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Geometric Aspects of Partial Differential Equations
Edited by: Bernhelm Booss-Bavnbek, Roskilde University, Denmark, and Krzysztof Wojciechowski, Indiana University-Purdue University, Indianapolis, IN
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Contemporary Mathematics
1999; 269 pp; softcover
Volume: 242
ISBN-10: 0-8218-2061-3
ISBN-13: 978-0-8218-2061-2
List Price: US$68 Member Price: US$54.40
Order Code: CONM/242

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.

Graduate students and research mathematicians interested in partial differential equations and global analysis.

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