Introduction to Fourier methods for polytopes and cones

Sinai Robins, Instituto de Matemática e Estatística, Universidade de São Paulo

This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.

Of the many applications of these techniques, we will focus on the following, as time permits:

1. The Fourier transform of a polytope, given its vertex description
2. Minkowski and Siegel's theorems in the geometry of numbers
3. Tilings and multi-tilings of Euclidean space by translations of a polytope
4. Discrete volumes of polytopes (Ehrhart theory)
5. The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.

We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.