Rational Generating Functions, Enumeration, and Polyhedral Geometry

This proposal focuses on three research topics in combinatorics involving multivariate integerpoint transforms of rational polyhedra, or specializations thereof. Section 1 contains background information regarding polyhedra, integer-point transforms, and Ehrhart theory. Section 2 contains two projects related to integer-point counts and normalized volumes of “combinatorial” polytopes. Section 3 contains three projects regarding connections between polyhedral geometry and partition theory, with an emphasis on MacMahon’s Partition Analysis. Section 4 describes two projects regarding combinatorial and geometric properties of reflexive polytopes. The intellectual merit of this proposal is supported by these sections, as is the broader impact of these projects to the research community.