Ehrhart Theory Project (Counting dots in shapes) - Chris Manon

Polytopes are generalizations of convex polygons in the plane. For those of you with experience with linear algebra, a polytope can be thought of as the solutions of systems of linear inequalities over the real numbers.  Roughly speaking, the Ehrhart function of a polytope \(P\) counts the number of integer points in a dilation \(nP\) of the polytope, where n is a non-negative integer.  For example, if we take \(P\) to the triangle with corners \((0,0), (1, 0), (0, 1)\), it is a not-too-bad exercise to show that the \(n\)-th dilate, namely the triangle with corners \((0, 0), (n, 0), (0, n)\), contains exactly \(\frac{1}{2}(n+1)(n+2)\) integral points.  Notably this function is a polynomial of degree 2, which happens to be the dimension of the triangle.  It turns out that this always happens. The consequent Ehrhart polynomial is a powerful invariant of the polytope.  This project will be concerned with finding the Erhart polynomial for a class of interesting polytopes which come from mathematical biology.