Many concrete deterministic dynamical systems exhibit apparently random
behaviour. This puzzle is studied my finding a time invariant probability
measure and discussing the statistical phenomenon using this measure. In
this way various systems (e.g. given by PDE's) can be said to be
"completely random" or "completely deterministic".

This leads to the project of classifying invariant probability measures.
The ergodic decomposition theorem shows that the basic building blocks of
these measures are the ergodic measures, which form a dense G_\delta set.
The equivalence relation of isomorphism is given by a Polish group action.
Thus the tools of descriptive set theory directly apply and one can show
that the action is "turbulent" and complete analytic. This precludes any
kind of recognizable classification.

This is joint work with Dan Rudolph and Benjy Weiss.

Mean-field theory is one of the most standard tools used by
physicists to analyze phase transitions in realistic systems. However,
regarding rigorous proofs, the link to mean-field theory has been
limited to asymptotic statements which do not yield enough control
of the actual systems. In this talk I will describe a new approach to
this set of problems -- developed jointly with Lincoln Chayes and
Nicolas Crawford -- that overcomes this hurdle in a rather elegant
way. As a conclusion, I will show that a general, ferromagnetic
nearest neighbor spin system on Z^d undergoes a first order phase
transition whenever the mean-field theory indicates one, provided
the dimension d is sufficiently large. Extensions to systems with non
nearest neighbor interactions will also be discussed.

We have developed and analysed a new class of discontinuous Galerkin
methods (DG) for wave equations. The new method
can be seen as a compromise between standard DG and finite element
method
(FEM) in the way that it is
explicit as standard DG and energy conserving as FEM.
There are in the literature many methods that achieves some of the
goals
of explicit time marching, unstructured grid, energy conservation and
optimal higher order accuracy, but as far as we know only our new
algorithms satisfy all the conditions.
Stability and convergence of the new method are rigorously analysed.
The convergence rate is optimal with respect to the order of the polynomials in the finite
element spaces.
Moreover, the convergence is described by a series of numerical
experiments.
This is a joint work with Bjorn Engquist.