Title: On some regularized nonlinear hyperbolic equations

Abstract: It is known that the solutions of hyperbolic partial differential equations develop discontinuous shocks in finite time even with smooth initial data. Those shock are problematic in the theoretical study and in the numerical computations. To avoid these shocks, one can add some small terms to the equation to obtain "smoother" solutions. The new equation is called a "regularization".
In this talk, we present and study some suitable regularizations that conserve the same properties of the original equations.

Title: Conformally flat spacetimes with complete lightlike geodesics

Abstract: C. Rossi proved that any maximal globally hyperbolic conformally flat spacetime which contains two homotopic lightlike geodesics with same extremities is a finite quotient of the Einstein universe. In the continuity of this result, I am interested in describing maximal globally hyperbolic (abbrev. MGH) conformally flat spacetimes with complete lightlike geodesics. In this talk, I will describe an example of such a spacetime, that we call Misner domain of the Einstein universe. Under some hypothesis, one prove that the universal covering of a MGH conformally flat spacetime with complete lightlike geodesics contains a Misner strip. The goal would be to prove that any MGH Cauchy compact conformally flat spacetime can be obtained by grafting (or removing) a Misner strip from another one. This would be the Lorentzian analogous of the grafting on hyperbolic surfaces introduced by Thurston

Title: On the analysis of some PDEs arising in Fluid Mechanics and Hydrodynamics

Abstract: I will be talking about some recent results on the existence, uniqueness, blow-up/global regularity and stability of solutions to the Navier-Stokes, Euler, MHD equations and some related systems.

Youtube link

Title: Strange duality at level one for the anti-invariant vector bundles

Abstract: For a smooth algebraic curve of positive genus over the field of complex numbers, the strange duality says that the space of sections of certain theta bundle on moduli of bundles of rank r and level k is naturally dual to a similar space of sections of rank \(k\) and level \(r\). In this talk, I will explain this duality and show that it remains true (at least at level one) on the moduli spaces of anti-invariant vector bundles.

Youtube link

Title: Regular projections and Lipschitz structures of real singular spaces

Abstract: The notion of regular projection is a strong tool in Lipschitz geometry of real singularities, it provides a way to prove metric properties of singular spaces by finding a finite number of directions which is transverse to the tangent space at the regular points of the singular set. We will be talking about how to prove the existence of regular projections for definable sets in polynomially bounded o-minimal structures (e.g., semi-algebraic sets) and its application to the study of the Lipschitz structures and Lipschitz cell decomposition of definable sets in o-minimal structures.

Title: Extending torsors via log schemes

Abstract: We present here an approach to the problem of extension of torsors defined over the generic fiber of a family of curves. The question is to extend both the structural group and the total space of the torsor above the entire family.
The origin of this subject is in the work of Grothendieck, who at the beginning of the years 1960, gave a good definition of the fundamental group of algebraic varieties, based on the notion of étale Galois covers. The problem of extending torsors under constant groups (and of orders prime to the characteristic of the residual field) has been solved, on a general basis, by Grothendieck's fundamental group specialization theory.
When we are interested in algebraic varieties from an arithmetic point of view, it is natural to consider also torsors whose structural group is finite but not necessarily a constant group. We then talk of fppf torsors, with reference to the theory of faithfully flat descent. The point of view defended here is that in order to study the problem of extension of torsors, it is better to place ourselves in a larger frame where we allow the torsors to admit ramification : these are log flat torsors. So, we first search for a log extension for the initial torsor and then see if it comes from an fppf one.

Youtube link

Title: Partially hyperbolic autonomous diffeomorphisms

Abstract: The aim of this talk is to introduce the notions of hyperbolic, partially hyperbolic, autonomous,... diffeomorphisms. The style is to be more into giving examples and raising questions. Finally, we present an algebraic classification of autonomous diffeomorphisms in dimensions two and three.

Youtube link

Title: On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

Abstract: Coleman and Mazur introduced the \(p\)-adic eigencurve, a rigid analytic space parametrizing the system of Hecke eigenvalues of p-adic modular forms of finite slope. I will present in this talk a joint work with Dimitrov and Pozzi in which we describe the geometry of the eigencurve at irregular weight one Eisenstein series. Such forms belong to the intersection of the Eisenstein locus and the cuspidal locus of the eigencurve. We proved that the cuspidal locus is étale over the weight space at any irregular weight one Eisenstein series. As a corollary, we gave some applications in Iwasawa theory.

Youtube link

Title: Introduction à l'étude des modèles physiques par des outils stochastiques

Abstract: Comprendre les comportements macroscopiques à partir des modèles microscopiques est un des objectifs de la physique statistique. Cependant, l'étude des modèles physiques non linéaires est extrêmement difficile, la chaîne anharmonique en est un parfait exemple. Un moyen de pouvoir étudier des modèles est de remplacer la non linéarité par un bruit stochastique. Dans cet exposé, je présenterai l'étude de la chaîne harmonique soumise à un bruit d'échange. Je montrerai comment à partir des équations microscopiques on peut observer le comportement macroscopique. L'exposé sera composé de deux parties ; la première partie sera composée des rappels de probabilités nécessaires pour comprendre la seconde. Dans celle-ci, j'étudierai la chaîne harmonique bruitée soumise à un champ magnétique. Je montrerai qu'à partir des équations de Newton on peut obtenir une équation de Boltzmann linéaire. Puis, au moyen d'un changement d'échelle temporelle et spatiale je montrerai comment on peut obtenir une équation de la chaleur fractionnaire dont l'exposant dépend de la présence ou non du champ magnétique introduit.

Youtube link: part 1, part 2

Title: Boundary controllability of coupled wave equations in 1-D

Abstract: In this talk, and after a brief introduction to control theory of PDEs, we present some results about boundary controllability of two coupled wave equations with first-order coupling in 1-D.

Youtube link: part 1, part 2

Title: An immersed discontinuous Galerkin method for elastic and acoustic-elastic wave propagation

Abstract: We present an immersed discontinuous Galerkin method for solving acoustic-elastic interface problems.
The method allows elements to be cut by the interface and thus leading to elements consisting of the union of an acoustic medium and an elastic medium. Thus, each interface element combines two separate models and is equipped with piecewise polynomial functions satisfying the interface jump conditions. The proposed discontinuous Galerkin formulation is stable and the IFE space contains optimally converging solutions. We present computational examples and results.

Youtube link: part 1, part 2

Title: Convex foliations of degree 4 on the complex projective plane

Abstract: In this talk, I will present the main results of a recent paper in collaboration with D. Marín. First, I will explain the outline of the proof of the result which states that up to automorphism of \( P_C^2 \) there are 5 homogeneous convex foliations of degree four on \( P_C^2 \). Second, we will see how to use this result to obtain a partial answer to a question posed in 2013 by D. Marín and J. Pereira about the classification of reduced convex foliations on \( P_C^2 \).

Title: Abelian varieties with complex multiplication: some theory and an application

Abstract: In this talk, I present a snapshot of the theory of abelian varieties with complex multiplication, or how the eigenvalues of possible automorphisms on an abelian variety may determine it up to isogeny or even up to isomorphism. I give a recent application of this theory: If \(A\) is an abelian variety and \(G\) is a finite group acting freely in codimension 2 such that the quotient \(A/G\) admits a resolution that is a Calabi-Yau variety, then \(A\) is isogenous to \(E^{dim A}\), where \(E\) is one of two possible elliptic curves.

Youtube link: part 1, part 2

Title: Analyse mathématique de modèles bioéconomiques: Équilibre de Nash généralisé

Abstract: Nous proposons d’analyser mathématiquement des modèles bioéconomiques concernant l’exploitation des populations marines.
Tout d’abord, nous présentons un modèle biologique général illustrant de façon simple et accessible les interactions entre les populations marines. Ensuite, pour concrétiser notre travail, nous choisissons les cinq petits pélagiques les plus connus du Maroc, à savoir, Sardina pilchradus, Sardinella, Engraulis encrasicolus, Scombercolias et Trachurus ; et nous donnons le modèle biologique représentant l’évolution de leurs biomasses sous forme d’un système à cinq équations différentielles. Nous introduisons par la suite l’activité de pêche afin de construire le modèle bioéconomique associé à ces populations. Nous calculons les points d’équilibre du modèle biologique et le point d’équilibre intérieur du modèle bioéconomique et nous étudions la stabilité locale du point d’équilibre intérieur de ce dernier modèle en utilisant l’analyse spectrale et le critère de Routh-Hurwitz.
En s’appuyant sur la théorie économique de Gordon, nous donnons l’expression du profit associé à chaque flottille de pêche, et nous écrivons le problème d’équilibre de Nash généralisé. La résolution de ce problème permet de déterminer l’effort de pêche en maximisant le profit de chaque flottille exploitant les cinq poissons petits pélagiques. Nous démontrons que la résolution de ce dernier problème mène à la résolution d’un problème de complémentarité linéaire. Ce dernier nous permet de donner l’expression mathématique de l’effort de pêche.
Finalement, nous considérons que l’effort de pêche représente le nombre de sorties de pêche, et nous cherchons à mettre en lumière l’impact de la variation des paramètres bioéconomiques sur les efforts de pêche, les captures et par conséquent les profits. Et afin de construire des modèles mathématiques qui décrivent le mieux la réalité, nous introduisons les effets m´météorologiques. Nous éclaircissons l’influence des changements météorologiques sur les efforts de pêche, les captures et les profits.

Youtube link

Title: Obstacle problems and Lewy-Stampacchia's inequalities

Abstract: In the first part, I will try to introduce some concepts about stochastic calculus and Stochastic PDEs.
In the main part, I present a result of existence and uniqueness, with the corresponding Lewy-Stampacchia’s (L-S) inequalities, of the solution to a stochastic obstacle problem with a nonlinear monotone operator associated with a random obstacle. By using a penalization method of the constraint, associated with a suitable perturbation of the stochastic reaction, we are able to prove on one hand the existence of a solution to the stochastic obstacle problem, and on the other hand, to prove the corresponding stochastic L-S inequalities.
Based on a joint work with Guy VALLET (UPPA).

Youtube link: part 1, part 2

Title: Geometric generalized wronskians and applications to hyperbolicity and foliations

Abstract: During this talk, we will recall the definition of generalized Wronskians, and exhibit a sub-family, whose elements are called geometric. Those geometric generalized Wronskians have two advantages: on the one hand, they allow global geometric constructions, that we will describe, and on the other hand, they still allow to detect linear independance of holomorphic functions (which is the fundamental property of generalized Wronskians, known since the work of Roth in the 1950s). We will then present applications of this construction in hyperbolicity (more precisely in the study of families of entire curves in Fermat hypersurfaces) and, if time allows, in foliation theory.

Youtube link: part 1, part 2

Title: Optimal Control Theory and Viability theory in help of decision making

Abstract: The continuous need for more accurate decision support have extensively developed the theory. However, Optimal control theory and Viability Theory can sound the same but at the same time be on opposite spectrum. In this talk, we will present the similarities and the difference of the two theories, while referencing interesting topics and published papers.

Youtube link: part 1, part 2

Title: Résolution des problèmes de programmation linéaires multi-objectifs à plusieurs niveaux

Abstract: L’optimisation à plusieurs niveaux semble être un outil très approprié pour modéliser les problèmes de prise de décision lorsque plusieurs décideurs interagissent dans une structure hiérarchique. Dans cet exposé je présenterai quelques méthodes de résolution de problèmes de programmation linéaire multi-objectif multi-niveaux (PPLMO-MN).

Youtube link: part 1, part 2

Title: Analysis of relativistic shell interactions

Abstract: In this talk we will discuss some recent work on Dirac Hamiltonians with delta-shell potentials \(H :=D+V_{\partial \Omega}\), where \(D=- i \alpha \cdot \nabla+ m \beta\) is the free Dirac operator in \(\mathbb{R}^3\) and \(V_{\partial \Omega}\) is a delta-type potential supported on the boundary of a domain \( \Omega \subset \mathbb{R}^3\). We will first explain how to rigorously define the Dirac Hamiltonian \(H\). Then we will consider issues of self-adjointness, structure of the spectrum, and quantum confinement under mild geometric measure theoretic assumptions on \(\Omega\). We will also discuss how the smoothness of \(\Omega\) affects the Sobolev regularity of the domain of \(H\) for different classes of domains.

Youtube link: part 1, part 2, part 3

Title: Simple semigroups in finite categories

Abstract: In the first part of the talk, we will define finite categories and show how we can associate them to matrices. The idea is to count finite categories with a certain size, we rely on the use of a program called PROVER9/MACE4 to construct models of finite categories. The data obtained shows that the algebraic nature of the endomorphism monoids of a category plays a very important role in the classification problem. In the second part of the talk, we classify finite categories with two objects such that one of the endomorphism monoids is a group. We prove that having a group on one side affects the structure of the other endomorphism monoid, and we prove that it is going to contain a simple semigroup. We also prove the other direction, that if we have a monoid (or a simple semigroup), then we can construct a category with two objects such that one of the objects is a group.

Youtube link: part 1, part 2

Title: Deformation of Holomorphic Branched Projective Structures on Riemann Surfaces

Abstract: Holomorphic projective structures are structures on Riemann surfaces that play an important role in the theory of differential equations on Riemann surfaces, as well as in the uniformization theorem. The notion of a branched holomorphic projective structure is a much more flexible concept. In particular, any representation of a surface group can be realized as the monodromy of a branched projective structure. One of the nicest feature of projective structures is the structure of the space of such structures on a given differential surface: it is a complex manifold with nice algebraic properties. We will show that most of these features extend to the moduli space of branched projective structures.

Youtube link: part 1, part 2

Title: Invariant topologique discret et problème géométrique des groupes

Abstract: Le covering type est un invariant combinatoire introduit par Karoubi & Weibel en 2016. Étudié sur les espaces topologiques \(K\) et noté \(ct(K)\), c'est le nombre minimal de sommets que contient la triangulation minimale d'un espace \(Y\) homotopiquement équivalent à \(K\). Dans cet exposé, je vais parler de la \(KW\)-complexité pour les groupes de présentations finies qui mesure la difficulté de ces groupes. Elle est définie comme étant le minimum de tous les \(ct(X)\) pour tout espace topologique \(X\) vérifiant \(\pi_1{X}=G\). On souhaiterait ensuite relier cette complexité simpliciale avec d'autres invariants de type géométrique tels que l'aire systolique et l'entropie volumique minimale des groupes.

Youtube link: part 1, part 2

Title: Coarse embeddings and homological filling functions

Abstract: Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. We will be particularly interested in the obstructions to the existence of such embeddings between spaces of non-positive curvature such as symmetric spaces of noncompact type and Euclidean buildings. We show that the rank (the maximal dimension of an isometrically embedded flat) is monotonous under coarse embeddings. First, we will start by introducing these spaces, and give some examples of known invariants that allow us to give obstructions to the existence of coarse embeddings. The second part of the talk will be dedicated to the main tools of the proof, which are the homological filling functions.

Youtube link: part 1, part 2

Title: Spectres de graphes

Abstract: On parlera de valeurs propres de la matrice d'adjacence, plus généralement d'opérateurs de Schrödinger, sur des graphes. Quand le graphe est infini, la notion pertinente est plutôt le spectre, une généralisation des valeurs propres qui sera définie. Certaines intuitions physiques nous permettent de prévoir à quoi devrait ressembler ce spectre, mais il y a beaucoup de problèmes ouverts. On parlera de graphes périodiques, d'arbres, ce qui arrive quand on ajoute des poids aléatoires. L'exposé se veut sans prérequis. (English Slides).

Youtube link: part 1, part 2

Title: Introduction to the Boltzmann equation and its formal compressible Euler hydrodynamic limit

Abstract: The problem of deriving macroscopic models from a microscopic description of matter originates from David Hilbert's sixth problem titled: "Mathematical treatment of the axioms of physics." In this talk, we will introduce the Boltzmann equation, which in the kinetic theory of gases is the equation that governs the evolution of molecules in an ideal gas, and then derive from it the compressible Euler equations, a macroscopic model of fluid dynamics.

Youtube link: part 1, part 2

Title: Second-Order Time and State-Dependent Sweeping Process in Hilbert Space

Abstract: see pdf file here

Title: 4-manifolds and exotic 2-spheres

Abstract: Dimension 4 is known to be the middle dimension that is high enough for complicated situations to arise, and low enough to prevent the resolution of these complications. It is therefore rich in “exotic” phenomena, notoriously: it being the only dimension \(n\) with manifolds homeomorphic but not diffeomorphic to \(\mathbb{R}^n\). We will begin by reviewing some classical results on 4-manifolds topology, then we will discuss a way of constructing families of infinitely many nullhomotopic “exotic” spheres in some 4-manifolds, these spheres are topologically unknotted but smoothly knotted.

Youtube link: part 1, part 2

Title: An introduction to birational geometry : running MMP, positivity of direct image sheaf

Abstract: We will give an introduction to birational geometry by recalling some basic definitions and natural transformations. We will discuss the history of birational classification of Complex projective varieties starting from Castelnuovo to Mori minimal model program. Thus, we will sketch the result of BCHM. For the second part, we will focus on the notion of the positivity of direct image sheaf, we will present our recent and ongoing work.

Youtube link: part 1, part 2

Title: Existence of SRB measures for hyperbolic maps with weak regularity

Abstract: In this talk, we will give an introduction to hyperbolic dynamics. Next, we will introduce the concept of SRB measures (named after Sinai, Ruelle, and Bowen) along with the necessary regularity conditions required for an Anosov diffeomorphism to possess an SRB measure. In the second part, we will concentrate on the main tools used to establish the existence of an SRB measure, which includes the regularity of unstable distribution, distortion...

Title: Action of multi-twists, examples, and non-examples

Abstract: It is known, due to the works of Goldman that the action of the mapping class group on the \(SU(2)\)-character variety of closed surfaces is ergodic. It is natural to ask whether subgroups of the mapping class group act ergodically or not.
In this talk, we discuss the case where the subgroup is generated by two filling multitwists. As a consequence, we provide invariant functions on the \(SU(2)\)-representation variety for some multi-twists on the genus two closed surface.

Youtube link

Title: Isometric actions on median spaces

Abstract: The aim of geometric group theory is the study of groups using geometric arguments. One way to go is to study the actions of a group on a space, featuring a particular geometry, to extract an algebraic property of the group. An interesting candidate for such spaces are median spaces.
In the first part of the talk, we will introduce the nomenclature and properties of median spaces as well as some well known results in order to display their role in the study of groups. In the second part of the talk, we will be investigating isometric action on locally compact median spaces of finite dimensions.

Title: Sobolev sheaves on the definable site

Abstract: We will be talking about sheafification of Sobolev spaces (in the sense of G. Lebeau) on the definable site (semialgebraic, subanalytic, ..etc). More precisely, we will try to discuss the motivation of this problem, the behavior of Sobolev spaces on singular definable spaces, the sensitivity of these spaces to the metric nature of singularities, and some partial answers on the transformation of these spaces into sheaves.

Title: Countable Lebesgue spectrum for analytic reparametrizations of irrational flows

Abstract: In classical ergodic theory it was proved that if a dynamical system has the K- property, then it has a countable Lebesgue spectrum. In this talk, we give examples of uniquely ergodic real analytic systems on the torus \(\mathbb{T}^5\) that have a Lebesgue spectrum with infinite multiplicity. In order to prove this, we are going to use a criterion for countable Lebesgue spectrum obtained by Fayad, Forni and Kanigowski which can be applied to zero entropy flows, and prove that some well chosen real analytic time-changes of irrational flows have countable Lebesgue spectrum.

Slides: see pdf file here

Title: Parabolic Hitchin connection

Abstract: Hitchin connection is a fundamental concept in mathematics that plays a key role in the study of moduli spaces, geometric structures, and their connections to other areas of mathematics and physics.
In the first part of the talk, we will present Hitchin's works, motivate the importance of the Hitchin connection and present an algebro-geometric criteria to the existence of such a connection based on the notion of Heat operators in algebraic geometry.
In the second part, we will show that the criteria are fulfilled over the moduli space of parabolic bundles on the vector bundle of parabolic non-abelian theta functions (parabolic conformal blocks).

Youtube link: part 1, part 2

Title: On third grade fluid equations: presence of a multiplicative noise, pathwise solution and invariant measures

Abstract: Most studies on fluid dynamics have been devoted to Newtonian fluids, which are characterized by the classical Newton’s law of viscosity. However, there exist many real fluids with nonlinear viscoelastic behavior that does not obey Newton’s law of viscosity. My aim is to present a recent results about the 2D/3D stochastic third grade fluids driven by a multiplicative Wiener noise. Namely, the strong local well-posedness, in PDEs and probabilistic senses, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space \(H^3\). Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. Finally, I will discuss the existence of an ergodic invariant measure for a subclass in 2D/3D.
The first part is devoted to explain "stochastic compactness method" and some PDEs results, which will be useful to present the main results of the talk.

Youtube link: part 1, part 2

Title: Sur certaines équations diophantiennes

Abstract: see pdf file here

Title: Lichnerowicz conjecture in the homogeneous pseudo-Riemannian setting

Abstract: In this talk we consider the so-called pseudo-riemannian Lichnerowicz conjecture. We prove in particular that if a pseudo-Riemannian compact manifold admits a transitive, conformal and essential action of a Lie group \(G\), then It is conformally flat.
This is a joint work with Deffaf, Raffed and Zeghib. In the first part, we will discuss motivations behind this conjecture and introduce some basic notions of pseudo-Riemannian conformal geometry. In the second part we will prove the Lichnerowicz conjecture under the additional hypothesis that the non-compact semi-simple part of \(Conf(M, g)\) is the Möbius group.

Youtube link: part 1, part 2

Title: Deep learning methods for stochastic dynamics

Abstract: In this talk I will presents deep learning methods for stochastic dynamics. The first dynamic is an application for solving initial path optimization of mean-field systems with memory where we consider the problem of finding the optimal initial investment strategy for a system modeled by a linear McKean-Vlasov (mean-field) stochastic differential equation with delay \(\delta > 0\), driven by a Brownian motion and a pure jump Poisson random measure. The problem is to find the optimal initial values for the system in this period \([-\delta,0]\) before the system starts at \(t=0\). Because of the delay in the dynamics, the system will after startup be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By using this equivalence we can find implicit expression for the optimal investment. We deep machine learning algorithms to solve explicitly some examples
The second type of dynamic is a second BSDE that represent a fully nonlinear second order PDE. As an application here we study \(\alpha\)-Hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connection between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using a deep learning based approximation of the underlying 2BSDE we can find the solution of our problem.

Title: In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory. In this spirit, we investigate the problem of the existence of infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake.
The first part of the talk will provide background in hyperbolic geometry and outline Mess's proof of Thurston's earthquake theorem. In the second part, I will explain the correspondence between surfaces in Half-pipe space and vector fields on the hyperbolic plane, followed by a sketch of the proof of our main theorem.

Youtube link

Title: A Bieberbach theorem for compact plane waves

Abstract: Any compact flat Riemannian manifold is finitely covered by the torus, by Bieberbach’s classical theorem. Similar classifications have been obtained for compact flat Lorentzian manifolds by Goldman-Fried-Kamishima. I will discuss the case of compact locally homogeneous plane waves, which are compact spacetimes whose universal cover can be thought of as a deformation as well as a generalization of the Minkowski spacetime. This is a joint work with Malek Hanounah, Ines Kath and Ghani Zeghib.

Youtube link: part 1, part 2

Title: On the Boundary of Convex Hyperbolic Manifolds

Abstract: Studying convex cocompact hyperbolic 3-manifolds from the data on their boundary has always been a topic of study. Bers' double uniformization theorem states that there is a full identification between such manifolds and the conformal structure on their ideal boundary. Thurston has conjectured that in the case when the boundary is a pleated surface, then there is a one-to-one correspondence between the bending locus and the deformation space of the 3-manifold. By the works of Labourie and Schlenker, it follows the existence of a one-to-one correspondence between the induced metric on the boundary (when it has Gaussian curvature > -1) and the deformation space of the 3-manifold. In this talk, we will explore more on the relation between convex cocompact hyperbolic 3-manifolds and the induced data on their boundaries. In particular, we will explore what happens when we mix these invariant data.

Youtube link: part 1, part 2

Title: Generalized mean curvature flows

Abstract: The mean curvature flow, a fundamental geometric concept for smooth surfaces, is designed to minimize surface area in the fastest possible way. In this talk, we present the theory of the mean curvature flow, and highlight some of its applications. We then explain the generalization, due to Brakke, of the mean curvature flow to singular submanifolds (and more generally to rectifiable sets), and we provide some examples of configurations that minimize the length (Steiner tree, Fermat's point..). Finally, we present the recent generalization of the mean curvature flow to general measures.

Youtube link: part 1, part 2, part 3

Title: (Symplectic) Determinant laws

Abstract: The notion of pseudo-representations was initially introduced for group algebras by Wiles (for \(GL_2\)) and by Taylor (for \(GL_d\)) in order to construct Galois representations associated to certain automorphic forms. Chenevier proposed an alternative theory of "determinant laws" which extends Wiles and Taylor’s definition to arbitrary rings. This theory has proved to be useful in the study of congruences between automorphic forms and in the deformation theory of residually reducible Galois representations.
In the first part of this talk, I will introduce this theory and highlight its main properties, aiming to illustrate its role in the study of moduli of Galois representations.
In the second part of the talk, I will present my joint work with Julian Quast on symplectic determinant laws, which adapts Chenevier’s framework to the symplectic group \(GSp_{2d}\).

Youtube link: part 1, part 2

Title: Systoles of hyperbolic manifolds

Abstract: The systole of a closed hyperbolic manifold \(M\) is the length of a shortest closed geodesic in \(M\). In the first part of the talk, we discuss low-dimensional constructions of closed hyperbolic manifolds with prescribed/small systole. In the second part of the talk, we prove using arithmetic techniques that, regardless of the dimension, systoles of closed hyperbolic manifolds are dense in the positive reals. This is joint work with Junzhi Huang.

Youtube link: part 1, part 2

Title: Fractional diffusion for Fokker-Planck equation via a spectral approach

Abstract: After a brief introduction to kinetic equations, I will provide motivation and explain the principle of diffusion approximation, which justifies that the solution of a kinetic equation can be approximated by an equilibrium profile with a density satisfying a macroscopic equation. I will then focus on the Fokker-Planck equation with heavy-tailed equilibrium handled by a spectral method.

Slides: see pdf file here

Youtube link: part 1, part 2

Title: Foliated geometric structures

Abstract: We explore the notion of tangential \((G,X)\)-foliations, which are a natural generalization of \((G,X)\)-structures. Roughly speaking we start by a foliated manifold \((M,\mathfrak{F})\) (say of codimension \(1\)), a tangential \((G,X)\)-foliation of \((M,\mathfrak{F})\) is a family of \((G,X)\)-structures that vary continuously from one \(\mathfrak{F}\)-leaf to another. The general question in this context is how the topology of the manifold \(M\) affects the geometry of the leaves. In this talk we show completeness of the leaves under the assumption that \(M\) is compact, for some particular geometry called the affine unimodular lightlike geometry. We also investigate natural relaxations and variants of the latter geometry. This is a joint work with Lilia Mehidi.

Youtube link: part 1, part 2

Title: Boundary stabilization of a fluid-structure interaction problem

Abstract: We show the stabilization by a finite number of controllers of a fluid-structure interaction system where the fluid is modeled by the Navier-Stokes system into a periodical canal and where the structure is an elastic wall localized on top of the fluid domain. The elastic deformation of the structure follows a damped beam equation. We also assume that the fluid can slip on its boundaries and we model this by using the Navier slip boundary conditions. Our result states the local exponential stabilization around a stationary state of strong solutions by using dynamical controllers in order to handle the compatibility conditions at initial time. The proof is based on a change of variables to write the fluid-structure interaction system in a fixed domain and on the stabilization of the linearization of the corresponding system around the stationary state. This is a joint work with Takéo Takahashi.

Youtube link: part 1, part 2

Title: Vortex patch motion in bounded domains

Abstract: We consider the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques combined with complex geometry tools. Additionally, we will present a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices.

Youtube link: part 1, part 2

Title: Conformal embeddings of Riemann surfaces in Lorentzian manifolds

Abstract: It has been known since the time of Gauss that conformal structures on a surface are in one-to-one correspondence with complex structures (Riemann surface structures). It is therefore natural to ask whether any Riemann surface has a conformal model embedded in a given Riemannian manifold \(M\). This problem was answered affirmatively by Garsia, Ruedy, and Ko. In this talk, we will extend this result to the case where \(M\) is pseudo-Riemannian. More precisely, we show that for any conformal structure on a closed surface \(\Sigma\), any spacelike embedding of \(\Sigma\) in \(M\) can be \(C^0\)-approximated by a smooth conformal embedding. Moreover, we show that if \(M\) is a quotient of the \((2+1)\)-dimensional solid timelike cone by a cocompact lattice of \(SO^{\circ}(2,1)\), then not all conformal embeddings can be convex.

Youtube link: part 1, part 2