Combinatorics, Algebra, and Geometry of Simplicial Complexes

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This project is devoted to the study of polytopes, as well as simplicial and polytopal complexes. Polytopes are geometric objects that include polygons, pyramids, cubes, octahedra, and their higher-dimensional analogs. They have been looked at and studied since antiquity; at present, they play a role in such diverse areas of pure and applied mathematics as optimization, statistics, combinatorics, representation theory, symplectic geometry, to name just a few. Using polytopes as building blocks and gluing them along their faces, one creates polytopal complexes. If the polytopes used are line segments, triangles, pyramids, and their higher-dimensional generalizations, one obtains a special class of polytopal complexes known as simplicial complexes. These objects appear naturally in robotics, discrete geometry, and topology, since they provide a simple way to approximate continuous spaces, such as manifolds, by discrete objects. Simplicial complexes are also useful in describing patterns of intersections of sets. Specifically, patterns of intersections of convex sets have applications in such subjects as neuro-biology (e.g., in the study of neurons which are simultaneously active in response to some stimulus). This research project aims to deepen our understanding of various aspects of polytopes and simplicial complexes. The award will also provide support of research training for graduate students. The primary aim of this project is to gain new insights and enhance our understanding of combinatorial, algebraic, geometric, and topological invariants of simplicial complexes and polytopes through the study of their face numbers, face rings, and stress spaces, and, in the process, to develop new tools to achieve this. Specifically, research on this project will attack several fundamental questions related to (1) the upper bound type problems originated in but going far beyond the classical upper bound theorem for spheres, (2) the lower bound type problems for simplicial complexes, especially simplicial spheres, with an additional structure such as flagness, and (3) finding new construction techniques to produce many simplicial polytopes and spheres with interesting extremal properties.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.