Geometric Combinatorics and Discrete Morse Theory

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Discrete geometry deals with objects that have corners and ridges, such as cubes or surfaces sewn out of triangles. Objects of this type are called polytopal complexes and provide a general approach to the study of other fields of mathematics, such as differential geometry and algebraic topology. The main advantage of discretizing is the possibility to leverage computational tools. Simple questions on the structure of polytopes, such as the Hirsch conjecture, have foundational importance in optimization. Meanwhile 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. Further tools arise from the study of networks, which are polytopal complexes of dimension one. A plan is to integrate in this line of work new random models, with the goal of revealing fundamental properties of intersection patterns of algebraic varieties.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 recently 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.