Positivity problems at the boundary between combinatorics and analysis

组合学是数学的一个分支，涉及对各种类型的有限结构(排列、图等)进行计数；它在计算机科学、统计物理、分子生物学和许多其他领域都有应用。相比之下，分析是与连续变化(即实数或复数的函数)有关的数学分支；它在科学和工程的几乎所有领域都有应用。拟议的研究位于组合学和分析之间：它涉及使用组合工具研究分析问题，反之亦然。更具体地说，这项研究包括三个主题，所有主题都旨在探索组合学和分析之间的新的正性性质。第一个主题涉及组合重要的多项式的逆幂具有正系数的泰勒展开的情况。第二个主题是研究由组合重要多项式序列组成的某些矩阵具有“全正”性质的情形。第三个主题是研究由组合重要多项式(例如连通图的计数多项式)组成的某些幂序列具有正系数的情况。后一个性质在很多情况下都被PI经验性地发现了，但大多数这些性质还没有被证明，其更深层次的意义还有待阐明。

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.