Le problème inverse de l’électrophysiologie cardiaque vise à reconstruire l’activité électrique du cœur à partir de mesures non invasives de potentiels électriques à la surface du torse. Ce problème est mathématiquement mal posé au sens de Hadamard, ce qui le rend particulièrement difficile à résoudre. Il est classiquement formulé comme un problème de minimisation contraint par une équation aux dérivées partielles. Dans un premier temps, je présenterai une approche visant à mieux contraindre ce problème par l’élaboration d’un nouveau modèle d’EDP de contrainte. Dans un second temps, j'introduirai une approche complémentaire, fondée sur le filtrage bayésien, permettant de mieux intégrer et visualiser les incertitudes associées à la résolution du problème et aux résultats obtenus.

Consider a particle randomly moving in a bounded (planar) domain starting at any given point within. Assume it bounces against the boundary and consider $\Sigma$, a small part of that boundary. What is the expected time we need to wait before the particle hits $\Sigma$ ? This question is known as the narrow escape problem. We can also consider the related question : what is the probability that the particle hits $\Sigma$ before another given subset of the boundary $\Gamma$ ? In this talk, I will address these questions and give quantitative answers in the asymptotic regime where the lengths of the windows tend to 0. To tackle the problem, I will prove a Feynman-Kac formula, linking the stochastic process studied to a deterministic PDE which has the form of a Poisson equation with mixed boundary conditions. Then, constructing appropriate quasimodes to this PDE, we are able to derive sharp asymptotics for the expected time and probabilities.

Pendant ma thèse j'ai construit l'espace des modules d'une certaine classe de connexions méromorphes et j'ai étudié le feuilletage isomonodromique induit par la fibration de Riemann-Hilbert. L'espace des modules en question est modelé sur un fibré en droites qui a pour base l'espace des modules des courbes rationnelles irrégulières. Après avoir donné quelques définitions, on étudiera l'espace des modules des courbes, sa compactification à la Deligne-Mumford, et son lien avec les connexions méromorphes en question.

In dimensions greater than four, the classification of smooth manifolds is an unsolvable problem, but manifolds can still be classified up to cobordism.

From this perspective, Liouville cobordisms provide a powerful tool for studying contact manifolds in high dimensions. In this talk, I will explain how Liouville cobordisms can be used to construct exact locally conformally symplectic (LCS) manifolds, in particular the LCS mapping tori associated with a contactomorphism. I will then use this construction to study the isomorphism classes of LCS mapping tori and explore their connections with the contact mapping class group.

Le patchwork combinatoire est une puissante méthode utilisée pour construire des hypersurfaces algébriques réelles avec du contrôle sur sa topologie. Dans cet exposé, je discuterai d'une généralisation de cette méthode en codimension supérieure grâce à la notion de structure de phase réelle. En codimension 2, on donne une nouvelle description explicite (basée sur des triangulations, distributions de signes et orientations d'arêtes) des T-variétés (les variétés obtenues par patchwork) proche de la description originale de Viro pour les hypersurfaces. Cette méthode permet d'obtenir une famille de T-courbes maximales dans l'espace projectif de dimension 3. En grande codimension, on présente de nouvelles bornes sur le nombre de composantes connexes qui montrent qu'on ne peut pas obtenir de T-courbes ou T-surfaces maximales

The study of complex structures on Lie groups provides a natural bridge between algebra, geometry, and complex analysis. In this talk, we focus on Lie algebras endowed with left-invariant complex structures, and on how these structures behave under degenerations and deformations.

A central question is how to understand the possible “limits” of such structures and which features remain stable under these processes. To address this, we introduce certain invariants that are well adapted to degenerations while preserving the complex structure.

We illustrate these ideas in the four-dimensional case, where a more concrete picture can be obtained.

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precision on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the well known Theory of Large Deviation.

Conway’s approach to enumerating knots from the 1970s highlighted an interesting ambiguity wherein pairs of tangles assembled in different ways can give rise to different knots. The process relating the resulting knots has come to be known as knot mutation, and because this often leads to a subtle and difficult-to-detect change to a knot, has received considerable attention ever since. This talk will focus on the history of knot mutation in the context of the Jones polynomial and its categorification known as Khovanov homology. The former highlights how one might view mutation as pointing to hidden symmetries in the definition of a knot invariant; the proof that the Jones polynomial is unchanged under mutation is surprisingly simple from this perspective. By contrast, the latter is a much more subtle story, which ultimately makes a surprising appeal to the homological mirror symmetry of the 3-punctured sphere. This last step is part of a project with Artem Kotelskiy and Claudius Zibrowius.