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.

« 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).

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

It has been realized recently that the c=1 conformal block of 4 point function in Liouville CFT is related to the Tau function of the Painlevé 6 integrable system. Here we propose a general construction: starting from a very general integrable system (a Hitchin system: the moduli space of flat G-connections over a Riemann surface, with G an arbitrary semi-simple Lie group), we define some « amplitudes », and we show that these amplitudes satisfy all the axioms of a CFT: they satisfy OPEs, Ward identities and crossing symmetry. The construction is very geometrical, by defining a notion of « quantum spectral curve » attached to a flat connection, defining homology and cohomology on it, and showing that amplitudes satisfy Seiberg-Witten like relations, and behave well under modular transformations.
So this link between CFTs and integrable systems unearths a new and beautiful quantum geometry.

This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.

Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds’ liftings.