[Facets-of-complexity] Invitation & Link to Monday Lecture & Colloquium - July 5th 2021

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

You are cordially invited to our next Monday Lecture & Colloquium on July 5th at 14:15 h & 16:00 h, online via Zoom.

Invitation link:
https://tu-berlin.zoom.us/j/69716124232?pwd=dzFlcTFHMmFXRTE5QmZLaEV5N0FRUT09
No password is required.

Online via:
Zoom - Invitation

Time: Monday, July 5th - 14:15 h

Lecture: Papa Sissokho (Illinois State University)

Title: Geometry of the minimal solutions of a linear Diophantine Equation

Abstract:

Let a₁,...,a_n and b₁,...,b_m be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x₁a₁+...+x_na_n=y₁b₁+...+y_mb_m. A solution (x₁,...,x_n,y₁,...,y_m) in S is called minimal if it cannot be expressed as the sum of two nonzero solutions in S.  For each pair (i,j), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, the solution whose only nonzero coordinates are x_i = b_j and y_j = a_i is called a generator.  We show that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Hosten-Sturmfels.

Break!

Time: Monday, July 5th - 16:00 h s.t.

Colloquium: Ansgar Freyer (Technische Universität Berlin)

Title: Shaking a convex body in order to count its lattice points

Abstract:

We prove inequalities on the number of lattice points inside a convex body K in terms of its volume and its successive minima. The successive minima of a convex body have been introduced by Minkowski and since then, they play a major role in the geometry of numbers.
A key step in the proof is a technique from convex geometry known as Blascke's shaking procedure by which the problem can be reduced to anti-blocking bodies, i.e., convex bodies that are "located in the corner of the positive orthant".

As a corollary of our result, we will obtain an upper bound on the number of lattice points in K in terms of the successive minima, which is equivalent to Minkowski's Second Theorem, giving a partial answer to a conjecture by Betke et al. from 1993.
This is a joint work with Eduardo Lucas Marín.