NCSU Algebra & Combinatorics Seminar

Spring 2024

Title: Friezes of Dynkin type

Abstract: 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 of the simplest questions was open until recently: are there finitely many such friezes? Time permitting, I will describe two proofs of finiteness, and on-going work into enumerating them.



Title: Tensor categories of Lie type G2


Abstract: Let C be a semisimple ribbon tensor category with the same tensor product rules as the Lie algebra of type G2. We show that if the braiding morphism c_{V,V} for the object V corresponding to the smallest nontrivial representation of G2 has mutually distinct eigenvalues, then C is equivalent to Rep U_qg(G_2) for q not a root of unity.



Title: Pop-stack sorting and pattern-avoiding permutations


Abstract: 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 connections to representation theory. This talk will be accessible to all.


Title: Partitioned fans, hyperplane arrangements and K(pi,1) spaces

Abstract: 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 we associate the category of a partitioned fan. The classifying spaces of these categories turn out to be cube complexes whenever the fan is simplicial, and hence we are interested in studying their topological properties.

On the other hand, there is often a natural poset on the maximal-dimensional cones of a fan. From this poset and the partition, we can construct a group we call the picture group. Our guiding question is whether the classifying spaces of the categories are K(pi,1) spaces, in other words have non-trivial homotopy group only in degree 1, for this picture group. We give an example of a partitioned hyperplane arrangement and prove that the classifying space is a K(pi,1) space for the picture group obtained from the poset of regions.

Title: Categorical valuative invariants of matroids

Abstract: 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 what we know about valuative polynomials: several known valuative polynomials can be seen as Hilbert series of some interesting graded vector space (e.g. Orlik-Solomon algebras and Chow rings) and we prove that these graded vector spaces let us define a valuative functor in the new sense. 

Lastly, we sketch how to categorify a Theorem by Ardila and Sanchez, which states that the convolution of two valuative invariants (respectively, valuative functors) is again valuative.

This is based on a joint project with Ben Elias, Dane Miyata and Nicholas Proudfoot.



Title: Weight multiplicities of some affine Lie algebra modules


Abstract: 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 subclasses called the set of maximal dominant weights are finitely many.  In this talk we will show that for $\lambda = k\Lambda_0$ the multiplicities of these maximal dominant weights are given by the number of certain pattern avoiding permutations.


Title: Quantum Computing meets Algebra: a physicists' perspective

Abstract: 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 to experiments. To make this connection, we often measure dynamical correlation functions -- a correlation between two operators at two separate space-time points.

In this talk, I will present an overview of how to obtain such quantities from a quantum computer. I will give a brief overview of some hardware platforms, outline why producing the desired dynamics can be difficult, and how Lie algebras arise naturally in this context.
Following that, I will highlight some of our recent work using Lie algebraic methods to simulate dynamics on quantum computers. Synthesizing the corresponding quantum circuit is typically done by breaking the operator into small circuit elements, named Trotter decomposition, which leads to circuits whose depth often scales unfavorably. We present two algorithms to help overcome these difficulties. First, it is possible to synthesize exact quantum circuit representations of the desired time evolution unitaries via Cartan decomposition.Second, when the circuit elements of the Trotter decomposition are limited to a subset of SU(4), or equivalently, when the Hamiltonian may be mapped onto free fermionic models, several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the circuit elements into a single block of quantum gates, resulting in a fixed depth time evolution for certain classes of Hamiltonians.

Title: Coparking functions for matroids

Abstract: 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 for matroids which admit “circuits system”. Our construction of coparking functions is the dual version of Postnikov and Shapiro’s construction of G-parking functions on graphs. In this talk, we will explain the connection with Postnikov and Shapiro’s construction and explain the main idea for the proof which utilizes the deletion/contraction relation of Tutte polynomials. This is a joint work with Anton Dochtermann.

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 .

Title: Duality and blow-up algebras.

Abstract: 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 study and determine the defining equations of the Rees algebra of certain families of ideals. This is joint work with Claudia Polini (University of Notre Dame) and Bernd Ulrich (Purdue University). Link: https://arxiv.org/abs/2205.03837.

Title:  Highly transitive groups, hyperbolicity, and random walks

Abstract:  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.