Fall 2022
12/03/2022 - Triangle Lectures in Combinatorics at UNC Greensboro
Please, check our website for more information and for the link to the registration form and to request funding.11/30/2022 - Mark Ebert (USC) Derived Superequivalences for Spin Symmetric Groups and Odd sl2-categorifications
Since Chuang and Rouquier's pioneering work showing that categorical sl(2)-actions give rise to derived equivalences, the construction of derived equivalences has been one of the more prominent tools coming from higher representation theory. In this talk, we explain joint work with Aaron Lauda and Laurent Vera giving new super analogues of these derived equivalences stemming from the odd categorification of sl(2). Just as Chuang and Rouquier used their equivalences to prove Broué's abelian defect conjecture for symmetric groups, we use our superequivalences to prove this long standing conjecture for spin symmetric groups.
11/23/2022 - Thanksgiving (No seminar)
11/16/2022 - Sinan Aksoy (PNNL) Non-reversible Markov chains and hypergraph data analysis
We explore tools for studying random walks on directed graphs and their application in hypergraph data analysis. We first highlight fundamental differences between random walks on directed versus undirected graphs via several results in extremal spectral graph theory. In particular, we focus on the stationary distribution and normalized Laplacian eigenvalues. We then apply these tools to develop a Laplacian based approach for hypergraph clustering. We discuss the challenges in devising tractable, applicable, and structurally faithful methods for analyzing hypergraph data, and conclude by mentioning future research threads.11/09/2022 - Qing Zhang (Purdue University) : Classification of Modular Categories by Galois Orbit Count
Given a modular category C, the irreducible characters of its fusion ring are in one-to-one correspondence with the set Irr(C) of isomorphism classes of simple objects of C. Consequently, the action of the absolute Galois group on these characters induces a permutation action on Irr(C). The analysis of this action is essential to the classification of modular categories, whether by rank or by other characteristics. Another essential property of a modular category is its associated (projective) SL(2,Z)-representation, whose kernel is a congruence subgroup. This gives another connection between the theory of modular categories and classical number theory. I will talk about the recent progress on the classification of modular categories using representation and number theoretic methods. Then I will discuss the results regarding the classification of modular categories according to the number of Galois orbits. The talk is based on joint work with Siu-Hung Ng and Yilong Wang; and separate joint work with Julia Plavnik, Andrew Schopieray, and Zhiqiang Yu.
11/02/2022 - Radmila Sazdanovic, NCSU: Categorification: knots, graphs and more
Categorification is a method that has many emanations hence eludes a precise definition. Therefore, we will discuss categorification through several examples of categorifying polynomials arising from different fields of mathematics, including knots, graphs, and orthogonal polynomials.
10/26/2022 - Sarah Mason, Wake Forest University: Combinatorics of the noncommutative inverse Kostka matrix
The inverse Kostka matrix transitions between two important symmetric function bases (Schur functions and complete homogeneous symmetric functions). Eğecioğlu and Remmel's combinatorial interpretation for its coefficients uses objects called "special rim hooks" to decompose partition diagrams. We generalize their construction to the space of noncommutative symmetric functions (NSym) by first extending composition diagrams to include all integer sequences and then introducing "tunnel hook fillings" which decompose these diagrams. Statistics on tunnel hook fillings provide the coefficients for expanding the "immaculate basis" (a generalization of Schur functions to NSym introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki) into the complete homogeneous noncommutative symmetric function basis. We show how to use our formula to expand monomial quasisymmetric functions into dual immaculates, and further apply our formula to expand upon Campbell's ribbon decomposition formulas.
10/19/2022 - Nathan Reading, NCSU: Noncrossing partitions of classical affine type
The story of noncrossing partitions starts with a Coxeter group W anda Coxeter element c. (If you're not familiar with Coxeter things, think: W is the symmetric group S_n and c is an n-cycle.) The noncrossing partitions in W are the elements of the interval [1,c]_T in a certain partial order on W (the absolute order). When W is a finite Coxeter group, [1,c]_T is a lattice. In the classical finite types, there are planar diagrams for noncrossing partitions. When W is infinite, [1,c]_T may not be a lattice. In the affine case, McCammond and Sulway embedded W into a larger group in which an analogous interval [1,c] is a lattice (thus proving some longstanding conjectures for Euclidean Artin groups). Their key idea was to factorthe translations in [1,c]_T. I'll discuss work (including work from Laura Brestensky's thesis) to construct planar diagrams for noncrossing partitions of classical affine types. The planar diagrams "know" how to factor translations, and thus yield new insights into the McCammond-Sulway construction. I'll discuss all of this (assuming no prior knowledge of Coxeter groups, Artin groups, or noncrossing partitions) and may mention some generalizations where the topology of the diagrams gets more interesting. Much of this is joint work with Laura Brestensky.10/12/2022 - Ashley Tharp, NCSU: Arcs and shards
The group of permutations is the canonical example of a finite Coxeter group, and each permutation can be represented visually by a noncrossing arc diagram. Each diagram encodes the canonical join representation of its permutation, and diagrams can be used to understand lattice congruences on the weak order of type A, equivalence relations that respect the meet and join. This talk will provide an overview of existing results on noncrossing arc diagrams for permutations and present two new constructions of noncrossing arc diagrams which relate a given pair of diagrams. We state dual theorems about the relevance of these constructions to the shard intersection order of type A, a partial order weaker (given by fewer relations) than the weak order. We also present analogs of noncrossing arc diagrams (joint with Barnard and Reading) for signed permutations (type B) and even-signed permutations (type D). These results lead to work in progress exploring sublattice relationships between shard intersection orders on different finite Coxeter groups.
10/05/2022 - Seth Sullivant, NCSU: Maximum Agreement Subtrees
Probability distributions on the set of trees are fundamental in evolutionary biology, as models for speciation processes. These probability models for random trees have interesting mathematical features and lead to difficult questions at the boundary of combinatorics and probability. This talk will be concerned with the question of how much two random trees have in common, where the measure of commonality is the size of the largest agreement subtree. The case of maximum agreement subtrees of pairs of random comb trees is equivalent to studying longest increasing subsequences of random permutations, and has connections to random matrices. This elementary talk will try to give a sense of what is known (not very much) and what is unknown (lots!) about this problem.
09/28/2022 - Nicholas Russoniello, William & Mary University: Contact seaweeds
A $(2k+1)-$dimensional Lie algebra $\mathfrak{g}$ is called contact if it admits a one-form $\varphi\in\mathfrak{g}^*$ such that $\varphi\wedge(d\varphi)^k\neq 0$; such a $\varphi$ is called a contact form and defines a contact structure on the underlying Lie group $G$. The construction and classification of contact structures on Lie groups is a basic problem in differential topology, and in this talk, we consider the classification problem restricted to ``seaweed" Lie algebras. A seaweed Lie algebra is one that can be most simply defined by a pair of compositions of $n,$ and each seaweed is equipped with an associated planar graph, called a meander. Our main results are a combinatorial method for constructing contact forms on seaweeds in types A and C, and a full, combinatorial and algebraic classification of contact seaweeds.
09/21/2022 - Jianping Pan, NCSU: A bijection between K-Kohnert diagrams and reverse set-valued tableaux
Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set- valued tableaux (RSVT) rule for Lascoux polynomials and reverse semistandard Young tableaux (RSSYT) rule for key polynomials. Besides, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with RSSYT. Ross and Yong introduced K- Kohnert diagrams, which are analogues of Kohnert diagrams. Ross and Yong conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between RSVT and K-Kohnert diagrams.
This is joint work with Tianyi Yu and the preprint can be found on arXiv:2206.08993.
09/14/2022- Bojko Bakalov, NCSU: An algebraic approach to 2-dimensional conformal field theory
In this talk, I will introduce several different algebras that appear in 2D conformal field theory (CFT). These include infinite-dimensional Lie algebras such as affine Kac-Moody algebras and the Virasoro algebra. Their commutation relations can be encoded in a Lie bracket depending on a formal variable, which leads to the notion of a Lie conformal algebra. As a further generalization, a vertex algebra axiomatizes the algebraic properties of quantum fields in 2D CFT. Finally, logarithmic CFT gives rise to logarithmic vertex algebras, which were recently developed in a joint work with Juan Villarreal.
09/07/2022- Corey Jones, NCSU: K_0 groups of associative algebras.
In this expository talk, we will give an elementary introduction to K_0 groups of associative algebras. We will consider several examples and sketch the proof of Elliott's theorem, which states that inductive limits of semi-simple algebras are completely classified by their (ordered) K_0 groups.
08/31/2022- Nick Mayers, NCSU: On toral posets and contact Lie poset algebras.
Type-A Lie poset algebras are subalgebras of sl(n) that are defined by an associated poset. Recent work has focused on using the associated poset to identify those type-A Lie poset algebras with certain algebraic properties, e.g., those which are “contact.” A (2k + 1)−dimensional Lie algebra is called contact if it admits a one-form φ such that φ ∧ (dφ)^k = 0. In this talk, we discuss results concerning a combinatorial recipe for constructing a large family of posets whose associated type-A Lie poset algebras are contact. Restricting attention to posets with chains of cardinality at most three, we find that the aforementioned combinatorial recipe provides a complete characterization of posets generating contact, type-A Lie poset algebras. In concluding, we will see how this investigation can be extended to the other classical families of Lie algebras.08/24/2022- Beginning of the semester meet and greet.