INTRODUCTION 3A topic of current interest in mathematical physics is arithmetic quantum chaos.
Quantum chaos is concerned with the study of the semiclassical limit of the quan-
tization of a classically chaotic Hamiltonian. There is ample numerical experimen-
tation in this direction in the physics literature. This has led to many observed
phenomena about which the usual methods of microlocal analysis can say very little.
For a system such as the geodesic flow on the unit tangent bundle of an arithmetic
hyperbolic surface, number theoretic techniques can be employed to resolve some
of the central problems (this study is now termed "arithmetic quantum chaos").
The input includes the problem of subconvexity for higher degree L-functions as
well as more traditional diophantine analysis. Terras' lectures, which are included
here, give an introduction to this subject as well as a description and analysis of
some finite field analogues. Rudnick's lectures, which are not included, described
some delicate applications of the Selberg Trace Formula to study fluctuations of the
smoothed remainder term in Weyl's law for the eigenvalue count for the modular
group.
A very powerful tool that has emerged in recent years in connection with prob-
lems of equi-distribution of measures in r \ G associated with arithmetic and spec-
tral geometry is ergodic theory. Specifically Ratner's Theorem gives a complete
and simple classification of such measures which are invariant under a unipotent
action. This allows one, in cases where such an invariance can be demonstrated, to
establish quite general and often striking equi-distribution results. In his lectures
Eskin gave an account of Ratner's work as well as a number of applications. These
included quantitative versions of Oppenheim's Conjecture due to Eskin, Margulis
and Mozes as well as arithmetical applications due to Vatsal and Cornut to ranks
of an elliptic curve, defined over Q, in anticyclotomic towers.
Since the time of the conference there have been many impressive advances
in the above topics. Noteworthy among them is the solution by E. Lindenstrauss
(Annals of Math., Vol. 163 (2006), 165-219) of the quantum unique ergodicity
conjecture for hyperbolic arithmetic surfaces. It combines the number theoretic
methods with recent advances by Einsiedler-Katok-Lindenstrauss on classificationof measures on r\C which are invariant under higher rank Cartan actions.
The success of the 2002 Graduate Summer School lies on the contributions
from many people. Our warmest thanks are due to our lecturers for their clear
and accessible presentations. We are particularly grateful to those who provided
us with their lecture notes. We also like to thank our teaching assistants for their
useful problem sessions as well as their help in preparing the lectures.
Finally thanks are due to Herb Clemens and the entire PCMI Steering Com-
mittee for asking us to organize this Summer School, and particularly John Morgan
and John Polking for all their help. Last, but not least, we like to thank Cather-
ine Giesbrecht and the entire IAS/PCMI staff for helping make this program the
success it became.
Peter Sarnak and Freydoon Shahidi,
Volume Editors and
Graduate Summer School Organizers
June, 2006