Ehrhart theory is the theory studying how many lattice points are contained in the dilate of a given (lattice) polytope; it has rich applications to combinatorial problems, and deep connections to algebraic geometry, representation theory and more. Some of the most fundamental problems remain open. 

We developed a new approach to understanding the fundamental class of associated semigroup algebras via Parseval-Rayleigh identities. This leads to the resolution of several combinatorial conjectures, such as unimodality of the $h^\ast$-polynomial for large classes of lattice polytopes, among them integrally closed Gorenstein polytopes. 

Based on joint work with Papadakis and Petrotou. If time permits, I will also give glimpses of upcoming work that involves in addition Katz and Oba.