PhD students working group

TBA

TBA

TBA

TBA

TBA

TBA

TBA

A fundamental question in geometry consists in understanding the effect of curvature on the shape of geometric spaces. In the case of surfaces, the Gauss-Bonnet Theorem establishes a link between the curvature of a surface and its topology. For example, it allows us to understand the topology of surfaces whose curvature is positive at every point.

When considering higher-dimensional geometric objects, called manifolds, we can define different notions of curvature. Scalar curvature is the weakest of these notions, and for this reason it is difficult to extract topological or geometric information from it. In particular, can we describe the topology of a manifold with positive scalar curvature?

In this talk, I will explain why this is an interesting question, and I will present a classification result for 3-dimensional manifolds with positive scalar curvature. This is a collaborative work with F. Balacheff and S. Sabourau.

In this talk, we begin by examining the application of curvature flows to a broad range of geometric problems. Following this, we introduce the essential geometric concepts required to understand these flows. Thereafter we focus on the mean curvature flow and its singularities. In particular, we give an intuitive and accessible proof of why singularities must occur if the initial surface is compact. After conducting a graphical analysis of various types of singularities, we describe how these singularities can be modeled by self-similar solutions of the mean curvature flow. Motivated by this, we conclude the presentation by exploring a current area of research: investigating the behavior of solutions that are in the proximity of such self-similar solutions.

Cela fait déjà plus de 150 ans que la recherche mathématique se casse les dents sur ce fameux problème appelé “Hypothèse de Riemann”. Portant sur les zéros non triviaux de la fonction Zêta de Riemann, elle est étroitement liée à la répartition des nombres premiers.

Journée conviviale d’exposés mathématiques pour les doctorants de l’IECL.

Programme :

Matin :

• 8h50 : Café d’accueil ;
• 9h20 : Karim Ramdani : Edition scientifique : un rapide survol des évolutions en cours ;
• 10h15 : Rodolphe Abou Assali : The Biharmonic Steklov Operator ;
• 10h55 : Pause ;
• 11h25 : Jérémy Dousselin : Arithmetic: from elementary statements to complex tools ;
• 12h15 : Pause repas

Après-midi :

• 14h : Aurélien Minguella : A brief introduction to stochastic partial differential equations ;
• 15h : Nathan Toumi : The level of distribution of the sum-of-digits function in arithmetic progressions ;
• 15h40 : Pause ;
• 16h10 : Valentin Schwinte : A minimization problem in the lowest Landau level, and centrosymmetric matrices ;
• 17h10 : Fin de la journée

This paper investigates the convergence time of log-linear learning to an $\epsilon$-efficient Nash equilibrium (NE) in potential games. In such games, an efficient NE is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. In this paper, we prove the first finite-time convergence to an $\epsilon$-efficient NE in general potential games. Our bounds depend polynomially on $1/\epsilon$, an improvement over previous bounds that are exponential in $1/\epsilon$ and only hold for subclasses of potential games. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning that requires a factor $A$ less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.

This work addresses the application of self-insurance in networks, where the network’s edges represent insured subjects facing losses. Each edge undertakes preventive efforts that influence the loss distribution, modeled as random variables. Insurance coverage is proportional, and a law-invariant coherent risk measure is considered to assess the network’s total risk. Furthermore, the work analyzes how preventive efforts impact the insurance cost and risk minimization. An optimization problem is proposed to determine the optimal levels of coverage and preventive effort, considering losses distributed according to a Pareto distribution. Through numerical techniques, specific cases, including global and local efforts, are studied to evaluate the model’s behavior in different scenarios.