In 50 hours, the Xenopus embryo turns from a single egg cell into a complex organism with highly differentiated tissues: beating heart, flowing blood, contracting muscles, functional sensory organs. This process has been examined by scientists for over 150 years yet we can not claim a thorough systems-level understanding of how it works. Today, embryology is enjoying a technological revolution. We are able to observe the molecular (DNA, RNA, metabolites and protein) makeup of life at unprecedented resolution: the complete genome sequence, expression level of all proteins at genomic scale, messenger RNA expression at a single cell level as it changes over time. This enormous amount of data already constitutes a treasure trove for embryologists working on particular molecular circuits, but we are only just beginning to sense paradigm shift towards mathematical description and modeling of embryonic development.

 

In this talk I will review these new data acquisition tools, explain what are the challenges and achievements, and discuss what we can expect in the near future and what we have learned already. I expect to combine slide presentation with chalk-talk and open discussion.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Mini-cours, 1ère partie

Mini-cours, 2ème partie

The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.

The local p-adic Simpson correspondence by Faltings asserts that there is a natural equivalence of categories between small generalized representations and small Higgs modules for an affine semi-stable scheme over a complete discrete valuation ring of mixed characteristic with algebraically closed residue field. However, in the case of rational coefficients, the construction of the functor from the former to the latter, reducing to the theory for integral coefficients, does not seem to work as it is written, as pointed out by Ahmed Abbes. In this talk, I give an alternative argument based on a generalized Sen's theory for the semi-stable scheme and complete the local theory for rational coefficients.

On expose les résultats récents sur la conjecture de Langlands locale pour les groupes classiques en caractéristique p, et les techniques de théorie des représentations utilisées. Nous commençons avec le cas d'une représentation générique. On discute les fonctions L et les facteurs locaux sur un corps local non archimédien. Nous présentons les résultats de Ganapathy-Varma sur le transfert de la caractéristique zéro à la caractéristique p pour les groupes classiques deployés. Nous donnons aussi une autre approche du global au local en collaboration avec Gan pour les groupes classiques quasi-deployés. Nous faisons une conjecture dans le cas général, mais le cas générique est complet. Nous donnons des détails pour l'étude des fonctions L avec la méthode de Langlands-Shahidi.