- 8h00: Petit-Déjeuner
- 09h30 : Johnatan ROTGÉ -
The middle prime factor of an integer : local laws and applications
In a recent work, McNew, Pollack and Singha Roy obtain several results regarding the distribution of the middle prime factor of an integer when multiplicity is taken into account. In this talk, we will discuss some of these results and some improvements that have been obtained regarding the local laws and some of their applications.
- 10h00 : Salim ALLOUN -
If you please... draw me the Klein quartic !
I will present the Klein quartic as a dessin d'enfant (=children drawing), in the context of a general theory aiming at describing the absolute Galois group of Q through its action on curves (projective smooth over number fields).
- 10h30 : Julia BUDZINSKI -
Simulation of diffusion with singular coefficients
TBA
- 11h00 : Goûter
- 11h30 : Thomas AGUGLIARO -
Exponential sums and six functors
Many problems in arithmetic can be translated to studying exponential sums. Whenever these exponential sums have a simple expression in terms of polynomials, there is a powerfull tool allowing to perform a fine study, namely the 6 functor formalism of l-adic sheaves.In this talk, we will give a gentle introduction to this « functors », which are nothing more than fancy pullback and pushforwards of measures.
- 12h00 : Mathilde GAILLARD -
Dimension reduction algorithms
Since datasets are growing much faster in terms of the number of features than the number of individuals, in order to avoid the curse of dimensionality, the assumption is made that the underlying structure of the data is in fact low-dimensional. In this context, dimension reduction algorithms have become essential for visualisation and preprocessing purposes. While linear methods such as principal component analysis (PCA) and multidimensional scaling (MDS) are mathematically well understood, they are limited in their ability to capture complex geometries. Conversely, non-linear methods (e.g. LargeVis, t-SNE and UMAP) based on pairwise similarities lacked a mathematical framework until a unifying one was proposed in 2022. This presentation aims to provide an intuitive understanding of some algorithms and present the unified framework.
- 12h30 : Repas + Pause
- 15h30 : Benjamin FLORENTIN -
Introduction to geometric inverse problems: The example of Steklov's problem.
Geometric inverse problems are an important class of problems in Fundamental Mathematics, originally derived from
concrete problems in Physics, and constitute a particularly active field of research at the crossroads of PDE and Geometry.
The question "Can you hear the shape of a drum?" formulated by Kac in the 60s,
is one of the best-known examples, and will serve to illustrate the philosophy behind geometric inverse problems.
The example of Steklov's inverse problem will then be presented, which can be interpreted as a generalization of Kac's problem
within the framework of the spectrum of the Dirichlet-à-Neumann operator. Some recent positive results will be briefly outlined
and compared with well-known results on the famous Calderón problem.
- 16h00 : Lucas NOEL -
Baire's pointwise limiting function theorem and application
Given a sequence of continuous functions on a metric space $(E,d)$ pointwise converging toward a function $f$, we can naturally wonder whether or not $f$ is contiuous. Of course, $f$ is (almost) never continuous. It not even sure that there exists points $f$ is continous at. However, with an additional assumption on the metric space $(E,d)$, we know that the set of continuity points of $f$ is a $G_\delta$-dense set. The aim of my talk is to prove this theorem and to give some application.
- 16h30 : Claire SCHNOEBELEN -
A (Very) Brief Introduction to $h$-Principle
The notion of $h$-principle was introduced by Gromov and Eliashberg in the early 1970s and, in geometry, constitutes a powerful tool for studying the existence of a wide range of applications satisfying different properties. In fact, many interesting properties for applications between manifolds are defined using algebraic relations involving the derivatives of these applications. Roughly speaking, proving the $h$-principle for a given differential relation amounts to establishing that there is no obstructions other than the algebraic ones to the existence of an application satisfying this relation.
In this talk, we will first define notions needed to formulate the $h$-principle, jets spaces and differential relations, before enunciating it. We will then briefly outline the very basic ideas behind the two proof techniques commonly used to establish $h$-principle in practice, through holonomic approximation theorem or by convex integration. Hopefully, we will have time to finish with an example.
- 17h00 : Goûter
- 17h30 : Marco ARTUSA -
Duality in arithmetic geometry
Duality is a mathematical principle allowing to give two points of view on the same object. It appears across different areas of mathematics - from geometry and algebra to analysis - where it allows to obtain information about a mathematical object thanks to its “dual”. In this talk, we begin with familiar examples of duality, and then explore a specific instance of this principle from arithmetic geometry: the local Tate duality. The hope is that this presentation will offer geometric intuitions for algebraic objects arising in number theory.
- 19h00 : Soirée tartes flambées