Katrina Barron

Irrational vertex operator algebras and graded pseudo-traces for indecomposable non-simple modules

It is known that the graded traces of vertex operator algebras exhibit modular invariance properties when the vertex operator algebra has two certain properties: rationality and $C_2$-cofiniteness. Relaxing the rationality condition and including graded pseudo-traces, the modular invariance property is retained in theory, although few examples are known.

We investigate questions of rationality for $\mathbb{C}$-graded vertex algebras and give applications to the Weyl vertex algebra under conformal flow. In addition, results on the computation of graded pseudo-traces for $C_1$-cofinite irrational vertex operator algebras are presented. These include graded pseudo-traces for all indecomposable modules for the Heisenberg vertex operator algebra and for all indecomposable modules induced from the level zero Zhu algebra for the universal Virasoro vertex operator algebras with certain central charges.

This work is part of the Women in Mathematical Physics initiative and involves joint work with Karina Batistelli, FLorencia Orosz Hunziker, Veronika Pedic Tomic, and Gaywalee Yamskulna.

Vyjayanthi Chari

Higher Rank Kirillov--Reshetikhin Modules and Monoidal Categorification.

Kirillov and Reshetikhin introduced a family of irreducible finite--dimensional modules for the Yangian of a simple Lie algebra and conjectured a character formula for these representations. Subsequently, these modules were also defined for the quantum loop algebra associated to a simple Lie algebra and these are now called the Kirillov--Reshetikhin (KR) modules. These modules have many nice properties and there is an extensive literature on the subject, which includes their connections with integrable systems, the combinatorics of crystal bases, the fermionic formula and more recently they have been shown to be connected to cluster algebras via the notion of monoidal categorification. In this talk we explain that the definition of these modules encodes essentially information from rank one and then discuss how to generalize this to higher rank. We discuss the combinatorics involved in proving certain classification results and applications of our theory. The talk is based on joint work with Matheus Brito.

Linda Chen

Quantum Hooks and the Plücker coordinate mirror of partial flag varieties

The quantum cohomology ring of a type $A$ flag variety $Fl(n,r_1,\ldots,r_p)$ is a deformation of the ordinary cohomology ring. I will discuss a natural map from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology ring of the partial flag variety. I’ll explain how for a large class of partitions, the image of the Schur polynomial is equal to a single Schubert class times a monomial in the deformation parameters $q_1, \dots ,q_p$; this is obtained by dividing the partition into a quantum-hook and smaller partitions. Surprisingly, this is the key result that proves a mirror theorem for type $A$ partial flag varieties. In particular, we use quantum hooks to prove that the Plücker coordinate mirror of the flag variety computes quantum cohomology relations.

This is joint work with Elana Kalashnikov.

Mode transition algebras and higher-level Zhu algebras

In today's talk I will describe how we can study the properties of a vertex operator algebra, and associated conformal blocks, through properties of an associative algebra associated to it, which we call the mode transition algebra (MTA). This algebra is particularly interesting when the vertex algebra is non rational. I will describe the properties of MTAs and show how they relate to higher-level Zhu algebras.

This is based on a joint work with A. Gibney and D. Krashen.

Laura Colmenarejo

Konhert diagrams and their not well-behaved poset structure

Given a diagram $D$ with unit boxes arranged in the first quadrant, we consider the set of diagrams formed from $D$ by applying certain moves, called “Kohnert moves,” which alter the position of at most one box. These diagrams defined the monomials appearing in the Kohnert polynomials, which generalized several families of well-known polynomials, including Key polynomials, Schur polynomials, and Schubert polynomials. In this talk, we will focus on combinatorial questions related to the sets of diagrams. In particular, we will present some enumeration results as well as some properties related to their poset structure, which is known to be not well-behaved. This is current work with Nick Mayers (postdoc at NCSU) and Etienne Phillips and Felix Hutchins (undergrad students at NCSU)

Rebecca Goldin

On positivity for flag manifolds and Hessenberg spaces

“Positivity” is a phenomenon involving the intersection of subvarieties in the presence of enough structure to ensure that intersection points are positively oriented, among other generalizations. It has a direct combinatorial interpretation, which has engaged mathematicians for at least 150 years, in one form or another. Such phenomena occur with frequency in the context of complex homogenous spaces such as flag varieties. In this talk, I will introduce and define positivity in a limited context, including some generalizations to the (torus) equivariant cohomology ring* of G/B, where G is a complex reductive group and B is a Borel subgroup. I will then turn to Hessenberg varieties, a special class of subvarietes of G/B. In some cases, these subvarieties also exhibit positivity properties, and have combinatorial formulas describing it. In other cases, positivity is unknown but lots of hints exist, leading to conjectural behavior. I will close with several open problems. The work I present consists of outcomes from various joint projects with L. Mihalcea, R. Singh, B. Gorbutt, M. Precup, and J. Tymoczko.

Seoyoung Kim

Diophantine $m$-tuples and elliptic curves

A set of positive integers is a Diophantine tuple of property $D(n)$ if the product of any two distinct elements in the set is n less than a square. Diophantine tuples were studied by a long list of authors that goes back to Diophantus who studied the quadruple $\{1,33,68,105\}$ with property $D(256)$. In this talk, I would like to discuss the recent progress and the generalizations of the problem. In particular, we also discuss a graph-theoretic approach to study the Diophantine tuples over a finite field.

Michael Lau

Georgia Benkart: A life in mathematics

Georgia Benkart was an exceptional mathematician and an exemplary member of the mathematical community. She is especially remembered for her fundamental work in algebra and combinatorics, and for her long and selfless service to the AWM, AMS, and other important institutions. On this sad anniversary of her passing, I will reflect on a few of her many contributions to research, to mentoring younger mathematicians, and to the growth and flourishing of the mathematical community in the U.S. and beyond.

Haydee Lindo

The Trace Property in Preenveloping Classes

In this talk we will discuss the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we use trace modules to characterize several classes of rings with a focus on the Gorenstein and regular properties.

Elizabeth Milićević

Folded Alcove Walks and their Applications

This talk will explain the tool of folded alcove walks, which enjoy a wide range of applications throughout combinatorics, representation theory, number theory, and algebraic geometry. We will survey the construction of both finite and affine flag varieties through this lens, focusing on the problem of understanding intersections of different kinds of Schubert cells. We then highlight several applications, including localizations in GKM theory, crystals for representations of the general linear group, and R-polynomials in Kazhdan-Lusztig theory.

Jennifer Morse

Hey Series, Have you herd of catalanimals?

We'll talk about the history of Macdonald symmetric functions and the Shuffle conjecture from the beginning to give inspiration for a more fundamental effort to develop symmetric function theory over the field $\mathbb Q(q,t)$. We will see how formulas for symmetric functions coming from infinite series have led us to solve a number of problems in this theory, as well as in quantum/affine Schubert calculus. We'll finish by showing some new combinatorial, representation theoretic and geometric questions naturally arise.

Joint work with J. Blasiak, M. Haiman, A. Pun, and G. Seelinger.

Rosa C. Orellana

Multiset tableaux and the Kronecker coefficients

The classical Schur-Weyl duality establishes a correspondence between the representation theory of the general linear group and the symmetric group via an action of both groups on tensor space. This duality establishes a correspondence between polynomial representations of the general linear group, symmetric functions and irreducible representations of the symmetric group.

Restricting the action of the general linear group to permutation matrices yields a Schur-Weyl duality between the irreducible representations of the symmetric group, the partition algebra, and symmetric function. This duality led to the discovery of a new basis of symmetric functions that evaluate to the irreducible characters of the symmetric group. The structure coefficients for this new basis are the stable Kronecker coefficients. In this talk I will use multisite tableaux to describe the coefficients that occur in products of several symmetric functions with our new basis. In particular, we describe a Pieri rule for the new basis of symmetric functions.

Joint work with Mike Zabrocki

Colleen Robichaux

Degrees of Grothendieck Polynomials and CM Regularity

We give an explicit formula for the degree of a vexillary Grothendieck polynomial. This generalizes a previous result of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt for degrees of symmetric Grothendieck polynomials. We further generalize this result for 321-avoiding unspecialized Grothendieck polynomials. We apply these formulas to compute the Castelnuovo-Mumford regularity of certain Kazhdan-Lusztig varieties and, in particular, the regularities of mixed two-sided ladder determinantal ideals. This is joint work with Jenna Rajchgot and Anna Weigandt.

Anne Schilling

Dimension of splines of degree two

Splines are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph $G$ by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form $(ax+by)^2$ and whose vertices are labeled by polynomials of degree at most two. We provide a formula for the dimension of splines in degree 2 when edge labels are distinct.

This is joint work with Shaheen Nazir and Julianna Tymoczko.

Julianna Tymoczko

The geometry and combinatorics of Springer fibers

The Springer fiber of a square matrix $X$ is a kind of generalization of an eigenspace: instead of looking at lines and asking which are fixed by $X$, we look at lines contained in planes contained in 3-dimensional linear spaces contained in (etc.), and ask when each of those nested linear subspaces is fixed by $X$. Springer fibers have beautiful geometry and also can be described very concretely in terms of the underlying linear algebra. At the same time, they are connected to deep mathematics: one of the classical examples of geometric representation theory shows that the cohomology of Springer fibers admits a representation of the symmetric group (or the Weyl group, in general Lie type). Like better-known Schubert varieties, the geometry of Springer fibers is deeply entwined with combinatorics. Unlike Schubert varieties, very little is known about even straightforward questions about this geometry. In this talk, we study the combinatorics and geometry of a particular family of Springer fibers that arise in combinatorics, representation theory, and knot theory. We give some results about how to partition these Springer fibers into affine (and non-affine) cells that are encoded by a kind of graph called a web.

Anna E Weigandt

Derivatives and Schubert Calculus

Schubert Calculus has its origins in enumerative questions asked by the geometers of the 19th century. Algebraic reformulations of these problems have led to a vast theory which studies symmetric polynomials and related tableau combinatorics. In this talk, we will discuss how to use derivatives to shed light on algebraic and combinatorial properties of families of polynomials.

The Isomorphism Problem for cominuscule Schubert Varieties

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. In this talk, I will report on joint work with E. Richmond and M. Tarigradschi showing Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic. Our proof is combinatorial in nature and type-independent. The talk is based on the arxiv preprint available at https://arxiv.org/abs/2302.11642