Future students

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.

Title: NC algorithm to find perfect matching in planar graphs

Abstract: Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution.