Geometric Combinatorics and Discrete Morse Theory

University Of Miami, Coral Gables FL

Investigators

Abstract

Discrete geometry deals with objects that have corners and ridges, such as cubes, pyramids, or surfaces sewn out of triangles. Objects of this type are called polytopal complexes and have been studied since the beginning of mankind. They also provide a schematic way to model real-world interactions. When the interactions are binary, i.e. when they involve only two agents at the same time, the resulting model is one-dimensional, and called a "network" or "graph"; when the interactions are not binary, the model has to be higher dimensional. For example, "friendship" is a binary interaction: That is why social networks are, indeed, networks. In contrast, "co-starring in a movie" is not a binary interactions: For a group of actors, the fact that "any two of them co-starred in a movie" does not mean that "they all co-starred in a movie". To distinguish the two situations, one should place a d-dimensional simplex whenever d+1 actors have co-starred in a movie, thereby creating a higher-dimensional model. The main advantages of using discrete models is that they adapt well to many deep areas of mathematics, bringing to such areas the option to leverage computational tools. Simple questions on the structure of polytopes, such as the Hirsch conjecture, have foundational importance in optimization. Discrete Morse theory, a reduction tool to simplify a given polytopal complex, is employed both in pure mathematics and in big data analysis, to understand high-dimensional shapes. The project builds on these tools and expands their application. Combinatorial and probabilistic approaches may shed light towards classical aspect of geometry, like intersection patterns of lines on smooth surfaces. Enumerative aspects on the number of polytopes and spheres with given number of facets have importance beyond pure mathematics, in Regge calculus and simplicial quantum gravity. Techniques from metric geometry can provide desired exponential upper bounds. Another problem with importance in applied mathematics is the polynomial Hirsch conjecture, which was proven with metric methods for flag polytopes. Techniques ranging from knot theory to differential and hyperbolic geometry may be applied to better understand obstructions and constructions in Discrete Morse Theory, thereby revealing when and how we can simplify a given shape. A new perspective in this project is to connect discrete Morse theory with the notion of embeddability. A further goal is to lift the classical theory of polytope graphs (for example Balinski's theorem or the Hirsch conjecture) into a more general theory of intersection patterns of algebraic varieties, where algebraic tools such as liaison theory and local cohomology can be employed. Integrating this theory with the study of random simplicial complexes may provide some new random models in commutative algebra. 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.