HAGFI - Hypoellipticity, Analysis on Groups and Functional Inequalities

Organisers: W. Hebisch (Wroclaw), B. Zegarlinski (London)

Invited Speakers

Jean-Philippe Anker (Orleans)

Dominique Bakry (Toulouse)

Martin Hairer (Warwick)

Krzysztof Oleszkiewicz (Warsaw)

Cedric Villani (Lyon)

Jean-Philippe Anker

Title : "Evolution equations on homogeneous spaces"
Abstract : We shall discuss the heat equation, the Schrödinger equation
and the wave equation in various settings. We shall first consider hyperbolic spaces
and present the state of the art in this model case. We shall next consider related results
for certain Lie groups (e.g. semisimple), homogeneous spaces (symmetric spaces),
or discrete structures (homogeneous trees, buildings).

Dominique Bakry

Title : "Gradient bounds for hypoelliptic heat equations"

Abstract : In this talk, we shall present some result on gradient bounds for different kind
of hypoelliptic heat equations, and describe some families of functional inequalities associated
with the corresponding heat kernels, such as spectral gaps, log-sobolev,  isoperimetry,
Li-Yau estimates, etc.. We shall concentrate mainly on the simplest model, the case
of Heisenberg groups, and show which part of these results which are valid
in the simplest situations may be extended to  a more general hypoelliptic setting

Martin Hairer

Title : "Slow energy dissipation in anharmonic chains"

Abstract : We study the dynamic of a very simple chain of three anharmonic oscillators with linear
nearest-neighbour couplings. The first and the last oscillator furthermore interact with heat baths
through friction and noise terms. If all oscillators in such a system are coupled to heat baths, it is well-
known that under relatively weak coercivity assumptions, the system has a spectral gap (even compact
resolvent) and returns to equilibrium exponentially fast. It turns out that while it is still possible to
show the existence and uniqueness of an invariant measure for our system, it returns to equilibrium
much slower than one would at first expect. In particular, it no longer has compact resolvent when
the pinning potential of the oscillators is quartic and the spectral gap is destroyed when the potential
grows faster than that.

Krzysztof Oleszkiewicz
Title "Noise stability of functions with low influences"

Abstract: We shall discuss stability properties of bounded functions on
the discrete cube under assumption that they have low influences
(i.e. by changing the sign of a single coordinate one does not
significantly change the function). This class of functions was considered
in the influential 1988 article by Kahn, Kalai and Linial and since then it became an
important subject in the dicsrete harmonic analysis and in the theoretical
computer science. An invariance principle allowing to transfer problems
from the dicsrete cube setting to a Gaussian space will be described.
We shall demonstrate its application to the proof of two stability type
results, obtained in the joint work with Elchanan Mossel and Ryan