RUI: Quantum, arithmetic, and categorial analysis of convex polytopes

San Francisco State University, San Francisco CA

Investigators

Abstract

Lattice polytopes and convex polytopes form a natural habitat for a major part of the contemporary combinatorics. The first research direction, proposed in this project, involves a radically new idea of using analogies with quantum probabilistic physics. It introduces a new topological context for normal lattice polytopes. The second direction develops essentially new techniques for studying general convex polytopes. It is based on advanced categorial and homological machinery. The first line of research aims at shedding new light and making substantial progress on several outstanding open problems, such as unimodular triangulations and covers, Caratheodory rank, higher syzygies of the corresponding toric rings. The new idea of attacking these challenges draws from a physical interpretation of special discrete point configurations. Concrete suggestions in this direction to the PI were made by theoretical physicists and further connections to the physical context will be paid special attention. The second direction focuses on the category of convex polytopes and their affine maps. The goal is to devise a universal technique, providing a general context for various recent important constructions and results in polytope theory. The categorial approach leads to very concrete problems in classical polytopal combinatorics. Combinatorics is the science of organizing, arranging and analyzing discrete data. An illustrative example is the set of integer points in a convex polygon in the plane or in a convex polytope in the space. Combinatorics of lattice polytopes studies such point configurations to encode important constructions in algebra, geometry, and topology, while combinatorial methods are well suited for related computations. The interaction of combinatorics and abstract mathematical techniques, central to the proposed research, over the last decades has resulted in a number of fundamental theorems in a variety of disciplines. The proposal explores some of the deepest problems in polytopal combinatorics, employs ideas from a variety of disciplines and unexpected links between them, and offers concrete strategies for attacking central open questions. Applications range from algebraic geometry (the science of solution sets to systems of multidimensional polynomial equations) to integer programming/computer science, probability theory, and physics. The progress would have been unimaginable without computer assisted investigation and experimentation, the increased importance of which is related to the demand for algorithmic understanding of discrete structures. Both proposed research lines incorporate a strong - sometimes, crucial - computational element. Developing and implementing related algorithms is an excellent possibility for involving beginning graduate students.