In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

Séminaire Laurent Schwartz — EDP et applications

Dans un premier temps, je présenterai des résultats qui quantifient le prolongement unique pour des équations de type
onde. Typiquement, est-ce que la petitesse d’une solution sur une portion définie de l’espace-temps implique la petitesse de la
solution globale? Dans un second temps, je présenterai des applications de ces méthodes à des opérateurs hypoelliptiques de type
« sommes de carrés de champs de vecteurs ». Il s’agit de travaux en collaboration avec Matthieu Léautaud.

Séminaire Laurent Schwartz — EDP et applications

In low dimension, it is expected that topological properties determine a natural geometry. In this spirit, several characterizations are conjectured for Kleinian groups, i.e. discrete subgroups of PSL(2,C). We will survey different methods that lead to their topological and dynamical characterizations, and point out their limits and the difficulties encountered in obtaining a complete answer.

In this talk I will describe the geometrical four-point functions (cluster connectivities) of 2d critical Q-state Potts model and its relation to minimal models four-point functions. By studying on the RSOS lattice of the ADE type, we provide a geometrical formulation of minimal models four-point functions of operators related to the Potts model and they involve the same types of cluster/loop expansions as that of the Potts model albeit with different weights. I will discuss how our results can be used to extract useful information about the Potts model from minimal models.

In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin’s crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

Séminaire Laurent Schwartz — EDP et applications

Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto’s theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.

Séminaire Laurent Schwartz — EDP et applications

Along with a strict determinism of early embryogenesis in most living organisms, the variability of cell fates and developmental pathways exhibits in some of them. We propose that both determinism and variability are established through cell interactions, with the important role played by cell potency, cell sensitivity and cell signaling. The sensitivity of embryonic stem cells is considered to be strong, while the sensitivity of fully differentiated cells is small. This means that high potency correlates with high sensitivity, and vice versa. Experimental data were obtained and analyzed for the early developmental stages of plant species with regular and  irregular types of embryogenesis, which explicit correspondingly determinism or variability of developmental pathways. For the irregular type the species with underdeveloped embryos in mature seeds were examined. As the result, we propose three conjectures for explanation the phenomenon of variability, leading to the invariant final embryo shape, and support each of  these cases with the example(s) of actual developmental pathways.

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard