RUI: Applications to Ehrhart theory

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A program of research related to discrete volume computation for rational polytopes is proposed for support under the NSF RUI program in Combinatorics. Here discrete volume refers to the number of integral points in a polytope P, typically in terms of an integral dilation parameter, giving rise to the Ehrhart (quasi-)polynomial of P. The proposed research will advance our understanding of fundamental structures of Ehrhart quasipolynomials, and it will provide innovative and novel applications of Ehrhart theory. Discrete volume computation for polytopes has been of great interest in recent years, partly because of applications to many mathematical fields, some of which seem distant from Geometric Combinatorics: Number Theory, Commutative Algebra, Algebraic Geometry, Optimization, Representation Theory, Statistics, and Computer Science. The goal of this project is to apply Ehrhart (quasi-)polynomials to various combinatorial problems; the PI proposes four concrete lines of problems to work on:- Classification of Ehrhart h-vectors for special families, including lattice polyhedral complexes and graphic polytopes.- Rational Ehrhart theory, expanding Linke's theory of enumerating lattice points in rational dilations of rational polytopes.- Applications of inside-out polytopes (integer-point enumeration in a polytope but off a hyperplane arrangement) to enumerative problems for coloring and flow constructions for simplicial complexes, antimagic graphs, and Golomb rulers.- Classic counting functions associated to partitions and compositions viewed from a discrete-geometric perspective, with connections to permutation statistics.Building on the PI's proven track record of mentoring research students and postdocs, a particular emphasis of the proposed program of research is on the active involvement of students. Two graduate students will be directly supported by this project each year; the requested funding will support their research activities and allow them to participate in conferences, giving them both first-hand exposure to the excitement of research and opportunities to network with mathematicians at other institutions. Some of the research problems have direct connections to fields outside of mathematics (e.g., computer science) and many are easily explained to a lay person (e.g., graph coloring). Both facts prove the high outreach potential of the proposed research. The PI will make use of this potential, in particular, to attract students to research. Continuing his leadership role in the San Francisco Math Circle, the PI will bring aspects of his research to various math circles throughout the Bay Area; these are after-school programs for K-12 students and teachers with the goal of drawing kids to the beauty of mathematics and motivate them to excel in the subject.