The objective of the summer school is to introduce students to a vital area of mathematics which exemplifies the interaction between different mathematical subjects, as well as the interaction between theoretical and computational aspects of mathematics. The participants will experience a deeper understanding of the wide range of mathematical fields and how they connect, and they will get first-hand experience of the excitement and challenges that are lurking behind computational experiments. While the roots for counting problems in polytopes reach far back (Pick's Theorem, e.g., is more than a hundred years old), the last decade has seen major breakthroughs in this area of discrete geometry, when researchers have concentrated on finding theoretical and computational methods to count integer points in polytopes. Only very recently have mathematicians started to apply these results to certain special classes of polytopes, often motivated by problems outside discrete geometry, e.g. representation theory, combinatorics, number theory, topology, and commutative algebra.
We plan to cover roughly the following:
- Introduction 1: The coin-exchange problem of Frobenius (2 lectures)
- Introduction 2: A gallery of polytopes (2 lectures) Counting lattice points in polytopes -- the Ehrhart Theory (4 lectures)
- Magic squares and the Birkhoff polytope (2 lectures)
- Finite Fourier series (2 lectures)
- Dedekind sums, the building blocks of lattice-point enumeration (4 lectures)
- The decomposition of a polytope into its cones -- Brion's Theorem (2 lectures)
- Euler-MacLaurin summation in R^d (2 lectures)
Lecture notes for the workshops will be available at http://math.sfsu.edu/beck/papers/ccd.html.