Positivity problems at the boundary between combinatorics and analysis

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Combinatorics is the branch of mathematics concerned with counting finitestructures of various types (permutations, graphs, etc.); it has applications in computer science, statistical physics, molecular biology,and many other fields. Analysis, by contrast, is the branch of mathematics concerned with continuous variation (i.e. functions of real or complex numbers); it has applications in nearly all fields of science and engineering.The proposed research lies at the interface between combinatorics and analysis: it involves using combinatorial tools to study analytic problems, and vice versa. More specifically, the proposed research comprises three themes, all of which are aimed at exploring novel positivity propertiesthat arise at the interface between combinatorics and analysis.The first theme involves studying situations in which inverse powersof combinatorially important polynomials have Taylor expansions withpositive coefficients. The second theme involves studying situations in which certain matrices formed from sequences of combinatorially important polynomials have a property called "total positivity".The third theme involves studying situations in which certain powerseries formed from combinatorially important polynomials (for example,the counting polynomials of connected graphs) have positive coefficients.This latter property was discovered empirically by the PI in manysituations, but most of these have not yet been proven, and their deeper meaning remains to be elucidated.