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.

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.

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).

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.

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.

The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open.

In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication.The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.

Let G be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(R^n) = O(n) ltimes R^n. By symmetries of G we understand a group Out(G). For any n ≥ 2, we shall construct a crystallographic group with trivial center and a trivial outer automorphism group. Moreover, we shall present properties of an example (constructed by R. Waldmuler in 2003) of the torsion free crystallographic group of dimension 141 with a trivial center and a trivial outer automorphism group. (It is a joint work with R. Lutowski.)

Ambitwistor strings are holomorphic string theories whose target space is the space of complex null geoedesics in a complexified space-times. I will explain how these theories explain the origin of the scattering equations in twistor strings and the CHY formulae in arbitrary dimensions and provide a reformulation of standard gauge, gravity and other theories in a holomorphic infinite tension analogue of conventional string theories. I will show how these results extend to 1-loop both on a torus and on a nodal Riemann sphere.

Second order operator $Delta$ on half-densities can be uniquely defined by its principal symbol $E$ up to a `potential' $U$. If $Delta$ is an odd operator such that order of operator $Delta^2$ is less than $3$ then principal symbol $E$ of this operator defines an odd Poisson bracket. We define the modular class of an odd Poisson supermanifold in terms of $Delta$ operator defining the odd Poisson structure. In the case of non-degenerate odd Poisson structure (odd symplectic case) the modular class vanishes, and we come to canonical odd Laplacian on half-densities, the main ingridient of Batalin-Vilkovisky
formalism. Then we consider examples of odd Poisson supermanifolds with non-trivial modular classes related with the Nijenhuis bracket.

The talk is based on the joint paper with M. Peddie: arXive: 1509.05686

« Return of the IHÉS Postdoc Seminar »

 

Abstract: In 1970, Manin observed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the lack of a K-rational point on X despite the existence of points everywhere locally is sometimes explained by non-trivial elements in Br(X). Since Manin's observation, the Brauer group and the related obstructions have been the subject of a great deal of research.

The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. The transcendental part of the Brauer group is generally more mysterious, but it is known to have arithmetic importance – it can obstruct the Hasse principle and weak approximation.
I will describe recent progress in computing transcendental Brauer groups and obstructions, and give examples where there is no Brauer-Manin obstruction coming from the algebraic part of the Brauer group but a transcendental Brauer class explains why the rational points of a variety fail to be dense in the set of its adelic points.

Séminaire Laurent Schwartz — EDP et applications

« 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.