A classical principle in deformation theory asserts that any formal deformation problem over a field of characteristic zero is classified by a differential graded Lie algebra. Using the Koszul duality between Lie algebras and commutative algebras, Lurie and Pridham have given a more precise description of this principle: they establish an equivalence of categories between dg-Lie algebras and formal moduli problems indexed by Artin commutative dg-algebras. I will describe a variant of this result for deformation problems around schemes over a field of characteristic zero. In this case, there is an equivalence between the homotopy categories of dg-Lie algebroids and formal moduli problems on a derived scheme. This can be viewed as a derived version of the relation between Lie algebroids and formal groupoids.

The talk is dedicated to flag manifold sigma-models, which are theories of generalized harmonic maps from a Riemann surface to a manifold of flags in $C^N$. These theories feature interesting geometric properties and are in certain cases examples of the so-called ‘integrable’ models. I will review some of these facts.

Séminaire Laurent Schwartz — EDP et applications

Dans le cas de coefficients indépendants du temps, le propagateur de Feynman peut être défini comme une valeur au
bord de la résolvante de l’opérateur d’onde (interprété comme un opérateur auto-adjoint). Un résultat célèbre de Duistermaat et
Hörmander en donne une caractérisation microlocale ainsi qu’une paramétrix qui se généralisent bien au cas des opérateurs de
type principal réel. Toutefois, la question d’existence d’un inverse canonique est longtemps restée ouverte. Le but de cet exposé
sera de présenter quelques résultats récents sur l’inversibilité de l’opérateur de Klein-Gordon sur des espaces-temps asymptotiquement
plats et d’expliquer comment interpréter l’inverse en termes de conditions aux limites globales à l’infini ainsi que dans un
langage plus « spectral». Quelques applications en théorie quantique des champs seront présentées. (travaux en collaboration avec
Christian Gérard et András Vasy)

Séminaire Laurent Schwartz — EDP et applications

Je discuterai d’un résultat récent obtenu avec Joackim Bernier et Benoît Grébert concernant la stabilité de petites solutions
régulières à l’équation de Schrödinger non linéaire (NLS) en dimension 1 sur un tore. Il s’agira de montrer un résultat du
type: « le plus souvent, les solutions de NLS sont stables sur des temps très longs » en précisant l’énoncé et les techniques utilisées.
Celles ci sont basées sur un nouveau type de formes normales dites rationnelles qui permettent de conjuguer le flot de NLS à
une dynamique intégrable sur des ensembles dont on mesure la taille par calculs probabilistes. La possible existence de contre
exemples et l’extension à d’autres situations, seront aussi évoquées et discutées.

Weight structures give certain filtrations of triangulated categories; the definition is a certain cousin of that of  t-structures. Particular cases of weight decompositions give stupid filtrations of complexes and cellular towers for spectra.  I will also mention interesting motivic and Hodge-theoretic examples of weight structures along with methods for constructing them. Weight structures yield weight filtrations and spectral sequences for cohomology as well as certain weight complex functors. In particular,  there exists an exact conservative functor from (relative) constructible Voevodsky motives into complexes of Chow motives, whereas the corresponding weight spectral sequences vastly generalize Deligne’s ones. Lastly, the relations between weight structures and t-structures yield new methods for constructing t-structures and proving their properties.

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.

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 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.

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.

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.

For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.