Geometric Combinatorics in Polytopes and Spheres

A polytope is the convex hull of finitely many points in the space. The ancient Greeks studied polytopes such as the Platonic solids as they are ideal to model nature. In modern days, scientists have found many applications of polytopes in diverse fields such as optimization and computer science. This research project focuses on the combinatorial “invariants” of polytopes. For example, count the number V of vertices, E of edges, and F of facets of an arbitrary 3-dimensional polytope. Then no matter which polytope we choose, we always end up with getting the identity “V-E+F=2”. The goal of this research project is to develop new methods to study various invariants of polytopes and spheres that arise from face numbers or other combinatorial data. These tools may further extend our understanding of the interplay between combinatorics, algebra, and geometry. One of the central conjectures in geometric combinatorics was the g-conjecture; that is, to characterize the face numbers of simplicial polytopes and spheres of all dimensions. This conjecture was only proved very recently, and its resolution requires deep results from other fields such as commutative algebra and algebraic geometry. This project is dedicated to new methods to study polytopes and manifolds with particular geometry or topology. One goal is to investigate various combinatorial models such as the Stanley-Reisner ring and the stress spaces, and how the algebra translates into combinatorial relations among the face number. Another goal is to investigate how preset geometry and topology (for example, central symmetry) affects the combinatorics of polytopes or polyhedral complexes, and vice versa. The project has applications to computer sciences and the PI also plans to develop lecture notes and work with students. 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.