# Current

Unless stated otherwise, the seminar takes place on Tuesdays, from 10 to 11, in room 385 at IMB. To get announcements, you can subscribe to the mailing-list.
• 2018-01-22
10:00
Salle 2
Philippe Moustrou (IMB)
On the Density of Sets Avoiding Parallelohedron Distance 1
Let $\Vert \cdot \Vert$ be a norm on $\mathbb{R}^n$. We consider the so-called unit distance graph $G$ associated with $\Vert \cdot \Vert$: the vertices of $G$ are the points of $\mathbb{R}^n$, and the edges connect the pairs $\{x,y\}$ satisfying $\Vert x-y\Vert=1$. We define $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ as the supremum of the densities achieved by independent sets of $G$. The number $m_1$ was introduced by Larman and Rogers (1972) as a tool to study the measurable chromatic number $\chi_m(\mathbb{R}^n)$ of $\mathbb{R}^n$ for the Euclidean norm.

The best known estimates for $\chi_m(\mathbb{R}^n)$ and $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ present relations with Euclidean lattices, in particular with the sphere packing problem.

The determination of $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ for the Euclidean norm is still a difficult question. We study this problem for norms whose unit ball is a convex polytope. More precisely, if the unit ball corresponding with $\Vert \cdot \Vert$ tiles $\mathbb{R}^n$ by translation, for instance if it is the Voronoi region of a lattice, then it is easy to see that $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)\geq \frac{1}{2^n}$.

C. Bachoc and S. Robins conjectured that equality always holds. We show that this conjecture is true for $n=2$ and for some families of Voronoi regions of lattices in higher dimensions.

• 2018-01-30
10:00
Salle 385
Jared Asuncion (IMB)
ECPP