Future students

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.

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!

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

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.

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.

Location: MC 5479

There will be a social starting at 1:00 pm.

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.

Location:  MC 5479

There will be a social starting at 1:00 pm.

This seminar will consist of 3 short talks by undergraduate students. There will be a coffee break at 1:30 pm.