Classification of Ehrhart quasi-polynomials of half-integral polygons

In the 1960s, Eugene Ehrhart developed Ehrhart theory to enumerate lattice points in convex polytopes. An important tool in Ehrhart theory is the Ehrhart quasipolynomial, which encodes information about continuous and discrete area, lattice boundary points, and lattice interior points. Here, we will give an introduction to Ehrhart theory and outline some of the methods used to characterize polytopes based on their corresponding Ehrhart quasi-polynomials. We will discuss work done recently, and then expand on this work to classify all half-integral polygons by the coefficients of their corresponding Ehrhart quasi-polynomials.