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DESCRIPTION: This talk has been canceled on short notice\, and will take p
lace at a later time. Ehrhart theory\, the art of counting lattice point
s in convex polytopes\, is a cornerstone of the interplay of combinatorics
and geometry. Many important combinatorial objects can be modelled as latti
ce points in polytopes and counting lattice points with respect to dilation
yields deep results in combinatorics. Conversely\, the combinatorics of po
lytopes 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 p
olytope. For $k=1$\, this is precisely Ehrhart theory. Determining these co
unting functions for $k >\; 1$ is quite challenging. For $k=2$ (free segm
ents)\, this is related to counting lattice points visible from each other.
In the talk I will discuss joint work with Sebastian Manecke on counting f
ree segments in dilates of unimodular simplices. Our main tool is a number-
theoretic variant of Ehrhart theory which can be computed using classical r
esults from geometry. The talk will be scenic tour (with impressions from t
he unusual summer 2020).
DTSTAMP:20210110T141300
DTSTART:20210111T141500
CLASS:PUBLIC
LOCATION:online
SEQUENCE:0
SUMMARY:CANCELED: Raman Sanyal (Goethe-Universität Frankfurt): From countin
g lattice points to counting free segments and back
UID:107739204@www.facetsofcomplexity.de
URL:http://www.facetsofcomplexity.de/monday/20210111-L-Sanyal.html
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