Topological and Algebraic Combinatorics of Posets and Stratified Spaces

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This project is in combinatorics, namely the area of math that provides the theoretical underpinnings for computer science and also arises naturally in other areas such as the study of DNA and RNA mutation in biology. This project in particular aims to develop new ways to combine combinatorial techniques with methods in topology. There is a rich tradition of such interactions that really began to flourish starting in the 1970's, the starting point being the observation that counting by inclusion-exclusion (namely counting things up via Venn diagrams and related enrichments) may be accomplished topologically. The PI has begun bringing more geometric topological tools to the mix in addressing offshoot questions about rather mysterious spaces, including spaces of real-valued matrices. Her techniques involve breaking the spaces into more manageable pieces and analyzing structures on these spaces called partially ordered sets (posets) for organizing the pieces, using geometric ideas to study how pieces glue together. The new work will focus on spaces arising from electrical networks, spaces coming from totally nonnegative matrices (i.e. matrices of real numbers where all subdeterminants are nonnegative), generalizations of these, and other related stratified spaces which seem to share important features also making them seem likely amenable to somewhat the same type of analysis. In addition to carrying out this research, the PI also will continue her efforts to encourage and train young mathematicians, including members of underrepresented groups, and to disseminate her results.Expressed in somewhat more technical terms, the PI will study the combinatorial and topological structure of partially ordered sets and of stratified spaces related to electrical networks, topological and combinatorial questions related to the Bruhat graph and generalizations of this, electrical analogues of subword complexes and Kazhdan-Lusztig polynomials, subword complexes and their interior dual block complexes and generalizations of these, fibers of important maps seemingly sharing with each other key structures encapsulated in part via the structure of subword complexes and generalizations thereof, as well as questions about polytope diameter and complexity of the simplex algorithm for linear programming approached via poset-theoretic methods. Other methods to be used include shellability, the application of poset-theoretic results of Dyer, Brenti and others as well as potential new extensions of these, methods the PI began to develop in her work on regular cell complexes in total positivity, new definitions suggested by the PI which she believes give helpful new perspective, and the PI's new techniques for combining poset-theoretic methods with tools of discrete geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.