Questions and Experiments in Geometric Combinatorics

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Polytopes have been studied by mathematicians, scientists, and artists for thousands of years. The modern theory of polytopes connects diverse areas of pure mathematics and is used in applied mathematics as well. Polytopes are used to determine solutions of transportation problems, model possible outcomes of elections, and investigate biological phenomena. The special class of lattice polytopes have deep connections with combinatorial enumeration, which is the precise counting of complicated but finite sets. This award will lead to a deeper understanding of lattice polytopes and their relation to enumeration and counting, using tools primarily from number theory and algebra. In addition to expanding basic research in mathematics, this award will broaden and strengthen the mathematical sciences workforce by supporting the training of graduate students in both pure and experimental approaches to mathematics.This research is focused on two broad goals: (A) prove theoretical results about combinatorial properties of lattice polytopes and (B) generate and analyze data about lattice polytopes resulting from computational experiments. The first project in this proposal focuses on unimodality problems for Ehrhart h-star-polynomials, with particular emphasis on challenging conjectures due to Stanley, Hibi-Ohsugi, and De Loera-Haws-Koeppe. The study of Ehrhart h-star-unimodality has led to a significant increase in our understanding of lattice polytopes, and focusing on these difficult conjectures will lead to further advances. The second project in this proposal involves a comprehensive study of geometric and algebraic aspects of a family of lattice simplices related to weighted projective spaces. By investigating Hilbert bases, Ehrhart h-star-vectors, Poincare series, and geometric h-star-polynomial factorizations for these lattice simplices, a refined understanding of their properties will emerge that can inform our study of lattice simplices in general. The third project focuses on the class of generalized permutahedra and their relationship to Hopf algebras. The combinatorial enumeration of faces of these polytopes is known to be a subtle and interesting problem, and this project will create an extension of traditional face enumeration to refined face enumeration using symmetric functions and combinatorial Hopf algebras for this class of polytopes.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.