Organiser: Gaëtan Borot, Alessandro Giacchetto
Dates: starting from October 8, 2019, usually once every week
Location: Seminarraum, Max Planck Institute for Mathematics, Bonn, Germany
We will follow the eponymous lectures Integer points in polyhedra of Barvinok (published in the series "Zurich lectures in advanced mathematics"). The goal is to learn the elementary but beautiful properties of generating series that can be used to count lattice points/compute volumes; we would like to arrive to Ehrhart theory and Euler-McLaurin type formulas.
1. The algebra of polyhedra, by Gaëtan Borot (Tuesday, 8 October, 10:00–11:30 am in Seminarraum.)
In the first session, we will introduce the algebra of (indicator functions) of polyhedra and study its properties. Linear forms on this algebra produce functions on the set of polytopes (i.e. bounded polyhedra), that are compatible with set-theoretic decompositions of polyhedra. For polytopes, the lattice point count or the volume are examples of such linear forms.
2. Volumes and lattice counts in polyhedra, by Séverin Charbonnier (Tuesday, 15 October, 10:00–11:30 am in Seminarraum.)
In the second session, we use the formalism exposed in the first talk to various valuations. The exponential valuation allows to compute the volume of polytopes through Brion's theorem, and allows to give a meaning to the notion of volume for polyhedra without lines. Next, lattice points count of polyhedra is considered, where a discrete analog of the exponential valuation is defined via generating functions. We recover then Brion's theorem for lattice point count.
3. Lattice points, Ehrhart theory, and the relation to volumes, by Reinier Kramer (Tuesday, 22 October, 10:00–11:30 am in Seminarraum.)
Continuing from the previous session, we will define the valuation for exponential sums, a discrete analogue of exponential integrals, and obtain another version of Brion's theorem. We continue to Ehrhart theory, showing polynomiality of lattice counts for polytopes with fixed cones of feasible directions. Finally, we relate exponential sums and exponential integrals via an Euler–Maclaurin type formula.
4. Counting lattice points via Hirzebruch–Riemann–Roch, by Alessandro Giacchetto (Wednesday, 13 November, 10:00–11:30 am in Seminarraum.)
Given a lattice polytope P, one can construct a toric variety XP , together with an ample line bundle LP on XP . It turns out that the Euler characteristic χ(XP , LP ) is equal to the number of lattice points contained in P. Moreover, the Hirzebruch–Riemann–Roch theorem tells us how to calculate this Euler characteristic in terms of the Todd class of the toric variety XP . This yields an efficient method for counting the lattice points in P , because there is a polynomial time algorithm that computes the Todd class of XP given the polytope P.