Polyhedral Subdivisions in Combinatorics and Geometry

This project will study polyhedral subdivisions and their applications to different areas of mathematics. Polyhedra are higher dimensional versions of polygons. They include cubes, pyramids, prisms, and many other shapes. These objects have been studied since antiquity. Gluing polyhedra together forms polyhedral complexes, which are ubiquitous throughout mathematics, computer science, and engineering. For example, polyhedral complexes are used to subdivide a space into smaller, more manageable pieces. This research aims to further our fundamental understanding of these objects and will include the involvement of students. Three specific problems that will be looked at are: (1) Spaces of subdivisions and the Baues conjecture: examining how well certain combinatorially defined spaces can approximate moduli spaces from geometry. (2) Unimodular triangulations and lattice polytopes: the existence and construction of unimodular triangulations, and their applications to resolutions of singularities in algebraic geometry. (3) Hadwiger's covering problem, as seen through the lens of subdivisions, and applications to convex geometry. The PI plans to develop new tools to help strengthen the interplay between combinatorics, geometry, and topology. 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.