## Grants.

• Projet Exploratoire Persyval-Lab AEPS. Period: January 2016 - September 2017. Funded by the French program Investissement d'avenir (ANR-11-LABX-0025-01). Project leader: Victor Magron. Participants: Victor Magron (CNRS Verimag, Grenoble), Bruno Gaujal (INRIA Mescal/CNRS Lig, Grenoble) and Panayotis Mertikopoulos (CNRS Lig, Grenoble). Topic: Algorithmes efficaces de programmation semidéfinie pour l'optimisation stochastique.
• PEPS-JCJC (Projet Exploratoire Pluridisciplinaire Jeunes Chercheurs) ACE. Period: January 2016 - December 2016. Funded by the French CNRS Institute of Information Sciences (INS2I). Project leader: Delphine Bresch-Pietri (CNRS Gipsa-Lab, Grenoble). Topic: Analysis and Control of Partial Differential Equations.

## Master/PhD/Postdoc projects.

We are seeking candidates for the following research projects:
• PhD (3 years): Analysis and Control of Partial Differential Equations using Occupation Measures, supervised by Victor Magron and Christophe Prieur. Please see the following description.
• PhD (3 years): Certified roundoff error bounds for programs with loops using polynomial optimization, supervised by Victor Magron and David Monniaux. Please see the following description.
• Master 2 (February-August 2018): Algorithms for exact sum of squares decompositions, supervised by Victor Magron and Mohab Safey El Din. Please see the following description.
• Master 2 (February-August 2018): Analysis and Control of Partial Differential Equations using Occupation Measures, supervised by Victor Magron and Christophe Prieur. Please see the following description.
• Master 2 (February-August 2018): Performance Analysis of Stochastic Descent Algorithms, supervised by Victor Magron and Bruno Gaujal. Please see the following description.

## Interesting Solved/Unsolved Conjectures.

• Conjecture $\frac{1}{n(n+2)}$: “For every even positive integer $n$, there exist polynomial sums of squares $\sigma_0 \in \mathbb{R}_n[x], \sigma_1, \dots, \sigma_n \in \mathbb{R}_{n-2}[x]$ such that $\prod_{i=1}^n x_i + \frac{1}{n(n + 2)} = \sigma_0 + \sum_{i=1}^n \sigma_i \, x_i \, (1 - x_i)$”. For more details, see the paper Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube, where the conjecture is stated by De Klerk and Laurent.

• Generalized Lax Conjecture: “Every hyperbolicity cone is spectrahedral”. For more details, see the slides by Tim Netzer.

• Kepler Conjecture: “The maximal density of sphere packings in 3D-space is $\frac{\pi}{\sqrt{18}}$”.
The formal verification project of the conjecture was completed on August 10, 2014.

• Reformulation of Bessis-Moussa-Villani (BMV) Conjecture by Lieb and Seiringer: “For all positive semidefinite matrices $A$ and $B$ and all $m \in \mathbb{N}$, the single variable polynomial $p(t) := \mathop{Tr}((A + tB)^m)$ has only nonnegative coefficients“. A proof of the conjecture by Herbert R Stahl is available here.