You are cordially invited to our next Monday Lecture in the New
Year. Next Monday Lecture, there will be again a regular schedule.
All Monday Lectures and Colloquia of winter term 2020/21 will be
given online via zoom.
You may find valid Invitation for zoom throughout all winter term
here:
http://www.facetsofcomplexity.de/monday/WS-2020-21/index.html
Invitation link:
https://tu-berlin.zoom.us/j/69716124232?pwd=dzFlcTFHMmFXRTE5QmZLaEV5N0FRUT09
Monday Lecture will be on January 11th 2021 at 14:15 h
& 16:00.
Online via:
Zoom - Invitation
Time: Monday, January 11th - 14:15 h
Lecture: Raman Sanyal (Goethe-Universität Frankfurt)
Title: From counting lattice points to counting free
segments and back
Abstract:
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: Monday, January 11th - 16:00
h
Lecture: Maria Dostert (Royal Institute of Technology)
Title: Exact semidefinite programming bounds for
packing problems
Abstract:
In the first part of the talk, I present how semidefinite
programming methods can provide upper bounds for various geometric
packing problems, such as kissing numbers, spherical codes, or
packings of spheres into a larger sphere. When these bounds are
sharp, they give additional information on optimal configurations,
that may lead to prove the uniqueness of such packings. For example,
we show that the lattice E8 is the unique
solution for the kissing number problem on the hemisphere in
dimension 8. However, semidefinite programming solvers provide
approximate solutions, and some additional work is required to turn
them into an exact solution, giving a certificate that the bound is
sharp. In the second part of the talk, I explain how, via our
rounding procedure, we can obtain an exact rational solution of a
semidefinite program from an approximate solution in floating point
given by the solver. This is a joined work with David de Laat and
Philippe Moustrou.
