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Abstracts for Reading group on "Integer points in polyhedra"

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The algebra of polyhedra

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Speaker: 
Gaëtan Borot
Affiliation: 
MPIM
Date: 
Tue, 2019-10-08 10:00 - 11:30
Location: 
MPIM Seminar Room

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.

Volumes and lattice counts in polyhedra

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Speaker: 
Séverin Charbonnier
Affiliation: 
MPIM
Date: 
Tue, 2019-10-15 10:00 - 11:30
Location: 
MPIM Seminar Room

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. 

Lattice points, Ehrhart theory, and the relation to volumes

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Speaker: 
Reinier Kramer
Affiliation: 
MPIM
Date: 
Tue, 2019-10-22 10:00 - 11:30
Location: 
MPIM Seminar Room

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.

Counting lattice points via Hirzebruch–Riemann–Roch

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Speaker: 
Alessandro Giacchetto
Affiliation: 
MPIM
Date: 
Wed, 2019-11-13 10:00 - 11:30
Location: 
MPIM Seminar Room

Given a lattice polytope P, one can construct a toric variety X, together with an ample line bundle L on X. It turns out that its Euler characteristic 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 X. 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 X given the polytope P.

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