Topological and algebraic combinatorics of posets and stratified spaces

North Carolina State University, Raleigh NC

Investigators

Abstract

This research project is in discrete mathematics, namely, the area of mathematics which provides the theoretical underpinnings for computer science as well as more recently for some substantial parts of biology. The PI particularly focuses on developing novel ways of combining geometric and topological techniques and intuition with combinatorial methods. In recent years, the PI has become particularly focused on finding effective ways to study topological-combinatorial structures on spaces of real-valued matrices satisfying naturally arising constraints, for instance matrices in which the determinant as well as all minors are nonnegative. Such spaces arise both in areas of theoretical mathematics such as representation theory and also in applications areas. For instance, they play an important role to our understanding of the relationship between current and voltage in electrical networks. The more theoretical results can sometimes give surprisingly powerful insights into such applications. The project also includes a study of how configurations of distinct points may move around in space without bumping into each other, taking an abstract, representation theoretic perspective. The PI will also continue her work in helping develop the STEM pipeline both through the training of graduate students in combinatorics and also through organizing workshops and other activities to help inspire and foster the development of the next generation of scientists. The specific projects include: (1) analysis of the homeomorphism type of fibers of maps to totally nonnegative varieties; (2) stability properties for configuration spaces related to the partition lattice via a mixture of poset topology and symmetric function theory; (3) analysis of combinatorial topological structure on spaces of electrical networks; and (4) development of poset-theoretic approaches to polytope diameter bounds for particularly nice classes of polytopes, motivated by complexity questions from operations research regarding linear programming. Many of these projects are collaborative. This work builds upon the PI's past research in topological combinatorics, and particularly in poset topology and in combining ideas of geometric topology with those of combinatorics to study combinatorial topological structure of stratified spaces.