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.it/GFO,
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.it/gfopisa

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 & {Malchiodi/Viaclovsky Lectures}
\hline
Monday 15 -- Friday 19 & {Ni Lectures}
\hline
Monday 22 -- Friday 26 & {Workshop on GeoFlows in Math/Physics}

\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}
}}
http://cvgmt.sns.it/event/98/