HAGFI - Hypoellipticity, Analysis on Groups and
Organisers: W. Hebisch (Wroclaw), B.
Krzysztof Oleszkiewicz (Warsaw)
Cedric Villani (Lyon)
Title : "Evolution equations on homogeneous
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
certain Lie groups (e.g. semisimple), homogeneous spaces (symmetric
or discrete structures (homogeneous trees, buildings).
Title : "Gradient bounds for hypoelliptic heat
Abstract : In this talk, we shall present
some result on gradient bounds for different kind
hypoelliptic heat equations, and describe some families of functional
corresponding heat kernels, such as spectral gaps, log-sobolev,
estimates, etc.. We shall concentrate mainly on the simplest model,
Heisenberg groups, and show which part of these results which are
simplest situations may be extended to a more general
"Slow energy dissipation in anharmonic chains"
: 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
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
resolvent) and returns to equilibrium exponentially fast.
It turns out that while it is still possible to
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
pinning potential of the oscillators is quartic and the spectral gap
is destroyed when the potential
grows faster than that.
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