This volume is an outgrowth of the AMS Special Session on Extremal Riemann Surfaces held at the Joint Mathematics Meeting in San Francisco, January 1995. The book deals with a variety of extremal problems related to Riemann surfaces. Some papers deal with the identification of surfaces with longest systole (element of shortest nonzero length) for the length spectrum and the Jacobian. Parallels are drawn to classical questions involving extremal lattices. Other papers deal with maximizing or minimizing functions defined by the spectrum such as the heat kernel, the zeta function, and the determinant of the Laplacian, some from the point of view of identifying an extremal metric.There are discussions of Hurwitz surfaces and surfaces with large cyclic groups of automorphisms. Also discussed are surfaces which are natural candidates for solving extremal problems such as triangular, modular, and arithmetic surfaces, and curves in various group theoretically defined curve families. Other allied topics are theta identities, quadratic periods of Abelian differentials, Teichmuller disks, binary quadratic forms, and spectral asymptotics of degenerating hyperbolic three manifolds. This volume: includes papers by some of the foremost experts on Riemann surfaces; outlines interesting connections between Riemann surfaces and parallel fields; follows up on investigations of Sarnak concerning connections between the theory of extreme lattices and Jacobians of Riemann surfaces; and, contains papers on a variety of topics relating to Riemann surfaces.

Table of Contents:
Extremal geometries by P. Sarnak Extremal Riemann surfaces with a large number of systoles by P. S. Schaller On arithmetic genus 2 subgroups of triangle groups by N. Naatanen and T. Kuusalo Some lattices obtained from Riemann surfaces by M. Bernstein and N. A. Sloane Jacobian of the Picard curve by J. R. Quine Fermat's quartic curve, Klein's curve, and the tetrahedron by R. E. Rodriguez and V. Gonzalez-Aguilera The splitting of some Jacobi varieties using their automorphism groups by J. X. Ries Riemann surfaces admitting large automorphism gorups by R. S. Kulkarni Number theory, theta identities, and modular curves by R. Brooks, H. M. Farkas, and I. Kra Uniformization of some quotients of modular curves by R. Brooks and Y. Kopeliovich Teichmuller disks and Veech's $\mathcal F$-structures by C. J. Earle and F. P. Gardiner On the geometry and spectral asymptotics of degenerating hyperbolic three manifolds by J. Dodziuk and J. Jorgenson Differential equations for the quadratic periods of abelian differentials by D. Curtis and M. Tretkoff Zeta functions on $S^2$ by C. Morpurgo A minimum problem for heat kernels of flat tori by A. B. II.