Lecture by Raman Sanyal (Goethe-Universität Frankfurt): From counting lattice points to counting free segments and back
This talk has been canceled on short notice, and will take place at a later time.
Ehrhart theory, the art of counting lattice points in convex polytopes, is a cornerstone of the interplay of combinatorics and geometry. Many important combinatorial objects can be modelled as lattice points in polytopes and counting lattice points with respect to dilation yields deep results in combinatorics. Conversely, the combinatorics of polytopes provides a powerful framework for the computation of these counting functions with numerous algebraic/combinatorial consequnces and challenges. A lattice polytope is free if does not contain lattice points other than its vertices. Klain (1999) suggested a generalization of Ehrhart theory by counting free polytopes with $k$ vertices contained in dilates of a given polytope. For $k=1$, this is precisely Ehrhart theory. Determining these counting functions for $k > 1$ is quite challenging. For $k=2$ (free segments), this is related to counting lattice points visible from each other. In the talk I will discuss joint work with Sebastian Manecke on counting free segments in dilates of unimodular simplices. Our main tool is a number-theoretic variant of Ehrhart theory which can be computed using classical results from geometry. The talk will be scenic tour (with impressions from the unusual summer 2020).
Time & Location
Jan 11, 2021 | 02:15 PM