This work covers 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.
Connections between random matrices and Szego limit theorems by E. L. Basor On a fourth order curvature invariant by S.-Y. A. Chang and P. C. Yang Small eigenvalues of the Hodge Laplacian for three-manifolds with pinched negative curvature by R. Gornet and J. McGowan Heating and stretching Riemannian manifolds by C. M. Judge Number theory zeta functions and dynamical zeta functions by J. C. Lagarias Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic by M. L. Lapidus and M. van Frankenhuysen High frequency cut-offs, trace formulas and geometry by K. Okikiolu Meromorphic continuation of the resolvent for Kleinian groups by P. Perry Variation of scattering poles for conformal metrics by Y. N. Petridis On Bilu's equidistribution theorem by R. Rumely Asymptotics of a class of Fredholm determinants by C. A. Tracy and H. Widom.