Starting with a polytope P in the Euclidean space, a high dimensional analogue of polygons on the plane. It contains some lattice points, i.e., points with integer coordinates. It turns out that if you double, triple, or quadruple (and so on) P, and record the number of lattice points contained in the sequence of ever larger polytopes, the sequence of numbers comes from one single polynomial, known as the Ehrhart polynomial of P. The major component of the project is to understand the formal statement and the proof of Ehrhart theorem, as well as some properties and examples of Ehrhart polynomials.
Moreover, it is often the case that when the polytopes are coming from other objects (such as graphs or hyperplane arrangements), or being used to construct other objects (such as ideals of polynomial rings), the Ehrhart polynomials of those polytopes provide information about the corresponding objects in an useful manner. As a teasing example, one can make sense of “the number of ways to colour the vertices of a graph with -1 colours so no adjacent vertices receive the same colour” using Ehrhart theory. Depending on the interest and background of student, the project might branch out into these connections and applications.
Reference:
- Computing the Continuous Discretely, Matthias Beck and Sinai Robins. Second edition. Available online here: https://matthbeck.github.io/papers/ccd.pdf