We will first introduce Summer UMS and then decide on the logistics for the ensuing weeks.

We will begin the talk by introducing an important tool in combinatorics known as the generating function and give some examples of how it can be used for problems such as coin counting and evaluating a formula for the fibonacci sequence. We will then introduce one of the most important tools that will guide us this summer: the Ehrhart Polynomial. For this lecture we will mostly focus on the example of the simplex, basic properties, and giving intuition for how the ehrhart polynomial relates the continuous and discrete. Lastly, we will finish the talk by introducing and proving Pick's theorem.

We will explore the connection between generating functions and volumes of polygons again through the Ehrhart theorem. This theorem will allow us to explore another connection between the continuous and the discrete alongside understanding the basis of open problems in the field.

We will explore the topic of reciprocity in Ehrhart Theory, in particular proving Ehrhart-Macdonald reciprocity and providing applications to the theory of reflexive polytopes. Time permitting we will also give a brief introduction Finite Fourier Analysis.

This week we’ll be exploring finite Fourier analysis, an analog of the continuous Fourier analysis you might already be familiar with, but for periodic functions defined on the integers. I’ll try to focus on examples and applications, rather than getting lost in detailed trigonometric proofs. We’ll start by studying representations of periodic functions in terms of finite Fourier series, and then move forward with a linear algebraic interpretation of these series, studying finite Fourier transforms and the finite Parseval identity. We’ll conclude with a discussion of finite convolutions, and time-permitting a brief introduction to Dedekind sums.

We will be tackling the problem of efficiently enumerating semimagic squares, square matrices whose entries are nonnegative integers and whose rows and column sum up to the same number. Along the way, we will be investigating properties of the Birkhoff-von Neumann polytope and use these properties to establish a connection between semimagic squares and the Ehrhart theory we have been exploring over the past few weeks. If time permits, we will close with a discussion of magic squares.

The main goal of this talk will be to prove the Dehn-Somerville Relations, which are a nice set of equations in Ehrhart Theory with implications regarding the faces of a polytope. In whatever time is remaining, we will give a survey of topics in chapters 9,10, and 11 of Computing the Continuous Discretely.

Differential Geometry is traditionally constructed using smooth maps in continuous spaces. However, representing differential forms one-to-one with their coordinate values leads to errors such that basic theorems such as Stokes theorem no longer holds numerically. These errors can lead to a loss of fidelity in certain conservation laws, like momentum. We will introduce a new formulation of differential geometry on discrete manifolds that will overcome this hurdle and explore topological consequences of this construction.