The world is continuous, but the mind is discrete. David Mumford We seek to bridge some
critical gaps between various? elds of mathematics by studying the interplay between the …

Ehrhart theory is the study of sequences recording the number of integer points in non-
negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is …

We study semigroup algebras arising from lattice polytopes, compute their volume
polynomials (particularizing work of Hochster), and establish strong Lefschetz properties …

The family of lattice simplices in $\mathbb {R}^ n $ formed by the convex hull of the standard
basis vectors together with a weakly decreasing vector of negative integers include …

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by
adjoining a cycle to a complete bipartite graph. The key observation is that this family admits …

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope
with the integer decomposition property (IDP) has a unimodal (Ehrhart) h⁎-polynomial. This …

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are
(1) to determine if its (Ehrhart) h^* h∗-polynomial is unimodal and (2) to determine if its …

In Ehrhart theory, the $ h^* $-vector of a rational polytope often provide insights into
properties of the polytope that may be otherwise obscured. As an example, the Birkhoff …

For an n-dimensional lattice simplex Δ (1, q) with vertices given by the standard basis
vectors and− q where q has positive entries, we investigate when the Ehrhart h*-polynomial …

The local $ h^* $-polynomial of a lattice polytope is an important invariant arising in Ehrhart
theory. Our focus in this work is on lattice simplices presented in Hermite normal form with a …