2 may 2009 - 31 jul 2009   [open in google calendar]

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 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 study the structure of Riemannian and K\"ahler 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 website of the trimester.

The official website of the trimester is http:/cvgmt.sns.itGFO, where you can find information about registration and application forms.

There is a mailing list for participants or interested people to get up--to--date information on the trimester at http:/cvgmt.sns.itgfopisa

If you have problems in registering to the list, please write an email to c.mantegazza@sns.it, possibly putting GFO in the subject.

{\bf{Scientific Committee}}

\noindent Luigi Ambrosio -- SNS, Pisa

\noindent Gérard Besson -- Institut Joseph Fourier, Grenoble

\noindent Jean-Pierre Bourguignon -- IHES, Bures-sur-Yvette

\noindent Sun-Yung Alice Chang -- Princeton

\noindent Gerhard Huisken -- Max-Planck-Institut für Gravitationsphysik, Albert Einstein Institute, Golm

{\bf{Organizing Committee}}

\noindent Luigi Ambrosio -- SNS, Pisa

\noindent Zindine Djadli -- Institut Joseph Fourier, Grenoble

\noindent Carlo Mantegazza -- SNS, Pisa

\noindent Andrea Malchiodi -- SISSA, Trieste

{\bf{Main Calendar}}

{\Large{

\begin{tabular}{
l
l
} \hline \em{Week} & \em{Events} \hline \multicolumn{2}{
c
}{\bf{May}}

\hline Monday 4 -- Friday 8 & {Gursky Lectures I} \hline Monday 11 -- Friday 15 & {Gursky Lectures II} \hline Monday 18 -- Friday 22 & {Carfora Lectures I} \hline Monday 25 -- Friday 29 & {Carfora Lectures II} \hline \multicolumn{2}{
c
}{\bf{June}}

\hline Monday 1 -- Friday 5 & \hline Monday 8 -- Friday 12 & {MalchiodiViaclovsky Lectures} \hline Monday 15 -- Friday 19 & {Ni Lectures} \hline Monday 22 -- Friday 26 & {Workshop on GeoFlows in MathPhysics}

\hline Monday 29 -- Friday 3 & {General Conference ``GFO in Pisa''} \hline \multicolumn{2}{
c
}{\bf{July}}

\hline Monday 6 -- Friday 10 & {WorkGroup on Mean Curvature Flow I} \hline Monday 13 -- Friday 17 & {WorkGroup on Mean Curvature Flow II} \hline Monday 20 -- Friday 24 & \hline \end{tabular} }}