Contemporary Mathematics 1999; 174 pp; softcover Volume: 237 ISBN10: 0821809407 ISBN13: 9780821809402 List Price: US$43 Member Price: US$34.40 Order Code: CONM/237
 These are the proceedings of the NSFCBMS Conference on "Spectral Problems in Geometry and Arithmetic" held at the University of Iowa. The principal speaker was Peter Sarnak, who has been a central contributor to developments in this field. The volume approaches the topic from the geometric, physical, and number theoretic points of view. The remarkable new connections among seemingly disparate mathematical and scientific disciplines have surprised even veterans of the physical mathematics renaissance forged by gauge theory in the 1970s. Numerical experiments show that the local spacing between zeros of the Riemann zeta function is modelled by spectral phenomena: the eigenvalue distributions of random matrix theory, in particular the Gaussian unitary ensemble (GUE). Related phenomena are from the point of view of differential geometry and global harmonic analysis. Elliptic operators on manifolds have (through zeta function regularization) functional determinants, which are related to functional integrals in quantum theory. The search for critical points of this determinant brings about extremely subtle and delicate sharp inequalities of exponential type. This indicates that zeta functions are spectral objectsand even physical objects. This volume demonstrates that zeta functions are also dynamic, chaotic, and more. Readership Graduate students and research mathematicians interested in number theory. Table of Contents  E. L. Basor  Connections between random matrices and Szegö limit theorems
 S.Y. A. Chang and P. C. Yang  On a fourth order curvature invariant
 R. Gornet and J. McGowan  Small eigenvalues of the Hodge Laplacian for threemanifolds with pinched negative curvature
 C. M. Judge  Heating and stretching Riemannian manifolds
 J. C. Lagarias  Number theory zeta functions and dynamical zeta functions
 M. L. Lapidus and M. van Frankenhuysen  Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic
 K. Okikiolu  High frequency cutoffs, trace formulas and geometry
 P. Perry  Meromorphic continuation of the resolvent for Kleinian groups
 Y. N. Petridis  Variation of scattering poles for conformal metrics
 R. Rumely  On Bilu's equidistribution theorem
 C. A. Tracy and H. Widom  Asymptotics of a class of Fredholm determinants
