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Spectral Problems in Geometry and Arithmetic
Edited by: Thomas Branson, University of Iowa, Iowa City, IA
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Contemporary Mathematics
1999; 174 pp; softcover
Volume: 237
ISBN-10: 0-8218-0940-7
ISBN-13: 978-0-8218-0940-2
List Price: US$43
Member Price: US$34.40
Order Code: CONM/237
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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

  • 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 three-manifolds 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 cut-offs, 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
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