CAREER: Discrete Geometry at the crossroads of Combinatorics and Topology

Many computational and combinatorial problems can be reduced to problems in geometry. For example, from a data set, we can generate families of points or lines in the plane. This leads us to study the structure of finite families of geometric sets. Surprisingly, many breakthroughs in this area use tools from equivariant topology, which may, at first glance, seem far detached from the original goal. This project will reinforce the connections between computational geometry, combinatorics, and topology. Success in the objectives of this project will illuminate new connections between these areas. The project will also establish a program to prepare students from minoritized groups for summer research opportunities in mathematics. The topological problems of this proposal deal with high-dimensional mass partition problems, such as the Stone-Tukey theorem. The project focuses on establishing connections with general fair partition problems (related to envy-free partitions) and problems related to the intersection patterns of convex sets. These are areas in which crucial results use similar methods. This project will show that said similarity is a consequence of a deeper connection between the families of problems listed. This work will yield new insight into how one can leverage topological tools to solve combinatorial problems. The project also includes computational applications to problems in supervised learning. 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.