**Contemporary Mathematics**

Volume: 237;
1999;
174 pp;
Softcover

MSC: Primary 11; 35; 53; 58;

Print ISBN: 978-0-8218-0940-2

Product Code: CONM/237

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

**Electronic ISBN: 978-0-8218-7828-6
Product Code: CONM/237.E**

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

# Spectral Problems in Geometry and Arithmetic

Share this page *Edited by *
*Thomas Branson*

These are the proceedings of the NSF-CBMS 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 objects--and 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

## Spectral Problems in Geometry and Arithmetic

- Contents vii8 free
- Preface ix10 free
- List of participants xi12 free
- Connections between random matrices and Szegö limit theorems 114 free
- On a fourth order curvature invariant 922
- Small eigenvalues of the Hodge Laplacian for three-manifolds with pinched negative curvature 2942
- Heating and stretching Riemannian manifolds 3952
- Number theory zeta functions and dynamical zeta functions 4558
- Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic 87100
- High frequency cut-offs, trace formulas and geometry 107120
- Meromorphic continuation of the resolvent for Kleinian groups 123136
- Variation of scattering poles for conformal metrics 149162
- On Bilu's equidistribution theorem 159172
- Asymptotics of a class of Fredholm determinants 167180