I will make a quick review of old and new results concerning the Lorentz gas; discuss new directions in which I’d like to proceed (e.g. non periodic obstacles, interacting particles, …) and some (very) partial results toward such directions.

The two-dimensional one-component Coulomb plasma is an ensemble of identical charges in the plane repelling each other via the logarithmic Coulomb potential and confined by an external potential. I will discuss results that show that the fluctuations of linear statistics of many particles are of order 1 and given by a Gaussian free field, at any temperature. This is joint work with P. Bourgade, M. Nikula, and H.-T. Yau.

Totally positive matrices are characterized by having all their minors positive. They appear in various areas of physics and mathematics, including oscillations in mechanical systems, quantum groups, and algebraic geometry. It has been known since the work of Fomin that the two-point correlations functions of the two-dimensional Gaussian free field satisfy total positivity. I will present an analogous result for the correlations of the planar Ising model. The idea is to prove that determinants of such correlations have interpretations in terms of probabilities of events in the random current model. A natural open question is to identify all planar totally positive spin systems.

We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. These log-correlated non-Gaussian random fields arise in different branches of statistical mechanics. Existing results were mainly focused on the CLT for the linear functionals. In this talk I will describe some recent progress on the global maximum and local CLT for the field, thus confirming they are in the Gaussian universality class in a very strong sense. The proof uses a random walk representation (a la Helffer-Sjostrand) and an approximate harmonic coupling (by J. Miller).

Affine manifolds, i.e. manifolds which admit charts given by affine transformations, remain mysterious by the very few explicit examples and their famous open conjectures: the Auslander Conjecture, the Chern Conjecture and the Markus Conjecture. I will discuss an intermediate conjecture, somehow between the Auslander Conjecture and the Chern Conjecture, predicting the vanishing of the simplicial volume of affine manifolds. In a joint work with Chris Connell and Jean-François Lafont, we prove the latter conjecture under some hypothesis, thus providing further evidence for the veracity of the Auslander and Chern Conjectures. To do so, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups in their fundamental group to have vanishing simplicial volume. This answers a special case of a question due to Lück.
Joint work with Chris Connell and Jean-François Lafont.

Quasifuchsian manifolds are an important class of hyperbolic 3-manifolds, classically parametrized by two copies of Teichmüller space. Their volume is infinite, but they have a well-defined finite « renormalized volume » which has nice properties, both analytic and « coarse ». In particular, considered as a function over Teichmüller space, the renormalized volume provides a Kähler potential for the Weil-Petersson metric; moreover, it is within bounded additive constants of the volume of the convex core and is bounded from above by the Weil-Petersson distance between the conformal structures at infinity. After describing these properties, we will outline some recent applications (by Kojima, McShane, Brock, Bromberg, Bridgeman, and others) to the Weil-Petersson geometry of Teichmüller space or the geometry of hyperbolic 3-manifolds that fiber over the circle. We will then explain how properties of the renormalized volume suggest new questions and viewpoints on quasifuchsian manifolds.
The talk will be accessible to nonexperts.

On ne présente pas Grothendieck dans ce qui fut son enceinte, mais son héritage mathématique est si vaste qu’on risque toujours d’en négliger une partie : analyse fonctionnelle, algèbre homologique, géométrie algébrique, théorie des nombres …
Toutes ces méthodes sont encore d’actualité, et j’essayerai d’en donner une description vivante.

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.