Volumes, Ehrhart polynomials and valuations of polytopes

University Of California-Davis, Davis CA

Investigators

Abstract

This project proposes research on polytopes, one of the central subjects of geometric combinatorics. There are many aspects of polytopes one can study. This proposal is focused on the volume and number of lattice points of polytopes as well as two tools for connecting these subjects: Ehrhart polynomials and valuations. Based on her previous work, the PI plans to investigate polytopes with the property that their Ehrhart coefficients can be written in terms of volumes or more generally are positive, study the Ehrhart coefficients of k-integral polytopes (a family of polytopes defined in the PI's recent work) from different perspectives, and further develop two methods for volume computations. The PI will also study the intermediate generating function, which is a valuation on polyhedra, and work on generalizing Linke's real Ehrhart theorem to a theory on valuations. Polytopes are higher-dimensional generalizations of polygons. An important topic of study for polytopes is their volume. Also important are their lattice points, i.e., points whose coordinates are whole numbers. Two tools for studying these concepts are Ehrhart polynomials and valuations. They have connections not only to combinatorics, but also to algebra, algebraic geometry, statistics and number theory. Progress in any direction the PI proposes can lead to either explicit descriptions of formulas for volumes or numbers of lattice points, or better understanding of other aspects of polytopes, and therefore benefits related areas. The proposed research has the potential to lead to new algorithms and applications outside of pure math. For instance, both lattice points counting and volume computations have direct relevance to aspects of statistical sampling, and some of the valuations discussed in the proposal have applications in optimization. In fact, the PI's recent work already leads to a new algorithm for computing volume. In general, these problems are sufficiently accessible that they may be integrated into course material and student research projects.