Events

Title: Control over the Kerov-Kirillov-Reshetikhin bijection with respect to the nesting structure on rigged configurations

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Abstract: The talk will discuss controlling the Kerov-Kirillov-Reshetikhin (KKR) bijection between semistandard tableaux and rigged configurations with a particular emphasis on the standard case. We introduce theorems and techniques to control the shape of the first rigged partition. We also introduce an operation on a standard tableau which induces a very small, very controlled change in the riggings of the corresponding rigged configuration. Despite how specific this operation seems, it can be used to manipulate all the riggings on the first rigged partition of a rigged configuration. It can also be used to give an alternate method to Kuniba et al. in order to "unwrap" the natural nesting structure on rigged configurations. The connection to the multi colour box ball system is discussed.

Title: Oddities in the pursuit of self-duality

Abstract: Certain important cones in conic optimization are self-dual, e.g., the positive semidefinite matrices and the nonnegative orthant. In this talk we will address the problem of determining which convex cones are self-dual in a broad sense, which allows changing the underlying inner product if necessary. We will describe some recent progress on this question for polyhedral cones and discuss some unexpected connections. In particular, we will show a curious result connecting self-dual polyhedral cones and extreme rays of doubly nonnegative matrices. We will also discuss how semidefinite programming can be used to search for self-dual polyhedral cones. Time allowing, we will describe new results on the related problem of determining the automorphisms of certain cones of interest.

This talk is based on joint works with João Gouveia and Masaru Ito.

Title: Lecture hall graphs and the Askey scheme

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Abstract: We establish, for every family of orthogonal polynomials in the Askey scheme and the q-Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall lattice. This generalizes to all families of orthogonal polynomials in the Askey scheme previous results of Corteel and Kim for the little q-Jacobi polynomials. This is joint work with Sylvie Corteel, Bhargavi Jonnadula, and Jon Keating.

Title: Everything is possible: constructing convex sets with prescribed facial dimensions, efficiently

Abstract: Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. In this work, we show that such sets can be realised as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral.

Ramsey numbers and configurations of finite polar spaces

Abstract: This talk is on some joint work with Anurag Bishnoi and Ferdinand Ihringer, about a simple observation on how Ramsey theory relates to certain induced subgraphs of collinearity graphs arising from finite polar spaces; the natural geometries for the finite simple groups of classical Lie type

A combinatorial proof of an identity involving Eulerian numbers

Abstract:In 2009, Brenti and Welker studied the Veronese construction for formal  power series which was motivated by the corresponding construction for  graded algebras. As a corollary of their algebraic computations, they  discovered an identity for the coefficients of the Eulerian polynomials.  The authors asked for a combinatorial proof of this identity given that  all of its ingredients are enumerative in nature. In this talk I will present one such combinatorial proof. I will gladly do a pre-seminar with the motivation and preliminaries for the talk!

Title: Stable Matchings and a Matroid Generalization

Abstract: Today we'll continue our theme of matchings and talk about stable matchings! We won't assume much previous experience with stable matchings, and we will (re)introduce what they are. After, we will talk about classic results and some more recent approximations for generalizations of the problem. In the classic stable matching problem, we are given a bipartite graph and for each vertex we are given a list of strict preferences over other vertices. The goal is to find a "stable" matching, where no two vertices would prefer being matched to other vertices. This can be accomplished using the classic Gale-Shapley algorithm, which we will review. We will also consider when ties and indifferences can be present in the list of preferences. With such preferences, the problem becomes APX-Hard. However, McDermid showed it is possible to achieve a 1.5 approximation. We will talk about this, and comment on a recent generalization to matroids from Csaji, Kiraly, and Yokoi.

Sum of squares of positive eigenvalues

The spectral Turán theorem says that if a graph has largest eigenvalue $\lambda_1$, $m$ edges and clique number $\omega$, then $\lambda_1^2 \leq 2m (1-\frac{1}{\omega})$. This result implies the classical Turán bound $m \leq (1-\frac{1}{\omega})\frac{n^2}{2}$.
In this talk, we present the proof of the Wocjan, Elphick and Anekstein conjecture in which, in the spectral Turán bound, the square of the first eigenvalue is replaced by the sum of the squares of the positive eigenvalues and the clique number is replaced by the vector chromatic number. 
We will also present recent progress towards a conjecture by Bollobás and Nikiforov in which, in the spectral Turán bound, the square of the first eigenvalue is replaced by the sum of the squares of the two largest eigenvalues. This is joint work with Gabriel Coutinho and Shengtong Zhang.

Tubings of rooted trees and resurgence

Abstract:

I will explain about how tubings of rooted trees can solve Dyson-Schwinger equations and how, when the Mellin transform is a reciprocal of a polynomial with rational roots, then one can extend the notion of, tubings to label the tubes with letters from some alphabets and from there just by standard generatingfunctionology obtain a system of differential equations that is perfectly suited to resurgent analysis.

Joint work with Michael Borinsky and Gerald Dunne, arXiv:2408.15883

There will be NO pre-seminar on September 19.

Title: LP based approximation algorithm for an stochastic matching problem

Abstract: Consider the random graph model where each edge e has a fixed weight w_e and it is independently present in the graph with probability p_e​. Given these probabilities, we want to construct a maximum weight matching in the graph. One can only determine if an edge is present by querying it, and if an edge is present, it must be irrevocably included in the matching. Additionally, each vertex i can be queried no more than t_i times. The goal is to device an adaptive policy (algorithm) to query the edges of the graph one by one in order to maximize the expected weight of the matching.

In this talk we present an elegant LP-based constant-factor approximation algorithm with respect to the optimal adaptive policy for the problem.

This is one of the results due to Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra, in their paper "When LP is the Cure for your Matching Woes" from 2011.