Room 208, School of Arts and Sciences
Central Campus
A matrix M with real entries is said to be totally positive (TP) if all its square submatrices have non-negative determinants. We can extend this definition to matrices with polynomial entries where a polynomial is said to be coefficientwise non-negative if all coefficients are non-negative. This talk will begin with a quick introduction to total positivity. We will then introduce Hankel matrices and provide a characterisation of totally positive Hankel matrices due to Gantmakher and Krein and show its relation to the Stieltjes moment problem and to continued fractions. Next, we will give examples of sequences occurring in combinatorics whose Hankel matrices are totally positive. We then generalise these notions to coefficientwise total positivity. We will conclude this talk by mentioning some of the proof techniques in this area. If time permits, we will mention some of our recent results. No prerequisites will be necessary to follow the talk.
Bishal Deb is a postdoctoral researcher based at Laboratoire de Probabilités, Statistique et Modélisation (LPSM) in Sorbonne University and is employed by CNRS. He finished his PhD in 2023 at University College London under the supervision of Alan Sokal. His primary research interest is in numerative combinatorics, in particular questions on total positivity in enumerative combinatorics. In his current position, he is learning about models in statistical physics and their interactions with combinatorics.
