Nous organisons un petit groupe de travail pour essayer de mieux comprendre les liens entre la méthode de BKW complexe, la correspondance de Hodge nonabélienne sauvage et la récursion toplogique d'Eynard-Orantin.

The basic aim is to try to better understand the relation between exact WKB, wild nonabelian Hodge theory and the topological recursion of Eynard-Orantin, as well as links to the (nonlinear) Stokes phenomenon.

I show how massless supergravity theories can evade the theorems that try to establish that there are « No Solitons without Horizons. » The explicit construction of smooth, horizonless BPS solitons will be reviewed and some of their physical properties will be discussed. The possible role of such solitons in describing black-hole microstates will be examined.

The arithmetic geometry of Shimura varieties has been intensively studied since, about twenty years ago, Kudla made some conjectures relating their arithmetic Chow groups with derivatives of Eisenstein series and of Rankin-Selberg L-functions. The conjectures concern special cycles in orthogonal and unitary Shimura varieties and predict in particular that Green currents for these cycles should exist satisfying some additional properties, including an explicit expression for archimedean height pairings.

I will explain how to attach a natural superconnection to each special cycle and how results of Quillen and further developments by Bismut, Gillet and Soule allow to define natural Green forms for special cycles. For compact Shimura varieties with underlying group O(p,2) or U(p,1) I will explain how to compute the resulting archimedean heights and relate them to derivatives of Eisenstein series, essentially settling the archimedean aspect of Kudla's conjectures in this case. This is joint work with Siddarth Sankaran.

The talk will review the birth of GR on the background of historical documents, and in particular on the basis of Einstein’s extant manuscripts. It will show how Einstein’s work emerged from a transformation of the knowledge of classical physics in a process that involved an interaction between the development of mathematical formalism and its physical interpretation. It will also emphasize the role of philosophical reflections for both the heuristics and the interpretation of the theory.

In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnorian. We will talk about the progress made so far towards classification of these non-Milnorian representations.

Journée autour des régulateurs et des valeurs de fonctions L.

A bi-Lagrangian structure on a manifold is the data of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, i.e. the para-complex analog of a Kähler structure. After discussing interesting features of bi-Lagrangian structures in the real and complex settings, I will show that the complexification of any Kähler manifold has a natural complex bi-Lagrangian structure. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which typically have a rich symplectic geometry. We will see that that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory while revealing new other features, and derive a few well-known results of Teichmüller theory. Time permitting, I will present the construction of an almost hyper-Kähler structure in the complexification of any Kähler manifold. This is joint work with Andy Sanders.

Nous organisons un petit groupe de travail pour essayer de mieux comprendre les liens entre la méthode de BKW complexe, la correspondance de Hodge nonabélienne sauvage et la récursion toplogique d'Eynard-Orantin.

The basic aim is to try to better understand the relation between exact WKB, wild nonabelian Hodge theory and the topological recursion of Eynard-Orantin, as well as links to the (nonlinear) Stokes phenomenon.

A classical result of Myers and Steenrod states that the isometry group of a compact Riemannian manifold is a compact Lie transformation group. Also classical is the fact that this compactness property fails for general pseudo-Riemannian manifolds, allowing interesting dynamics for the group of isometries. In this talk, we will be interested by the topological, and dynamical consequences of the noncompactness of the isometry group. We will especially focus on the case of Lorentz manifolds, and we will present a complete topological classification of 3-dimensional closed Lorentz manifolds having a noncompact isometry group.

We will first explain the concepts of relatively hyperbolic group and the Bowditch boundary. We will then give some interesting examples of groups whose boundaries embed in the two-sphere. The most prominent family of this type is the class of geometrically finite Kleinian groups. However, we show that there are lots of relatively hyperbolic groups with planar boundaries that are not virtually Kleinian. We formulate a conjecture about which groups with planar boundary are virtually Kleinian, and prove this in a certain case. This is joint work in progress with Chris Hruska.

Two versions of the SYZ conjecture proposed by Kontsevich and Soibelman give a differential-geometric and a non-Archimedean recipes to find the base of the SYZ fibration associated to a family of Calabi-Yau manifolds with maximal unipotent monodromy. In the first one this space is the Gromov-Hausdorff limit of associated geodesic metric spaces, and in the second one it is a subset of the Berkovich analytification of the associated variety over the field of germs of meromorphic functions over a punctured disc. In this talk I will discuss a toy version of a comparison between the two pictures for maximal unipotent degenerations of complex curves with flat metrics with conical singularities, and speculate how the techniques used can be extended to higher dimensions.

Séminaire de Probabilités et de Physique Théorique

 

We present results on the critical behaviour of long-range models of multi-component ferromagnetic spins and weakly self-avoiding walk in dimensions 1, 2, and 3. The range of the interaction is adjusted so that the models are below their upper critical dimension.  Critical exponents are computed for the susceptibility, specific heat, and critical two-point function, using a renormalisation group method to perturb around a non-Gaussian fixed point.  This provides a mathematically rigorous version of the epsilon expansion.