CAREER: Geometric and Topological Combinatorics

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Geometry and topology are mathematical branches that provide methods that measure phenomena that are global instead of local. If a problem's resolution depends on the aggregate of its information then geometric methods are useful in detecting global obstructions. Numerous problems across mathematics and its applications are global in this sense, ranging from data science to economics. The research supported by this grant will develop new topological and geometric methods to tackle problems further afield, primarily discrete, non-continuous problems. This promises new insights at the confluence of combinatorics, geometry, and topology. Students at all stages will be involved in the research effort supported by this grant. Among the general goals of the research program are the following: In applications of equivariant topology one often requires that a certain parameter has to be a prime power, and methods fail outside of this prime power case. Recent research of the PI has suggested that one may effectively circumvent this prime power requirement via a synthesis of topological and combinatorial techniques. This will be further developed. The application of topological methods brings about geometric generalizations of combinatorial problems. To understand the limitations of topological techniques one has to study those types of problems, where geometric results deviate considerably from their combinatorial special cases. A central goal will thus be to delimit rigid combinatorial results from their flexible geometric counterparts. In addition to showing the existence of a solution to some combinatorial problem, one is often interested in quantifying how rich the space of solutions is. General topological methods will be developed that provide lower bounds for the topology of the space of solutions of a given combinatorial problem. 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.