5ECM
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
O'Donnell.