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.

I will talk about a method of constructing virtual fundamental cycles on moduli spaces of J-holomorphic maps.  The construction uses quite a bit of homological algebra, in particular homotopy colimits and homotopy sheaves, and most of the action happens "at the chain level".  I will also mention some applications to existing and conjectural enumerative invariants in symplectic and contact geometry.

Séminaire Laurent Schwartz — EDP et applications

The black hole firewall problem is a conflict between three important physical principles: causality, unitarity, and the equivalence principle. I will describe how the conflict arises in the description of Hawking radiation from a black hole, and explain why resolving the problem is crucially important for our understanding of quantum gravity. I will briefly describe possible resolutions of the conflict. Solving the firewall problem is likely to teach us something important about quantum gravity.

Dans ce cours nous étudierons plusieurs modèles de graphes aléatoires allant du plus classique (le modèle d'Erdös-Renyi introduit en 1960) aux plus récents (les cartes planaires aléatoires étudiées depuis le début des années 2000). Le fil conducteur du cours sera la notion de convergence locale et les propriétés des graphes limites dits dilués.

Contenu du cours :

– Modèle d'Erdös-Renyi, transition de phase et propriétés de base
– Convergence locale et "méthode objective" d'Aldous et Steele
– Arbre couvrant minimum et théorème de Frieze 
– Graphes aléatoires unimodulaires
– Limites locales d'arbres aléatoires
– Limites locales de cartes aléatoires (construction, épluchage, théorème de Benjamini-Schramm)

The goal of this talk is to give a pedagogical introduction to Bell’s theorem and its implication for our view of the physical world, in particular how it establishes the existence of non-local effects or of actions at a distance. We also discuss several misunderstandings of Bell’s result and we will explain how the de Broglie-Bohm theory allows us to understand, to some extent, what non-locality is.

Dans cet exposé, je présenterai certains résultats mathématiques  récents concernant la limite de champ moyen pour des systèmes quantiques en  interaction, le lien avec la condensation de Bose-Einstein et la théorie de  Bogoliubov.