VT Algebra Seminar

Fridays 2:30-3:30 eastern time

McBryde 209

Fall 2024

September 6

Quantum K Rings of Partial Flag Varieties

Irit Huq-Kuruvilla (VT)

Gu-Mihalcea-Sharpe-Xu-Zhang-Zhou gave a conjectural presentation of the quantum K ring of type-A partial flag varieties based on some ideas from quantum field theory. We prove these conjectures using the newly developed abelian/non-abelian correspondence for quantum K rings, and in the process give a geometric explanation for the appearance of the Bethe Ansatz equations from quantum-many body systems in the conjecture.
September 13

Feigin-Loktev fusion product, quantum loop group, and coherent Satake category

Ilya Dumanski (MIT)

The Feigin-Loktev fusion product of cyclic graded modules over the current Lie algebra was introduced in 1998. We propose a way to study it geometrically, relating it to the geometry of affine Grassmannian. This allows us to compute it in some new cases. Moreover, this establishes a connection between two categories with cluster structure in representation theory: the one of modules over the affine quantum group and the one of perverse coherent sheaves on the affine Grassmannian. Based on arXiv:2308.05268.
September 20

A little introduction to skein algebras

Daniel Douglas (VT)

Skein algebras (resp. modules) are algebraic objects associated to surfaces (resp. three-dimensional manifolds) that reflect the geometry and topology of the space. They can be thought of as a generalization of the concept of the Jones polynomial of a knot in R^3. In this talk, I will give a light introduction to some of these ideas.
September 27

Arithmetic Symmetric Functions

Milo Bechtloff Weising (VT)

In this talk I will introduce arithmetic symmetric functions. These are algebraic objects built from ordinary symmetric functions which encode arithmetic data about number fields. We will build an analogy between the theory of arithmetic symmetric functions and algebraic number theory exhibiting symmetric function versions of many fundamental objects in number theory including the Artin L-functions. We will also discuss some analytic results regarding these special functions.
October 4

The log Grothendieck group of varieties

Leo Herr

The ordinary Grothendieck group of varieties is a group with simple generators and relations built out of varieties analogous to the construction of the integers from the natural numbers. Called the "poor man's motives," this group recovers Hodge numbers, even for open, singular varieties, and point counts over finite fields. Log schemes are morally pairs of a scheme with a divisor (X, D). Used for compactifications and degenerations of varieties, they admit a log Hodge theory due to Kato-Usui. This theory is not yet understood for open or singular varieties, so we define a log Grothendieck group to get log Hodge numbers. Unfortunately, this does not work! We modify log Hodge numbers to get weaker well-defined invariants from our theory, leaving lots of open questions. (joint with David Holmes, Pim Spelier, Jesse Vogel)
October 25

An introduction to polysymmetric functions and their bases

Aditya Khanna

Partitions and symmetric functions are the bread and butter (not necessarily in that order) of algebraic combinatorialists. We begin the talk by introducing these notions. In their preprint, Asvin G and Andrew O'Desky generalize the above ideas to splitting types and polysymmetric functions (in that order). The bases of the space of polysymmetric functions that we work with either arise by taking tensor products of bases of symmetric functions or are some non-classical bases as defined in the preprint. I will define these notions fully in the talk and mention some results we have about relating the non-classical bases to the tensor bases.
October 25

Counting Pattern Avoiding Permutations by Big Descents

Johnny Rivera

One can define various statistics on permutations and study the distribution of these statistics over the symmetric group. Moreso, one may seek to describe the distribution of a permutation statistic over a set of restricted permutations which avoid predetermined patterns. In this talk, I will provide an introduction both to permutation patterns and permutations statistics. Further, I will highlight new enumerative results regarding the big descent statistic on pattern avoiding permutations.
November 1

Moduli of arbitrarily singular reduced curves via subrings

Sebastian Bozlee (Fordham)

Moduli spaces are spaces whose points parametrize geometric objects. There is a moduli space of arbitrarily singular reduced algebraic curves, yet it is generally considered too frightening of a space to work with. In this talk, we will explain an approach that allows us to think of moduli of such curves as coming from two simpler parts: moduli of smooth curves and moduli of subrings of certain finite algebras. We will then describe a stratification analogous to the stratification of stable curves by dual graph and an application of the theory to connectedness of the moduli space of all reduced curves.
November 15

Centers of quantum affine algebras

Naihuan Jing (NCSU)

Affine Lie algebras usually have trivial centers except at the critical level, where the centers are called the Feigin-Frenkel centers. I will talk about the recent joint work with Ming Liu and Alexander Molev on the quantum Feigin-Frenkel center. The quantum center in type A is shown to contain a family of special central elements indexed by the primitive idempotents of the Hecke algebra and parameterized by Young diagrams. The Harish-Chandra images of the Sugawara operators can be identified with the eigenvalues of the operators acting in the q-deformed Wakimoto modules, where the eigenvalues are given by certain deformed Schur functions.
November 22

Title TBA

Chi Nguyen

TBA
November 22

Title TBA

Wendi Gao

TBA
December 6

Title TBA

Andrey Smirnov (UNC Chapel Hill)

TBA