Geometry, Algebra, and Topology of Face Numbers

University Of Washington, Seattle WA

Investigators

Abstract

From antiquity, people looked at and studied objects such as polygons, pyramids, cubes, and their higher-dimensional generalizations called polytopes. Using these objects as building blocks and gluing them face-to-face, one may construct more involved objects known as polytopal complexes. For instance, any object obtained by gluing triangles, pyramids, and their higher-dimensional analogs is known as a simplicial complex, while an object obtained by gluing cubes of various dimensions is known as a cubical complex. The reason to study polytopes and polytopal complexes is explained by the fact that many continuous objects can be approximated by these more discrete structures. These objects also often show up in problems related to optimization, statistics, and engineering. For instance, the space of motions of a robot sometimes can be described by a cubical or simplicial complex. Simplicial complexes are also useful in describing patterns of intersections of convex sets, which, in turn, has 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 understanding of various aspects of simplicial complexes. The award provides support of research training of graduate students. The research project aims to attack several fundamental questions related to (1) studying the face numbers of centrally symmetric simplicial polytopes and spheres (2) extending the results on the face numbers and Stanley--Reisner rings of triangulations of manifolds to the setting of pseudomanifolds, and (3) exploring the effect of various topological invariants (beyond the usual Betti numbers) on the face numbers of manifolds and pseudomanifolds. For instance, while (as of a few months ago), the upper bound problem for centrally symmetric simplicial spheres is now completely resolved, there is not even a plausible upper bound conjecture for centrally symmetric polytopes; while we understand reasonably well the face-vectors of simplicial manifolds with and without boundary, very little is known about face-vectors of triangulations of spaces with singularities. While Betti numbers play a prominent role in the study of face-vectors, our knowledge on how other topological invariants (e.g., the fundamental group, characteristic classes, etc.) affect the face-vectors is next to nothing. The aim of this project is to attack these and related problems and in the process develop the necessary new combinatorial, geometric, and algebraic tools. 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.