**Luigi
Ambrosio** (Scuola Normale Superiore di Pisa)

*A crash course on Gamma convergence*

The theory of Gamma-convergence, introduced by E.De Giorgi in the '70,
is a powerful mathematical tool that allows to study the limiting behaviour
of variational problems. The great flexibility of this concept allows
to treat in a rigorous manner many convergence problems: sharp phase transition
models as limits of diffuse phase transition models, the derivation of
continuum laws by discrete (atomistic) ones, homogenization, dimension
reduction, singular perturbation, etc. In the course the basic concepts
of the theory will be presented and then a series of examples will be
given, mostly in the 1-dimensional case to avoid technical issues. If
time allows, also the convergence of evolution problems (gradient flows)
will be illustrated. The course will be based on A.Braides' monograph
"Gamma-convergence: a beginner's guide" and, for the evolution
problems, on its more recent monograph.

Reference: A. Braides. Gamma-convergence for Beginners.

Oxford University Press, 2002

**Robert Jerrard**
(University of Toronto)

*Variational methods for effective dynamics*

in many problems, one wants to show that the evolution of a complicated
physical system can be reduced, at least under certain circumstances,
to a lower-dimensional problem. For example, one might want to prove that
2d fluids, under suitable conditions, possess objects one may call "point
vortices"; that the evolution of the fluid is largely determined
once one knows how these point vortices evolve; and that one can derive
an equation governing point vortex dynamics. This course will examine,
largely by consideration of several concrete model problems, ways in which
techniques from the calculus of variations can be useful in problems of
this type. We will focus on problems of hyperbolic and Schroedinger type,
for which it is not a priori clear that variational methods are relevant
or useful.

**Felix
Otto** (Max-Planck Institute for Mathematics in the Natural Sciences
at Leipzig)

TBA:

**Mary Pugh*** (University
of Toronto)

TBA:

**Robert
Seiringer**** **(Institute for Science and Technology, Austria)

*Structure of the excitation spectrum for many-body quantum systems*

Many questions concerning models in quantum mechanics require a detailed
analysis of the spectrum of the corresponding Hamiltonian, a linear operator
on a suitable Hilbert space. Of particular relevance for an understanding
of the low-temperature properties of a system is the structure of the
excitation spectrum, which is the part of the spectrum close to the spectral
bottom. We present recent progress on this question for bosonic many-body
quantum systems with weak two-body interactions. Such system are currently
of great interest, due to their experimental realization in ultra-cold
atomic gases. We investigate the accuracy of the Bogoliubov approximations,
which predicts that the low-energy spectrum is made up of sums of elementary
excitations, with linear dispersion law at low momentum. The latter property
is crucial for the superfluid behavior the system.

* to be confirmed