14 July - 19 December 2008

**Organisers:** Professor Y Fyodorov *(Nottingham)*, Professor I Goldsheid *(Queen Mary, London)*, Professor T Spencer *(Institute for Advanced Study)* and Professor MR Zirnbauer *(Institut für Theoretische Physik, Köln)*

In his seminal paper *Absence of diffusion in certain random lattices*
(1958) Philip W. Anderson discovered one of the most striking quantum
interference phenomena: particle localization due to disorder. Cited
in 1977 for the Nobel prize in physics, that paper was fundamental for
many subsequent developments in condensed matter theory. In
particular, in the last 25 years the phenomenon of localization proved
to be crucial for the understanding of the Quantum Hall effect,
mesoscopic fluctuations in small conductors as well as some aspects of
quantum chaotic behaviour.

Random Schrödinger operators are an area of very active research in mathematical physics and mathematics. Here the main effort is to clarify the nature of the underlying spectrum. In particular, it has been proved that in dimension one all states are localized, and in any dimension the random Schrödinger operator has dense point spectrum for large enough disorder. Some open mathematical problems of major importance include the long-time evolutions of a quantum particle in a weakly disordered medium and existence of absolutely continuous spectrum in three dimensions. The expected transition from localized (point spectrum) to extended eigenstates (absolutely continuous spectrum) will also be addressed.

The goal of the program is to bring together the world leaders in
spectral theory of random Schrödinger operators and theoretical
physicists successfully working on the problem of Anderson
localization. Among the topics that will be addressed during the
program are: The nature of critical phenomena associated with
localization-delocalization transitions; The existence and
statistical properties of extended states for *D* > 2 and the
behaviour in the critical dimension *D* = 2; rigorous version of
supersymmetric methods and of the nonlinear *σ*-model
techniques; the localization-delocalization phenomena associated
with the Integer Quantum Hall effect; rigorous mathematical
understanding of the relation between magnetic Schrödinger
operators and network models, and the connection with quantum
percolation; localization in the presence of a random magnetic
field; behaviour of products of random matrices and associated
Lyapunov exponents; localization and delocalization in disordered
systems characterised by non-selfadjoint operators; dynamical
localization in Quantum Chaotic systems; localization in systems
with aperiodic potential, as well as in models with correlated or
long-ranged disorder; localization in systems with nonlinearities,
and localization-delocalization phenomena in disordered systems of
interacting quantum particles.