**Contemporary Mathematics**

Volume: 242;
1999;
269 pp;
Softcover

MSC: Primary 53; 58; 81;
Secondary 35

Print ISBN: 978-0-8218-2061-2

Product Code: CONM/242

List Price: $72.00

Individual Member Price: $57.60

**Electronic ISBN: 978-0-8218-7832-3
Product Code: CONM/242.E**

List Price: $72.00

Individual Member Price: $57.60

# Geometric Aspects of Partial Differential Equations

Share this page *Edited by *
*Bernhelm Booss-Bavnbek; Krzysztof Wojciechowski*

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

# Table of Contents

## Geometric Aspects of Partial Differential Equations

- Contents v6 free
- Preface vii8 free
- Part I. Index and Small Eigenvalues 110 free
- Part II. Eta-Invariants, Spectral Flows, and Seiberg-Witten Monopoles 5160
- Symplectic functional analysis and spectral invariants 5362
- 1. The Hörmander Index under Transversality Condition 5564
- 2. The Maslov Index for Paths 5665
- 3. The Fredholm Lagrangian Grassmannian 5867
- 4. The Topology of the Fredholm Lagrangian Grassmannian 5968
- 5. The Hörmander Index in Infinite Dimensions 6473
- 6. An Example 6574
- 7. General Boundary Data and Cauchy Data Spaces in Distribution Space 6776
- 8. Generalized Yoshida–Nicolaescu Formula in H–½ (Σ) 7483
- Appendix A. A Characterization of Lagrangian Fredholm Pairs 8089
- References 8392

- Eta invariants, spectral flows, and finite energy Seiberg-Witten monopoles 8594

- Part III. Heat Kernels, Determinants, Torsion 105114
- Heat asymptotics with spectral boundary conditions 107116
- Heat content asymptotics 125134
- Extremal Kähler metrics and Ray-Singer analytic torsion 135144
- Noncommutative residues, Dixmier's trace, and heat trace expansions on manifolds with boundary 161170
- Introduction 161170
- 1. The Classical Results for Closed Manifolds 163172
- 2. Boutet de Manvel's Calculus 170179
- 3. The Noncommutative Residue for Manifolds with Boundary 173182
- 4. Dixmier's Trace for Boundary Problems 176185
- 5. Heat Trace Asymptotics 180189
- 6. Remarks and References to Further Work 184193
- References 184193

- Part IV. Generalizations 187196

#### Readership

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