Geometric Aspects of Partial Differential Equations: Proceedings of a Minisymposium on Spectral Invariants, Heat Equation Approach, September 18-19, 1998, Roskilde, Denmark

This collection of papers by leading researchers gives a broad picture of current research directions in geometric aspects of partial differential equations. Based on lectures presented at a Minisymposium on Spectral Invariants - Heat Equation Approach, held in September 1998 at Roskilde University in Denmark, the book provides both a careful exposition of new perspectives in classical index theory and an introduction to currently active areas of the field.Presented here are new index theorems as well as new calculations of the eta-invariant, of the spectral flow, of the Maslov index, of Seiberg-Witten monopoles, heat kernels, determinants, non-commutative residues, and of the Ray-Singer torsion. New types of boundary value problems for operators of Dirac type and generalizations to manifolds with cuspidal ends, to non-compact and to infinite-dimensional manifolds are also discussed. Throughout the book, the use of advanced analysis methods for gaining geometric insight emerges as a central theme. Aimed at graduate students and researchers, this book would be suitable as a text for an advanced graduate topics course on geometric aspects of partial differential equations and spectral invariants.

Table of Contents:
Part I. Index and small eigenvalues: The Dirac operator and the scalar curvature of continuously deformed algebraic varieties by C. Bar and D. Bleecker The geometry of the Riemann-Hilbert problem by B. Bojarski Chiral anomaly and Grassmannian boundary conditions by T. Kori The index of cone Mellin operators by G. Rozenblum Part II. Eta-invariants, spectral flows, and Seiberg-Witten monopoles: Symplectic functional analysis and spectral invariants by B. Booss-Bavnbek and K. Furutani Eta invariants, spectral flows and finite energy Seiberg-Witten monopoles by L. I. Nicolaescu Part III. Heat kernels, determinants, torsion: Heat asymptotics with spectral boundary conditions by S. Dowker, P. Gilkey, and K. Kirsten Heat content asymptotics by P. Gilkey Extremal Kahler metrics and Ray-Singer analytic torsion by W. Muller and K. Wendland Noncommutative residues, Dixmier's trace, and heat trace expansions on manifolds with boundary by E. Schrohe Part IV. Generalizations: Spectral invariants and geometry of mapping spaces by A. Asada Spectral theory of boundary value problems for Dirac type operators by J. Bruning and M. Lesch Ellipticity and parametrices on manifolds with cuspidal edges by B.-W. Schulze and N. Tarkhanov Classical and quantum completeness for the Schrodinger operators on non-compact manifolds by M. Shubin.