# Geometric Flows and Geometric Operators

## In collaboration with INdAM

2 May 2009 - 30 July 2009

# Aims

The trimester will concentrate on two main topics, related to series of lectures at introductory level.

The first one is the study of geometrically invariant operators, possibly of higher order. In recent years, there has been an intensive study of the relations between this kind of operators (which satisfy some invariance property under conformal change of metric), their associated scalar invariants and the study of the related partial differential equations. In two dimensions for example, the integral of the Gauss curvature (whose transformation law under conformal changes of metrics is related to the Laplace operator) determines completely the topology of the surface. In four dimensions there are partial analogues, involving the Q-curvature and the Paneitz operator, but most results are limited to the case of manifolds with positive curvature: it would be interesting to relax these assumptions and consider more general situations. Lectures related to this research direction will be given by Matt Gursky, Jeff Viaclovsky and Andrea Malchiodi.

The second topic will concern geometric flows, in particular, higher order flows, "coupling" of flows and relations to theoretical physics. Lei Ni will present recent progresses in applying Ricci flow to the study of the structure of Riemannian and Kähler manifolds. Mauro Carfora will discuss some recent ideas in setting connections between geometric flows and theoretical physics, in particular with quantum field theories, string theory and renormalization group techniques. A workshop putting together leading theoretical physicists and mathematicians is planned for the second part of June. In July we plan to organize a workgroup on mean curvature flow with the aim to follow the classical'' line of analysis of the evolution till the first singular time (the works of Ecker, Huisken, Hamilton, etc).

The recent results about Ricci flow gave a strong impulse to the field of evolution of geometric structures, but not so much is known at the moment about other "natural" flows which can be used analogously to find out canonical metrics or to investigate the geometry and topology of manifolds, in particular in higher dimensions. One of the goal of the trimester is to try to generalize these techniques, considering other flows associated to geometric quantities like functionals of the curvatures, with special care for the ones which are conformally invariant, for instance, the Q-curvature. This analysis leads to higher order flows and it is one of our motivations to put together people working on higher order differential operators and on geometric flows.

At the end of June there will be the general conference of the trimester.

We plan, in parallel to the series of lectures, to organize daily seminars given by the guests at the Center De Giorgi. All the people will be invited to give a talk on their research activities.

Financial support is available for young students. Interested people need to apply at the link above.

The trimester is organized in collaboration with INdAM, GNFM, Institut Joseph Fourier, Grenoble, Project ANR "Flot et Operateurs Geometriques" ANR-07-BLAN-0251-01, SISSA - Trieste, FIRB Ideas "Analysis and Beyond".

Some photos from the workshops by Sylvain Maillot. Photos by Leili Shahriyari. Other photos can be found in the Documents section. Reto Muller has a photo album dedicated to the trimester in his facebook page.