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.

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The talk will review the birth of GR on the background of historical documents, and in particular on the basis of Einstein’s extant manuscripts.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.

Journée autour des régulateurs et des valeurs de fonctions L.

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators.  We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE and is analytic in s, analogous to the Caron-Huot formula for the four-point function. Some important assumptions are made in deriving this result: we comment on them.

Hall algebras play an important role in representation theory, algebraic geometry and combinatorics. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. We consider twisted and extended Hall algebras of triangulated categories and note that in some cases they are well-defined even when their non-extended counterparts are not. We show that each exact category with weak equivalences with an appropriate extra structure naturally gives rise a twisted extended Hall algebra of its homotopy category. If time permits, we will discuss the relation of this construction to graded quiver varieties and to categorification of modified quantum groups.

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.

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.

(Joint work with Yi Huang) Goncharov and Shen introduced a Landau-Ginzberg potential on the Fock-Goncharov $A_{G,S}$ moduli space, where $G$ is a semisimple Lie group and $S$ is a ciliated surface. They used the potential to formulate a mirror symmetry via Geometric Satake Correspondence. This potential is the markoff equation for $A_{ PSL(2,R), S_{1,1} }$. When $S=S_{g,m}$, such potential can be written as a sum of rank $G*m$ partial potentials. We obtain a family of generalized Mcshane's identities by splitting these partial potentials for $A_{PSL(n,R),S_{g,m}}$ by certain pattern of cluster transformations with geometric meaning. We also find some interesting new phenomena in higher rank case, like triple ratio is bounded in mapping class group orbit. As applications, we find a generalized collar lemma which involves $lambda 1 / lambda 2$ length spectral, discreteness of that spectral etc. In further research, we would like to ask how can we integrate to obtain the generalized Mirzakhani's topological recursion with $mathcal{W}_n$ constraint?

Degenerations of maximal representations of a surface group may be seen as maximal representations in Sp(2n,F) for some non-Archimedean real closed field F. We associate to every such maximal representation a geodesic current whose intersection number is the length function of the representation for the L1 norm. When  the current is a measured lamination, we reconstruct an equivariant isometric embedding of the dual real tree in the Bruhat-Tits  building of Sp(2n,F).  This involves a general construction of an intersection current associated to a non necessarily continuous positive cross-ratio.

This is joint work with Marc Burger, Alessandra Iozzi, and Beatrice Pozzetti.