Parreau compactified the Hitchin component of a closed surface S of genus greater or equal to two in such a way that a boundary point corresponds to the projectivized length spectrum of an action of pi_1(S) on an R-building. We will explain how to use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.

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.

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.

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.

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.

I will present some history and histories around the first years of the IHES and answer (briefly) the questions: By whom? Why? How? Where? When? was the IHES created, with documents from the archives – letters, administrative as well as mathematical documents – and people's memories.

Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of BunG, and on trace formulas. Work with Will Sawin.

The notion of Cohomological Hall algebra (COHA) was introduced in our joint paper with Maxim Kontsevich 10 years ago. It can be thought of as a mathematical incarnation of the notion of BPS algebra envisioned by physicists Harvey and Moore in the 90's.

Mathematically, COHA is an associative algebra structure on the cohomology of the moduli stack of objects of a 3-dimensional Calabi-Yau category with coefficients in a certain constructible sheaf. Interesting categories can be of geometric or algebraic origin (sheaves on Calabi-Yau 3-folds, quivers with potential, etc.).

In the talk I plan to discuss actions of COHA on the cohomology of certain instanton moduli spaces (spiked instantons of Nekrasov). This gives a relationship of COHA with affine Yangians and more recent "vertex algebras at the corner" introduced  by Gaiotto and Rapcak.

Recent advances in two-loop superstring theory will be discussed, including the structure of supermoduli space, the spontaneous supersymmetry breaking on Calabi-Yau orbifolds, and the matching of the D6 R4 effective low energy corrections to supergravity with predictions from supersymmetry and duality.