Mon, 20 Jan 2020

15:45 - 16:45
L3

Recent developments in random geometry

JEAN-FRANCOIS LE GALL
(Universite Paris-Sud)
Abstract

We discuss the models of random geometry that are derived
from scaling limits of large graphs embedded in the sphere and
chosen uniformly at random in a suitable class. The case of
quadrangulations with a boundary leads to the so-called
Brownian disk, which has been studied in a number of recent works.
We present a new construction of the Brownian
disk from excursion theory for Brownian motion indexed
by the Brownian tree. We also explain how the structure
of connected components of the Brownian disk above a
given height gives rise to a remarkable connection with
growth-fragmentation processes.

Mon, 18 Feb 2019

16:00 - 17:00
L4

Hypoelliptic Laplacian, Brownian motion and the trace formula

Jean-Michel Bismut
(Universite Paris-Sud)
Abstract

The hypoelliptic Laplacian is a family of operators indexed by $b \in \mathbf{R}^*_+$, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as $b \to 0$ and the generator of the geodesic flow as $b \to +\infty$. These operators are not elliptic, they are not self-adjoint, they are hypoelliptic. One can think of the total space of the tangent bundle as the phase space of classical mechanics; so that the hypoelliptic Laplacian produces an interpolation between the geodesic flow and its quantisation. There is a dynamical counterpart, which is a natural interpolation between classical Brownian motion and the geodesic flow.

The hypoelliptic deformation preserves subtle invariants of the Laplacian. In the case of locally symmetric spaces (which are defined via Lie groups), the deformation is essentially isospectral, and leads to geometric formulas for orbital integrals, a key ingredient in Selberg's trace formula.

In a first part of the talk, I will describe the geometric construction of the hypoelliptic Laplacian in the context of de Rham theory. In a second part, I will explain applications to the trace formula.

 

Thu, 31 May 2018

16:00 - 17:00
L6

Coherent sheaves on arithmetic schemes and basic results on arithmetic ampleness

François Charles
(Universite Paris-Sud)
Abstract

We will discuss a basic framework to deal with coherent sheaves on schemes over $\mathbb{Z}$, involving infinite-dimensional results on the geometry of numbers. As an application, we will discuss basic results, old and new, on arithmetic ampleness, such as Serre vanishing, Nakai-Moishezon, and Bertini. This is joint work with Jean-Benoît Bost.

Tue, 17 Jun 2014

15:45 - 16:45
L4

Torus action and Segre classes in the context of the Green-Griffiths conjecture

Lionel Darondeau
(Universite Paris-Sud)
Abstract

The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Berczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Beside the original argument coming from equivariant geometry, we will explain our alternative proof of such a formula and we will particularly be interested in the interplay between the two approaches.

Tue, 17 Jun 2014

14:00 - 15:00
L4

Jet techniques for hyperbolicity problems

Lionel Darondeau
(Universite Paris-Sud)
Abstract

Hyperbolicity is the study of the geometry of holomorphic entire curves $f:\mathbb{C}\to X$, with values in a given complex manifold $X$. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^{n}\setminus X_{d}$ of projective hypersurfaces $X_{d}$ having sufficiently high degree $d\gg n$.

Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.

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