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.

My presentation is set within the framework of inverse problems. The main objective is to determine initial conditions, the state, or parameters of a system from available observations, with a particular focus on biological applications. We concentrate on sequential methods in data assimilation, where observations are incorporated as they become available. In this context, I present two examples: the reconstruction of a source term in a wave equation, and the determination of both state and parameters in a PDE system modeling tumor growth. For the first problem, we define a Kalman estimator in infinite dimensions that sequentially estimates the source term. We show that this sequential estimator is equivalent to minimizing a functional, which allows us to perform convergence analysis under observability conditions. The second project studies the evolution of non-spherical tumor growth by combining mathematical modeling with data assimilation from biological measurements. The general strategy is to extract relevant information from images of spheroids, formulate a PDE model for tumor evolution, and then reduce it to an ODE model. A reduced ROUKF coupled with a Luenberger observer is then used to estimate both the state and the parameters.

L’objectif de cet exposé est l’étude d’une équation de diffusion non linéaire fractionnaire posée sur un domaine bornée. Cette équation constitue une variante de l’équation des milieux poreux ou de l’équation de diffusion rapide, dans laquelle le laplacien est remplacé par un laplacien fractionnaire. Dans le cas des milieux poreux, les solutions présentent une décroissance à vitesse algébrique, tandis que dans le cas de la diffusion rapide, elles présentent un phénomène d’extinction en temps fini: à partir d’un certain temps elles sont uniformément nulles. Par ailleurs, les solutions convergent vers les solutions à variables séparées, lorsque le temps tend vers l’infini dans le cas des milieux poreux, ou lorsqu’il tend vers le temps d’extinction dans le cas de la diffusion rapide. Nous introduirons ensuite un schéma numérique qui préserve ces propriétés qualitatives. Ce schéma permettra de déterminer numériquement le temps d’extinction ainsi que le taux de convergence vers les solutions à variables séparées.

Dans cette présentation, nous nous intéresserons à l'équation de Schrödinger semi-classique, une équation fondamentale de la mécanique quantique, qui décrit l'évolution temporelle des particules quantiques. Comme les solutions exactes de cette équation sont rarement explicites et que les méthodes numériques classiques se révèlent souvent trop coûteuses, notre objectif est de développer des stratégies alternatives, à la fois plus faciles à mettre en œuvre et suffisamment précises, pour approcher les solutions de l'équation considérée. Pour cela, nous étudierons des fonctions particulières, appelées paquets d'ondes, qui représentent des états quantiques localisés et qui sont caractérisées par plusieurs paramètres. Dans un premier temps, nous montrerons comment, à partir d'une donnée initiale définie par un paquet d'onde, il est possible de construire une bonne solution approchée pour l'équation de Schrödinger semi-classique scalaire. Dans un second temps, nous expliquerons comment cette approche peut être généralisée à des équations plus complexes, à valeurs vectorielles, où de nouveaux phénomènes apparaissent.

Let $(M,g,X)$ be a complete gradient Kähler–Ricci expander with quadratic curvature decay (including all derivatives). Its geometry at infinity is modeled by a unique asymptotic cone, which takes the form of a Kähler cone $(C0,g0)$. In this talk, we will show that if there exists a solution to the Kähler–Ricci flow on $M$ that desingularizes this cone, then it necessarily coincides with the self-similar solution determined by the soliton metric $g$. Furthermore, if one perturbs the soliton metric in a suitable manner, the resulting initial data generates an immortal solution to the Kähler–Ricci flow which, after appropriate rescaling, converges to an asymptotically conical gradient Kähler–Ricci expander.