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.

The shortest path approach is a well known approach in systems biology to the search of connections between nodes in molecular networks. However, little has been shown about the biological relevance of such approach. The first part of the talk will be devoted to the analysis of the shortest paths between proteins in canonical molecular pathways built in human interactome. One can enhance this approach by taking into account the betweenness centrality of nodes or paths in order to predict the unknown members of molecular pathways from the results of the genome-scale functional screens. This will be shown in the second part of the talk.

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.)

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.

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.

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).

Séminaire de Relativité Générale Mathématique

 

We will discuss dynamical properties of the Schwarzschild interior, backwards and forwards (in time) with respect to the initial value problem for the Einstein vacuum equations.

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

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

Let $G$ be GL$(2n)$ over a totally real number field $F$, $ngeq 2$. Let $Pi$ be a cuspidal automorphic representation of $G(mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square $L$-function of $Pi$ has a pole at $s=1$. In this talk, we present a rationality result for the residue of the exterior square $L$-function at $s=1$ and also for the holomorphic value of the symmetric square $L$-function at $s=1$ attached to $Pi$. As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square $L$-functions and a product of Gauss ~sums of Hecke characters.
 

In their paper, "On the motivic class of the stack of bundles",  Behrend and Dhillon were able to derive a formula for the class of a stack of vector bundles on a curve in a completion of the K-ring of varieties.  Later, Mozgovoy and Schiffmann performed a similar computation in order to obtain the number of points over a finite field in the moduli space of twisted Higgs bundles.  We will briefly introduce motivic classes.  Then, following Mozgovoy and Schiffmann's argument, we will outline an approach for computing motivic classes for the moduli stack of vector bundles with connections on a curve. This is joint work with Roman Fedorov and Yan Soibelman.