A simple étude on the mathematics of twistor integrals.

Il s‘agit d’un travail en commun avec Jeremy Kahn et Shahar Mozes. Nous montrons que les réseaux dans certains groupes de Lie non compacts — en particulier tous les groupes complexes — possèdent des sous-groupes isomorphes à des groupes de surfaces. Nous montrons de plus que ces sous-groupes sont « quasi-symmétriques » par rapport à un choix préalable d’un SL(2) en un sens à préciser. Nous donnerons quelques idées de la preuve, qui suit le schéma du travail fondateur de Kahn—Markovic, en insistant sur le nouvel outil : l’étude de triangles dans les variétés de drapeaux.

Nous discuterons de tores d’Einstein et de leurs intersections, à l’aide de la correspondance entre l’univers d’Einstein de dimension trois et l’espace des plans lagrangiens dans R^4 symplectique. Comme application, nous discuterons d’un critère pour que des surfaces appelées « surfaces croches » soient disjointes, en reliant ce critère à celui de Danciger-Guéritaud-Kassel.

(Travail conjoint avec Jean-Philippe Burelle, Dominik Francoeur et Bill Goldman.)

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments also establish the existence of big Cohen-Macaulay algebras for complete local domains of mixed characteristics.

For many years researchers have tried to glean hints about quantum gravity from black hole thermodynamics. We try a different approach using a minimal statistical mechanical model for the horizon fluid based on Damour-Navier-Stokes (DNS) equation. For asymptotically flat black hole spacetimes in General Relativity, we show explicitly that at equilibrium the entropy of the horizon fluid is the Bekenstein-Hawking entropy. Further we show that, for the bulk viscosity of the fluctuations of the horizon fluid to be identical to Damour, a confinement scale exists for these fluctuations, implying quantization of the horizon area.

Thanks to recent progress in biotechnologies, the high-throughput data
in molecular biology are getting more
 and more interesting from the point of view of application of advanced
methods for data analysis, aimed
 at finding non-trivial topological characteristics in the data such as
branching points and holes. Existence
 of such non-trivial structures in the data can have direct biological
interpretations such as the process
 of cell fate decisions during cell differentiation or existence of
cyclic processes in a cell (e.g., cell cycle).

 I will present examples of application of such methods in studying
various biological systems and explain their general
 principles. I will focus on the universal method of elatic principal
graphs for topological data analysis developed by us.
 The method is based on application of the notion of harmonic graph
embedding into a multi-dimensional space,
 minimization of graph elastic energy and using graph grammars defining
a family of possible graph structures (such as trees).
 Simplest implementations of the approach already give very usefull
data approximators such as principal curves,
 principal closed curves, principal manifolds and principal trees.
Several ideas for making these data approximators
 robust to the noise and outliers in the biological data will be
presented.

I will discuss several questions and some results about algebraic flows, o-minimal flows and holomorphic flows on abelian varieties and Shimura varieties.

In this talk, we show that the elliptic deformation of W-algebra is naturally realized using quiver gauge theory in six dimensions compactified on a torus. This construction is based on the gauge theoretical realization of W-algebra proposed in our previous study [arXiv:1512.08533]. In particular, double quantization of Seiberg-Witten geometry for Γ-quiver gauge theory provides a generating current of W(Γ)-algebra in the free field realization. We also show that the partition function is given by a correlator of the corresponding W(Γ)-algebra, which is equivalent to the AGT relation under the gauge/quiver (base/fibre; spectral) duality. This talk is based on a collaboration with V. Pestun [arXiv:1608.04651]