Speaker’s webpage: https://math.sciences.ncsu.edu/people/jpmather/

This talk will introduce cluster algebras, with an emphasis on their combinatorics, and describe a recent joint result with Vincent Pilaud and Sibylle Schroll. At the heart of a cluster algebra is a complicated, branching recursion that defines cluster variables (certain rational functions organized into finite sets called clusters). The recursion looks bizarre at first…

Quantum cellular automata (QCA) are models of discrete-time unitary dynamics of quantum spin systems. They can be characterized algebraically as certain automorphisms of the associative algebra generated by local observables of a spin system. We will give a gentle introduction to this topic, and explain some of our recent contributions to the problem of classification…

Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231.  The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C'_w(q)$ of the (type-A) Hecke algebra. In particular, for…

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t…

In 1995, Stanley introduced the chromatic symmetric function of a graph, a symmetric function analog of the classical chromatic polynomial of a graph. The Stanley-Stembridge e-positivity conjecture is a long-standing conjecture that states that the chromatic symmetric function of a certain class of graphs, called incomparability graphs of (3+1)-free posets, has nonnegative coefficients when expanded…

We will define quivers of type A-tilde, their representations, and exceptional collections of these representations. We will then introduce a combinatorial model of these representations, based on the one constructed by Garver, Igusa, Matherne, and Ostroff for type A, by drawing strands on a copy of the integers. We will see that collections of strands…

Lam, Lee, and Shimozono (LLS) recently introduced backstable double Schubert polynomials to represent classes in the cohomology ring of the infinite flag variety. Using these polynomials, they introduce double Stanley symmetric functions, which expand into double Schur functions with polynomial coefficients called double Edelman--Greene coefficients. They prove that double Edelman--Greene coefficients are Graham positive. For…

A group is highly transitive if it admits a faithful, highly transitive action, that is an action which is k-transitive for all k>0. We will discuss some algebraic properties of these groups, as well as constructions of highly transitive actions for hyperbolic groups (and a wide array of generalizations of hyperbolic groups) using random walks.…

We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein. As a main consequence, we obtain a new general framework to investigate blowup algebras. We use our results to…

Fun activity: Prepare a 5-10 minutes presentation on a math object that you want others to know about. That could include their definition, a couple of examples, some fun fact or property, an open problem, etc. This is meant to be an informal and fun discussion .

In Stanley’s seminal work “Cohen-Macaulay Complexes”, Stanley conjectured that all h vectors of matroid complexes are pure O-sequences. We constructed coparking functions on matroids with extra restrictions and showed that the degree sequences of coparking functions are the same as h vectors of matroid complexes. By this construction, we proved that Stanley’s conjecture is true…

 Quantum hardware has advanced to the point where it is now possible to perform simulations of small physical systems. Although the current capabilities are limited, given the rapid advancement it is an opportune time to develop novel algorithms for the simulation of quantum matter, and to develop those that make it possible to make connections…

Consider the affine Lie algebra $\mathfrak{g}$ associated with the simple Lie algebra $sl(n)$ consisting of $n\times n$ trace zero matrices over the field of complex numbers. For every dominant integral weight $\lambda$ there is a unique (upto isomorphism) irreducible highest weight $\mathfrak{g}$ module $V(\lambda)$. Although there are infinitely many weights of this module, certain important…

Speaker’s webpage: https://spdaugherty.github.io/

In this talk we define a new category of matroids, by working on matroid polytopes and rank preserving weak maps. This lets us introduce the concept of  categorical valuativity for functors, which can be seen as a categorification of the ordinary valuativity on matroid polytope decompositions. We also show that this new theory agrees with…

Speaker's website

Polyhedral fans are geometric objects, which arise naturally in many areas of mathematics, for example in toric geometry, the theory of hyperplane arrangements and representation theory. In many cases, there are natural ways of identifying some of the polyhedral cones defining a fan, thus giving a "partition of the fan". To each such partitioned fan…

The pop-stack sorting method takes an ordered list or permutation and reverses each descending run without changing their relative positions. In this talk we will review recent combinatorial results on the pop-stack sorting method, and we will extend the pop-stack sorting method to certain pattern avoiding permutations, called c-sortable. If time permits, we will describe…

A "frieze" is an infinite strip of numbers satisfying certain determinantal identities, or any one of several generalizations of this idea. In this talk, I will give an introduction to friezes whose shape is determined by a Dynkin diagram (motivated by their exceptional properties as well as connections to representation theory and cluster algebras). One…