After a series of partial results, Ilya Kapovich proved in 2009 that if G  is a non-elementary geometrically finite Kleinian group acting on a finitely-dimensional unit ball by hyperbolic isometries, then either the Hausdorff dimension of the limit set L(G ) of G  is strictly larger than its topological dimension k , or else, L(G )  is a geometric k -sphere.

Using the methods, different than Kapovich's ones, stemming from the theory of conformal iterated function systems and geometric measure theory (recifiability), we shall formulate and discuss the proof of an extension of Kapovich result to the case, where geometric finiteness is replaced by the weaker requirement that the Hausdorff dimension of the limit points that are not radial, is smaller than the Hausdorff dimension of the set of radial points.

A counterpart of this theorem for rational functions of the Riemann sphere will be also discussed.

Finally, the case of Kleinian groups acting on the unit ball of infinitely dimensional separable Hilbert space will be considered.

Tits defined Kac-Moody groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Tits' presentation can be simplified considerably when the Dynkin diagram is hyperbolic and simply laced.  Over finitely generated rings R, we give finitely many generators and defining relations parametrized over R and we describe a further simplification for R = Z. We highlight the role of the group E10(R), conjectured to play a role in the unification of superstring theories.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: La catégorie dérivée des faisceaux cohérents d'une variété projective lisse est souvent considérée comme un invariant algébrique très riche de cette variété. Les travaux de Bondal et Orlov montrent par exemple que deux variétés projectives dont les catégories dérivées sont équivalentes ont même anneau (anti)-canonique. La catégorie dérivée encode donc des informations birationnelles importantes. On pourrait se demander si elle contient des informations de nature topologique ou analytique. On souhaiterait par exemple savoir si deux variétés dont les catégories dérivées sont équivalentes ont même nombres de Hodge.

Dans cet exposé, je ferai un état de l'art rapide lié a cette question et je développerai quelques résultats que j'ai obtenus et qui suggèrent qu'une certaine sous-algèbre de l'algèbre de Hodge est invariante par équivalence dérivée. La définition de cette algèbre s'étend au cadre non-commutatif et sa construction permet d'obtenir une grande partie de la structure de Hodge conjecturale d'une variété Calabi-Yau de dimension 3 non-commutative.

After reviewing the well known fact that bosonic « Starobinsky » R + f(R) models of inflation are equivalent to standard single-scalar slow-roll theories, we show how to embed R + f(R) models in supergravity, in an efficient way, using supergravity’s « new minimal » formulation. We also discuss how supergravity can improve the naturalness of such models and conclude presenting a rather general class of R + f(R) models coupled to matter.

« Return of the IHES Postdoc Seminar »

 

Abstract: I will start by reviewing the main features of Kuznetsov's Homological projective duality (HPD for short).  Then I will explain how, inspired by Calabrese and Thomas' construction of derived equivalent Calabi-Yau 3-folds, one gets the idea of considering a natural HPD problem for linear systems with base loci. Inparticular, this yields a process for constructing new HP duals from old. Finally, we will explicitly apply the procedure in the example of singular cubic 4-folds. This is joint work with Zak Turcinovic (arXiv:1511.09398).

We define pentagram maps on polygons in any dimension, which extend R. Schwartz’s definition of the 2D pentagram map, as well as describe recent results on integrable cases for these higher-dimensional generalizations. The corresponding continuous limits of such maps coincide with equations of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We discuss their geometry and a numerical evidence of non-integrability of certain cases. This is a joint work with Fedor Soloviev (Univ. of Toronto).

Séminaire Laurent Schwartz — EDP et applications

I will talk about recent progress in understanding quantum Hall states on curved backgrounds and in inhomogeneous magnetic fields and their large N limits, N being the number of particles. The large N limit of the free energy of the Laughlin states in the integer Quantum Hall is controlled by the Bergman kernel expansion, and, in a sense, is exactly solvable to all orders in 1/N. For the fractional Laughlin states, the large N limit can be determined from free field representation. The terms in the large N expansion are given by various geometric functionals. In particular, the Liouville action shows up at the order O(1) in the expansion, and signifies the effect gravitational anomaly. The appearance of this term leads us to argue for the existence of a third quantized kinetic coefficient, precise on the Hall plateaus, in addition to Hall conductance and anomalous viscosity. Based on: 1309.7333, 1410.6802, 1504.07198 and upcoming work.

Séminaire Laurent Schwartz — EDP et applications

The goal of this talk is to discuss some properties of line defects in certain supersymmetric QFTs, the so-called theories of class S. I will spend some time reviewing some old work of Gaiotto Moore and Neitzke on the emergence of Hitchin systems in these theories, and on certain coordinates on the associated moduli spaces. Line defects can then be understood as certain functions on these moduli spaces. If there is time I will present new results. The tone of the discussion will be informal.

Séminaire Laurent Schwartz — EDP et applications