Date: August 23rd
Time: 10:30 am to 11:50 am
Venue: Lecture Hall 6
Combinatorics and Total positivity
University College London, UK.
A matrix of real numbers is said to be totally positivity if all its minors (determinants of all square submatrices) are non-negative and a matrix of polynomials with real coefficients is said to be coefficientwise totally positive if all its minors are polynomials with non-negative coefficients. We begin the talk by introducing the notion of total positivity and look at some basic properties and examples. We shall then look at three different settings that one is interested in and mention proof techniques for each: Toeplitz matrices, Hankel matrices and lower triangular matrices.
We conclude by looking at the lower triangular matrix of Eulerian numbers whose total positivity was conjectured by Brenti in 1996 and is still widely open! We look at some generalisations of the conjecture that we obtained using extensive computer experimentation. This work is based on ongoing joint work with Xi Chen, Alexander Dyachenko, Tomack Gilmore and Alan D. Sokal.
The talk will be largely accessible to undergraduates.
Bonus: There may may be memes!