Fall 2022

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.


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.

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.

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.

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.


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.

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.

 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.


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.