**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)

Title : "Evolution equations on homogeneous spaces"

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).

Title : "Gradient bounds for hypoelliptic heat equations"

Abstract : In this talk, we shall present some result on gradient bounds for different kind

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.

Title "Noise stability of functions with low influences"Abstract: We shall discuss stability properties of bounded functions onthe discrete cube under assumption that they have low influences(i.e. by changing the sign of a single coordinate one does notsignificantly change the function). This class of functions was consideredin the influential 1988 article by Kahn, Kalai and Linial and since then it became animportant subject in the dicsrete harmonic analysis and in the theoreticalcomputer science. An invariance principle allowing to transfer problemsfrom the dicsrete cube setting to a Gaussian space will be described.We shall demonstrate its application to the proof of two stability typeresults, obtained in the joint work with Elchanan Mossel and RyanO'Donnell.