[Facets-of-complexity] Invitation and link to Monday Lecture - January 11th 2021 - online via zoom

[Ita Brunke <i.brunke@inf.fu-berlin.de>]

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.