The classical Weyl problem asks whether any Riemannian metric of positive curvature on the sphere S2 can be uniquely realized as the induced metric on the boundary of a convex domain in Euclidean space. In hyperbolic space, there is an analogue which was solved by Alexandrov in the 1950s, but also a dual statement describing the possible third fundamental forms of the boundaries of bounded, convex domains.We will describe those classical results, as well as some conjectural statements and partial results extending them either to convex domains in hyperbolic manifolds, or, more generally, to unbounded convex domains in H3.

I will explain and give some motivation for the notion of quantum ergodicity in an elementary manner, using wave packets. I will then describe an attempt at understanding the defect of uniqueness in this setting, first exhibited for the quantisation of the cat map on the torus. What will be discussed here is the product of a collective work under the alias JMP.

A celebrated result by Benoist in the 90s asserts that if G is a semi-simple real-algebraic group and Γ < G is a Zariski-dense semigroup, then the smallest closed cone that contains the Jordan projections {λ(γ) : γ ∈ Γ} is convex and has non-empty interior. In this talk we will focus on analogous concepts for tangent vectors to the character variety Hom(Γ,G)/G, and if time permits we will treat some applications to higher rank Teichmüller theory.

Given a closed topological surface S of genus at least two and an integer d, the Teichmüller space of S embeds into the space of conjugacy classes of representations of the fundamental group of S to PSL(d,R) by postcomposing by the irreducible representation from PSL(2,R) to PSL(d,R). The connected component of the Teichmüller space in this space is called the Hitchin component.A few years ago, Bridgeman, Canary, Labourie and Sambarino constructed on this component several metrics called pressure metrics. The construction is inspired by a characterization of the Weil-Petersson metric due to Thurston and Wolpert, and work of McMullen. These metrics are not yet fully understood. We will describe their restriction to a subspace of the Hitchin component consisting of deformations by bending of points of the Teichmüller space. This will allow us to show that the Teichmüller is distorted, in the sense that there is a sequence of points in the Teichmüller space whose Weil-Petersson distance to the origin diverges while their pressure 

Probability and analysis informal seminarIn this talk, we will investigate geometric properties of random planar triangulations coupled with an Ising model. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. I will briefly explain this phenomenon, and why it hints at a different universality class than the Brownian sphere.In the second part of the talk, we will focus on the geometry of spin clusters in the infinite volume setting. We will exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. A lot of precise information can be derived in all regimes. In particular, we will see that in the whole supercritical temperature regime, critical exponents for spin clusters are the same as for critical Bernoulli site percolation on uniform planar triangulations.Based on joint works with Marie Albenque and Gilles Schaeffer.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarMultiple SLEs come naturally as the scaling limit of multiple interfaces in 2-dimensional statistical physics models. Dyson Brownian motion usually describes the movement of trajectory of independent Brownian motions under mutual repulsion. In this talk, we will describe the connection between multiple SLEs and Dyson Brownian motion. The talk has two parts. In the first part, we take critical FK-Ising model as an example and explain the emergence of multiple SLEs. We give the connection probabilities of multiple SLEs. Such probabilities are related to solutions to BPZ equations in conformal field theory. In the second part, we explain the connection between multiple SLEs and Dyson Brownian motion. It turns out that, under proper time-parameterization, and conditioning on a rare event, the driving function of multiple SLEs becomes Dyson Brownian motion. Using such a connection, we may translate estimates on Dyson Brownian motion to estimates on multiple SLEs. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarRandom matrices over the integers and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory, topology and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices. I will outline this area (no background in discrete random matrix theory will be assumed), discuss exact results and their parallels with classical random matrix theory, and give probabilistic results for products of random matrices. The latter yield interesting new local limit objects analogous to the extended sine and Airy processes in classical random matrix theory. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Theories of quantum gravity are believed to have no global symmetries. We will check this conjecture for the U-duality symmetry in 4d N=8 supergravity. Since this theory arises from string theory, the ‘t Hooft anomaly for this symmetry ought to vanish. We then perform a bordism computation to classify the anomaly of U-duality and notice an interesting fact about the particular Thom spectrum that we use for the computation. Finally we show that evaluating the anomaly for our particular theory on the generating manifold of the bordism group leads to it vanishing.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The notion of pseudo-representations was initially introduced for group algebras by Wiles (for GL2) and by Taylor (for GLd) in order to construct Galois representations associated to certain automorphic forms. Chenevier proposed an alternative theory of « determinant laws » which extends Wiles and Taylor’s definition to arbitrary rings. This theory has proved to be useful in the study of congruences between automorphic forms and in the deformation theory of residually reducible Galois representations. 
In this talk, I will present my joint work with Julian Quast on symplectic determinant laws, which adapts Chenevier’s framework to the symplectic group GSp2d. I will give the definition and highlight some of its key properties, and then explain its connection to Geometric Invariant Theory. In particular, I will show that in characteristic zero, the space of symplectic determinant laws recovers the GIT quotient of the  space of symplectic representations by the conjugation action. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Séminaire informel sur les intersections atypiquesIn this informal talk, we will illustrate the theory described by Ziyang Gao in the previous talk by investigating the existence of quadratic homogeneous relations between holomorphic periods of anti-Weyl CM abelian varieties. This is a work in progress with Ziyang Gao. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Séminaire informel sur les intersections atypiquesThe goal of this talk is to propose a possible framework to study quadratic relations among holomorphic periods of CM abelian varieties. We define, for each Shimura variety and a CM point, a bi-$overline{mathbb{Q}}$-structure on the tangent space. Then we explain that in the case of the Siegel moduli variety, the numbers comparing the two $overline{mathbb{Q}}$-structures are precisely the products of the holomorphic periods of the CM abelian varieties parametrized by the CM point (up to $2pi i$). Next we propose a hyperbolic analytic subspace conjecture, which is the analogue of Wüstholz’s analytic subgroup theorem in this context, and explain why it implies the desired consequence on the quadratic relations among these holomorphic CM periods. This is joint work with Emmanuel Ullmo and Andrei Yafaev. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

We consider representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems, due to Meyer and Atiyah, to Burger-Iozzi-Wienhard’s Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer’s rho invariant, initially defined on U(p), to discontinuous class functions, first on U(p,q), and then on other classical groups via embeddings into U(p,q). As an application, we obtain a Milnor-Wood type inequality which slightly differs from, and sometimes improves upon Burger-Iozzi-Wienhard’s version. This is joint work with P. Pansu and X. Wan.