Les représentations de Hitchin sont un exemple paradigmatique de représentations d'Anosov et ont été largement étudiées comme analogues en rang supérieur de l'espace de Teichmüller. Labourie a prouvé l'existence de courbes équivariantes dans l'espace des drapeaux complets. La trace de ces courbes dans l'espace projectif est toujours de classe C1 mais les courbes ne sont en général pas plus lisses que Hölder. Dans cet exposé on va donner une preuve du fait suivant : si la courbe est lisse alors la représentation est Fuchsienne. Les techniques sont aussi importantes pour des résultats de rigidité pour l'exposant critique. Cela fait partie d'un travail en collaboration avec A. Sambarino.

Parmi les représentations de groupes de surfaces à valeurs dans le groupe de Lie hermitien SO(2,n), celles dont l'invariant de Toledo est maximal forment une famille de représentations d'Anosov, dont les nombreuses propriétés géométriques et dynamiques ont été mises en évidence par les travaux de Labourie, Guichard et Wienhard. Dans un travail en commun avec Brian Collier et Jérémy Toulisse, nous étudions plus en détail l'action de ces représentations sur les différents espaces homogènes de SO(2,n). Nous démontrons en particulier que l'exposant critique de ces représentations est majoré par 1, et qu'elles préservent une unique surface minimale dans l'espace symétrique de SO(2,n).

The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.

I’ll present a general prescription for the 4d-2d partition function of half-BPS surface defects in d = 4, N = 2 gauge theories in Omega-background which is applicable for any surface defect obtained by gauging a 2d flavour symmetry using a 4d gauge group and reproduces known results obtained via the Higgsing procedure and Kanno-Tachikawa orbifold calculation for Gukov-Witten defects. The role of “negative vortices” which appear in the background of instantons will be emphasized.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Nous montrerons que, pour de telles représentations ρ : π1Σ → G, l'adhérence de Zariski de ρ(Γ) contient le SL2 principal sauf lorsque Γ est un sous-groupe monogène de π1Σ, auquel cas cette adhérence de Zariski est un sous-groupe abélien régulier connexe. Ceci entraîne la classification des adhérences de Zariski possibles puisque la classification des groupes algébriques contenant le SL2 principal est connue. Enfin dans le cas où Γ = π1Σ (ou un sous-groupe d'indice fini) les paramètres de Hitchin déterminent l'adhérence de ρ(Γ).

We investigate the degree to which the geometry of a compact real projective manifold with boundary is reflected in the associated holonomy representation, a representation of the fundamental group in the projective general linear group PGL(n,R) which in general need not have any nice properties.

We show that if the projective manifold is strictly convex, then its holonomy representation is projective Anosov, a condition which generalizes the dynamical properties of convex cocompact representations in rank one (e.g. hyperbolic) geometry. Conversely, a strictly convex projective manifold may be constructed from a projective Anosov representation that preserves a properly convex set in projective space. Applications include new examples of both convex projective manifolds and Anosov representations. Joint work with François Guéritaud and Fanny Kassel.

In this talk I will survey some recent work on coarse geometry of transformation groups, specifically, groups of homeomorphisms and diffeomorphisms of manifolds. Following a framework developed by C. Rosendal, many of these groups have a well defined quasi-isometry type (despite not being locally compact or compactly generated). This provides the right context to discuss geometric questions such as boundedness and subgroup distortion — questions which have already been studied in the context of actions of finitely generated groups on manifolds.

At present, the mod $p$ (and $p$-adic) local Langlands correspondence is only well understood for the group $mathrm{GL}_2(mathbb{Q}_p)$. One of the main difficulties is that little is known about supersingular representations besides this case, and we do know that there is no simple one-to-one correspondence between representations of $mathrm{GL}_2(K)$ with two-dimensional representations of $mathrm{Gal}(overline{K}/K)$, at least when $K/mathbb{mathbb{Q}}_p$ is (non-trivial) finite unramified.

 

However, the Buzzard-Diamond-Jarvis conjecture and the mod $p$ local-global compatibility for $mathrm{GL}_2/mathbb{Q}$ suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic $p$ coefficients (cut out by some modular residual global representation $bar{r}$). Moreover, the work of Gee, Breuil and Emerton-Gee-Savitt show that, to get information about the $mathrm{GL}_2(K)$-action on the cohomology, one could instead study the geometry of certain Galois deformation rings of the $p$-component of $bar{r}$.

In a work in progress with Haoran Wang, we push forward their analysis of the structure of potentially Barsotti-Tate deformation rings and, as an application, we prove a multiplicity one result of the cohomology at full congruence level when $bar{r}$ is reducible generic emph{non-split} at $p$. (The semi-simple case was previously proved by Le-Morra-Schraen and by ourselves.)

Séminaire de Relativité Générale Mathématique

 

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this talk I will describe how one proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture in general relativity.