Professor Alan David Sokal
University College London, New York University
My current research centers on topics at the interface between combinatorics and analysis. In particular, I am interested in novel and unexpected positivity properties that arise for some combinatorially defined polynomials and power series. Some of these properties are closely connected with the Stieltjes moment problem and with continued fractions.
I am also interested in the real and complex zeros of combinatorially defined polynomials such as the Tutte polynomial (and its specializations such as the chromatic polynomial and the flow polynomial), and their relation to phase transitions in statistical mechanics. Some of this work leads to the investigation of "classical" problems in complex analysis.
# Time
Tuesday, 15:30-17:30
June 17, 2025
# Venue
C654, Shuangqing Complex Building
# Abstract
A matrix $M$ of real numbers is called {\em totally positive}\/ if every minor of $M$ is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ of real numbers is totally positive if and only if the underlying sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence, i.e.~the moments of a positive measure on $[0,\infty)$. Moreover, this holds if and only if the ordinary generating function $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type continued fraction with nonnegative coefficients:
$$
\sum_{n=0}^{\infty} a_n t^n
\;=\;
\cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 - \cfrac{\alpha_3 t}{1- \cdots}}}}
$$
(in the sense of formal power series) with all $\alpha_i \ge 0$. So totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions.
Here I will introduce a generalization: a matrix $M$ of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive}\/ if every minor of $M$ is a polynomial with nonnegative coefficients. And a sequence $(a_n)_{n \ge 0}$ of polynomials will be called {\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficient wise Hankel-totally positive. In some cases this can be proven using continued fractions, by either combinatorial or algebraic methods; I will sketch how this is done. There is also a more general algebraic method, called {\em production matrices}\/. In a vast number of cases, however, the conjectured coefficientwise Hankel-total positivity remains an open problem.
