Combinatorics, Algebra, and Topology of Stanley-Reisner Rings

University Of Washington, Seattle WA

Investigators

Abstract

Algebraic and geometric combinatorics, and, in particular, the study of discrete objects that approximate high-dimensional shapes, is a rapidly developing field that has close connections to optimization, computer science, engineering, statistics, and mathematical biology. For instance, in engineering and computer science, e.g., robotics, one often needs to describe a space of "allowed motions." This space can usually be approximated by a collection of points, segments, triangles, pyramids, and higher dimensional analogs of pyramids nicely glued together -- an object known as a simplicial complex. A simplicial complex, in turn, can be encoded in a certain algebraic structure that involves polynomials. Study of this algebraic structure led to several spectacular developments in the field. This research project aims to deepen understanding of various aspects of simplicial complexes and the corresponding algebraic structures. The project aims to deepen our understanding of algebraic, combinatorial, and topological invariants of simplicial complexes through the study of their face numbers and Stanley-Reisner rings. Specifically, research on this project will attack fundamental questions related to (1) tracing various topological invariants (beyond the usual simplicial homology) in the Stanley-Reisner rings, (2) extending results on face numbers and Stanley-Reisner rings of triangulations of manifolds to the setting of normal pseudomanifolds, and (3) studying face numbers of simplicial complexes with an additional structure such as balancedness or symmetry. Thus, this project seeks to vastly improve our understanding of Stanley-Reisner rings (and modules) of complexes, especially when they are not Cohen-Macaulay or even not Buchsbaum. Consequently, it is expected that results of the project will impact not only algebraic and geometric combinatorics, but also commutative algebra, discrete geometry, and potentially even algebraic topology.