Contemporary Mathematics Volume 237, 1999 Spectral problems in geometry and arithmetic: Preface Thomas Branson In August, 1997, more than 50 participant.s came together for the NSF-CBMS conference "Spectral problems in geometry and arithmetic," with principal speaker Peter Sarnak. Among other things, the conference explored some of the remarkable new connections that have arisen among seemingly disparate mathematical and sci- entific disciplines. Some of these links are surprising even to veterans of renaissance in Physical Mathematics forged by gauge theory in the 1970's. Our principal speaker reported on some of these developments, to which he was a central contributor. Numerical experiments have recently shown that the local spacing between zeros of the Riemann zeta function is modelled by spectral phenomema - the eigenvalue distributions of random matrix theory in particular the Gaussian unitary ensemble (GUE). This phenomenon has become known as the Montgomery-Odlyzko law. In [4, 3] it is shown that this is a more general feature of automorphic L-functions. One of the interpretations of such functions is as spectra of certain invariant operators. Related phenomena have recently been observed from the point of view of dif- ferential geometry, and global harmonic analysis. The zeta function of an elliptic differential operator allows one to define the operator's functional determinant, a quantity familiar to quantum theorists in connection with evaluation of functional integrals. The search for critical points of this quantity, as the underlying mani- fold's geometry is varied, puts one in contact with extremely subtle and delicate sharp inequalities of exponential type. These are not just the Sobolev imbedding inequalities that comprise the heart of familiar geometric problems like Yamabe's, but rather endpoint derivatives (in the formulation of Beckner [1]) of a series of such Sobolev inequalities. This circle of ideas is producing new discoveries about 4-dimensional geometry [2]. The paper of Chang and Yang in this volume is an excellent point of entry to the subject. The moral: zeta functions are spectral objects, even physical objects. As the authors of this volume ably demonstrate, they are also dynamical, chaotic, and more. More is being discovered all the time about the ways in which fundamen- tal scientific issues are connected to each other. Perhaps, given any two truly fundamental questions about Nature, there must be a significant overlap to the corresponding two realms of Mathematics that can be productively applied. 1991 Mathematics Subject Classification. llF72,llM36,35P20,53A30,58C40. Supported in part by NSF Grant DMS-9612075. © 1999 American Mathematical Society ix

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