We discuss some recent work on a family of weighted
Hardy-Sobolev inequalities due to Caffarelli-Kohn-Nirenberg
(1984),
including symmetry property and symmetry breaking of extremal
functions,
improved Hardy inequalities, as well as bound state solutions to
the associated nonlinear PDEs.

We study well-posedness and asymptotic behaviour of the
Cahn-Hilliard
equation with dynamic boundary condition. This modification of the
usual
non-flux condition has been introduced to incorporate surface
effects. By
means of optimal regularity results in the L_p-setting for the
linearized problem, well-posedness and the global semiflow in an
appropriate phase space are obtained. We also show convergence of
the
solutions to equilibrium states in energy norm. This result is
proved
via the recent Lojasiewicz technique.

In 1967 Furstenberg discovered a very surprising phenomenon:
while both $T: x \to 2 x \bmod 1$ and $s: x \to 3 x \bmod 1$ on $\R / \Z$ have many closed invariant sets, closed sets which are invariant under both $T$ and $S$ are very rare (indeed, are either finite sets of rationals or $\R / \Z$). Furstenberg also conjectured that a similar result holds for invariant measures. This conjecture is of course still open.
As has been shown by several authors, including Katok-Spatzier and Margulis this phenomenon is not an isolated curiosity but rather a deep property of many natural $\Z ^ d$ and $\R ^ d$ actions ($d > 1$) with many applications.
Recently, there has been substantial progress in the study of measures invariant under such actions. While we are at present still far from full resolution of this intriguing question, the partial results we currently
have are already powerful enough to prove results in other fields. In particular, these techniques enable proving a special but important case of Rudnick and Sarnak's Quantum Unique Ergodicity Conjecture, as well as a partial result towards Littlewood's Conjecture on simultaneous diophantine approximations (the later is in a joint paper with M. Einsiedler and A. Katok).