Spring 2025 Seminar

Title:  Cantor Dynamical Systems, Bratteli Diagrams, and Their Associated Full Groups

Abstract:  If a countable group  G  acts on a Cantor set  X, one can enlarge  G  to a (still countable) group  F(G)  by adding homeomorphisms of  X  that act locally as elements of  G. This group, known as the full group of the dynamical system  (X, G), completely determines the system up to topological orbit equivalence—any abstract isomorphism between full groups is spatially realized by a homeomorphism implementing orbit equivalence. Full groups can be viewed as generalized symmetric groups associated with the orbit equivalence relation of  (X, G). More generally, full groups can be associated with étale groupoids.

Originally introduced in the study of operator algebras via the closed product construction, full groups have recently played a significant role in geometric group theory. By adjusting the dynamical properties of  (X, G), one can construct new classes of countable groups  F(G), leading to solutions of several open problems in the field.

In this talk, we will explore the connection between the dynamical properties of Cantor systems and the algebraic structure of their associated full groups. We will primarily focus on the case of integer actions, where additional tools are available. Specifically, we will introduce the notion of a Bratteli diagram—an infinite graded graph that encodes the structure of  Z-actions. We will explain how Bratteli diagrams can be used to compute the  K_0-group of the system, find invariant measures, and investigate the connection between characters of full groups and invariant measures.

Title:  A Comparison of Two Supercharacter Theories

Abstract:  In 2008, P. Diaconis and I.M. Isaacs introduced a generalization of classical character theory, which they called supercharacter theory.  Let X be a partitioning of the irreducible characters of a finite group G such that the trivial character is in its own block of this partition.  Let Y be a partitioning of the conjugacy classes of G such that identity element of G is in its own block of this partition.  Define the supercharacters of G to be the sums of the irreducible characters in each block in X, weighted by their degree.  Define the superclasses of G to be the unions of the elements in each block in Y.  If the values of the supercharacters are constant on the superclasses, this is called a supercharacter theory of G.  As a supercharacter is determined by these partitions, the structure of a group allows for multiple supercharacter theories.  In this talk we will compare the construction of two different supercharacter theories, one based on the degrees of the irreducible characters of G and one based on the size of the conjugacy classes of G.  In particular, we will demonstrate necessary and sufficient conditions that will force these two supercharacter theories to coincide when the groups considered are semidirect products of cyclic groups.

Title:  Two-row Delta Springer varieties

Abstract:  I will discuss the geometry and topology of a certain family of so-called Delta Springer varieties from an explicit, diagrammatic point of view. These singular varieties were introduced by Griffin—Levinson—Woo in 2021 in order to give a geometric realization of an expression that appears in the t = 0 case of the Delta conjecture of Haglund, Remmel, and Wilson. In the two-row case, Delta Springer varieties generalize both ordinary Springer fibers as well as Kato's exotic Springer fibers. Moreover, the homology of two-row Delta Springer varieties has a diagrammatic description and can be equipped with an action of the degenerate affine Hecke algebra. This recovers and upgrades the action of the symmetric group obtained by Griffin—Levinson—Woo and yields a skein theoretic description of said action. This is joint work with A. Lacabanne and P. Vaz.

Title:  Monoidal complete Segal spaces

Abstract:  Viewing a monoid as a category with a single object allows us to encode the binary operation using the properties of composition and associativity inherent in any category. In this talk, we use this idea to explore the relationship between (∞,1)-categories with a monoidal structure and (∞,2)-categories with one object. This exploration relies on the model structure of simplicial and ⍬2-spaces.  The talk is designed to be self-contained, requiring no prior knowledge of the aforementioned categories.

Title:  Join us for plenty of pizza and a great selection of pies!

Abstract:  RSVP with the Department of Mathematics front desk, i.e., Jordan White, by filling out the Pi Day Invitation form.

No seminar today. 

Title:  Poisson geometry and Azumaya loci of cluster algebras

Abstract:  Roughly speaking, cluster algebra is a commutative algebra obtained from taking the intersection of Laurent polynomial rings associated a "seed". When a cluster algebra satisfies some compatibility condition, M. Gekhtman, M. Shapiro, and A. Vainshtein showed that one can connect a Poisson bracket to the cluster algebra. In this talk we will explain the global geometry picture of the affine Poisson varieties associated to a cluster algebra and its quantization, root of unity quantum cluster algebra. In particular, we prove that the prime spectrum of the upper cluster algebra, endowed with the GSV Poisson structure, has a Zariski open orbit of symplectic leaves and give an explicit description of it. Our result provides a generalization of the Richardson divisor of Schubert cells in flag varieties. Further, we describe the fully Azumaya loci of the root of unity upper quantum cluster algebras, using the theory of Poisson orders. This classifies their irreducible representations of maximal dimension. This is a joint work with Greg Muller, Kurt Trampel and Milen Yakimov.

Title:  Motives in geometric representation theory

Abstract:  The spherical and Iwahori-Hecke algebras of a reductive group are of great importance in the Langlands program. They are categorified by equivariant sheaves on the affine Grassmannian and affine flag variety respectively, as introduced in the previous talk. Similarly, the Satake and Bernstein isomorphisms are categorified by the Satake equivalence and Gaitsgory's central functor, for which one needs a choice of cohomology theory. In this talk, I will explain that the choice of cohomology theory is irrelevant, in the sense that the Satake equivalence and central functor are of motivic origin.

Title:  The Gen and Cogen Formulas for Torus Knot Homology

Abstract:  In recent work on the Oblomkov–Rasmussen–Shende conjecture about algebraic links, Oscar Kivinen and I observed that there are two similar, but inequivalent, closed formulas for the triply-graded link homology of a torus knot. We call them the Gen and Cogen formulas. They respectively arise from works of Mellit and Hogancamp–Mellit. I will explain what they are, how they differ, and how they relate to our proof of some new cases of the ORS conjecture. Time permitting, I will also mention an interesting speculation about the formulas that is implicit in writings of Ivan Cherednik.

Title:  Spectral algebraic geometry and topological modular forms

Abstract:  An important area of study in homotopy theory is the study of elliptic cohomology. In this talk we briefly review the context of elliptic cohomology and its interpretation in terms of spectral algebraic geometry. Furthermore, we construct the Deligne-Mumford compactification of the stack of oriented elliptic curves. If time permits we discuss some applications of the construction.

Title:  Twisted graded categories

Abstract:  Given a presentably symmetric monoidal ∞-category C and a commutative monoid M, we introduce and classify twisted graded categories, which generalize the Day convolution structure on Fun(M, C). We apply this framework to study cyclotomic-closedness, or orientability, in ∞-semiadditive categories. We show that a category is orientable if and only if a twisted graded category over it identifies with the Day convolution. We present results concerning the categories of E_n-modules and super vector spaces.

This is joint work with Shaul Ragimov.

Title:  Multiplicative universal centralizer: Bruhat stratification, cluster structure and applications

Abstract:  The universal centralizer of a complex reductive group plays an important role in geometric representation theory. Aside from the standard group scheme structure, it possesses a natural Bruhat decomposition (and consequently a ''parabolic induction" structure) that makes the geometry quite explicit, and has many applications.

The multiplicative version of the universal centralizer possesses a similar feature but has much richer (and more complicated) algebraic geometric structures. I will talk about recent results on several geometric features of the multiplicative universal centralizer. These include (a complete description of) a natural Bruhat stratification and the cluster structure on it. I will also talk about several applications. This is based on joint work with Ben Webster.

Title:  Awards Day 2025

Abstract:  This event features catering. Please respond to the Front Office if you plan to attend.

Title:  Preparing for the Academic Job Market

Abstract:  Join a discussion with a panel of postdocs and faculty to learn about their recent experiences navigating the academic job market. They will discuss preparation of CV, research statements and related documents, expectations for interviews and more. Please come with questions! There will be time for Q&A.