant regular functions, different from but related to the basis constructed by Steinberg in 1975.
February 18, 2025
Geometry & Topology Random quotients of hierarchically hyperbolic groups 4:00pm -
KT 207

Hierarchically hyperbolic groups (HHGs) expand on work of Masur and Minsky on mapping class groups of compact surfaces to provide a geometric model for certain acylindrically hyperbolic groups Gromov hyperbolic groups, and subgroups of right-angled Artin groups.
This geometric frame work has been especially useful for verifying finer properties of acylindrically hyperbolic groups such as the Tits alternative and uniform exponential growth.
I will describe why quotients obtained from subgroups normally generated by finitely many independently distributed random walks preserve hierarchical hyperbolicity.
These ideas also let us understand the geometry of random quotients of hyperbolic, and relatively hyperbolic groups. This is joint work in progress with Abbott, Berlyne, Mangioni, and Rasmussen.
February 20, 2025
Analysis A generalized Legendre duality relation and Gaussian saturation 4:00pm -
Zoom

https://yale.zoom.us/j/95303636613
This talk is based on joint works with Hiroshi Tsuji (Saitama Japan).
The Blaschke—Santal\'{o} inequality describes a correlation between a convex body and its dual object (polar body). Motivated by the recent studies in convex geometry, optimal transportation theory, as well as information theory, a problem of extending the inequality to multiple convex bodies was proposed by Kolesnikov--Werner. Their formulation of the problem naturally involves some generalization of the (functional) Legendre duality. In this talk, we are going to establish a genuine Gaussian saturation principle for the generalized Blaschke--Santal\'{o}-type inequality, and in particular give an affirmative answer to the conjecture of Kolesnikov--Werner.

Our novel observation is a simple but crucial link between the above problem and the inverse form of the Brascamp--Lieb (multilinear) inequality (IBL inequality). The study of the IBL inequality was initiated by Chen--Dafnis--Paouris, and then later Barthe--Wolff developed its theory in more general framework, but under a certain non-degeneracy condition.
Our second main result is about the Gaussian saturation principle for the IBL inequality beyond the framework of Barthe—Wolff.
The above result on the generalized Blaschke—Santal\’{o}-type inequality is a consequence of this second result.
There are further fruitful consequences from our study of the IBL inequality, which we will present as long as time permits.
Quantum Topology and Field Theory A quantum N-dimer model for ribbon graphs 4:30pm -
KT 801

We associate to each ciliated bipartite ribbon graph in R^3 an isotopy invariant Laurent polynomial in a single variable q^(1/N), called the SL_N quantum trace, which can be expressed as a quantum deformation of the partition function for the N-dimer model.  The construction is based on Sikora’s SL_N quantum traces for N-webs in R^3.  For planar graphs, the quantum trace is moreover a symmetric Laurent polynomial in q, which can be expressed as a quantum deformation of the Kasteleyn determinant of the graph equipped with the trivial connection.  We also provide a similar expression for planar graphs equipped with a general quantum matrix connection (subject to a relatively strong commutativity constraint).  This is joint work with Richard Kenyon, Nicholas Ovenhouse, Sam Panitch, and Sri Tata. 
February 21, 2025
Friday Mornings Webs and multiwebs for the symplectic group 10:00am -
KT 801

The dimer model is a statistical mechanical model that studies random dimer covers (perfect matchings) of a graph. Web categories are developed to compute the Witten-Reshetikhin-Turaev quantum invariants and to study the representations of quantum groups. 

Kasteleyn’s theorem computes the number of dimer covers of a graph by calculating the determinant of a modified adjacency matrix. The generalizations of the theorem connect generalized dimer models to type A web categories. I will talk about further generalizations to the type C cases, relaxing the bipartiteness condition of the underlying graph. This talk is based on joint work with Richard Kenyon. 
Algebra and Geometry lecture series Vertex algebras and moduli of Higgs bundles I 3:00pm -
KT801

Motivated by a physics construction in 4d N=2 superconformal field theories, mathematicians have studied a class of vertex algebras with a number of interesting relations with the cohomology of moduli spaces of stable Higgs bundles of a curve C. We will discuss basic definitions and constructions in the theory of vertex algebras, and discuss examples, including Arakawa's construction of class S VOAs.