This volume is a collection of surveys of diverse topics in geometry related to the eigenvalues of the Laplacian and other geometric operators. A good understanding of the spectral properties of differential operators has deep consequences in geometry, physics, number theory, probability theory, etc. Analytic methods have penetrated also into discrete mathematics, where the study of the spectral properties of difference operators leads to a deeper understanding of the combinatorial question on graphs. The present collection represents a substantial part of the subject, and it will be beneficial to geometers and experts from the adjacent area.

A publication of International Press. Distributed worldwide by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in geometry, topology, and differential equations.

Table of Contents

• M. T. Barlow -- Anomalous diffusion and stability of Harnack inequalities
• G. Besson -- From isoperimetric inequalities to heat kernels via symmetrisation
• F. Chung -- Discrete isoperimetric inequalities
• T. H. Colding and W. P. Minicozzi II -- An excursion into geometric analysis
• A. Grigor'yan, Y. Netrusov, and S.-T. Yau -- Eigenvalues of elliptic operators and geometric applications
• M. Ledoux -- Spectral gap, logarithmic Sobolev constant, and geometric bounds
• L. Lovász -- Discrete analytic functions: An exposition
• W. H. Meeks III and J. Pérez -- Confromal properties in classical minimal surface theory
• R. Neel and D. Stroock -- Analysis of the cut locus via the heat kernel
• L. Saloff-Coste -- Analysis on Riemannian co-compact covers
• F. Shahidi -- Functoriality and small eigenvalues of Laplacian on Riemann survaces
• S. Zelditch -- The inverse spectral problem