Geometric Aspects of Partial Differential Equations.

Intro -- Contents -- Preface -- Part I. Index and Small Eigenvalues -- The Dirac operator and the scalar curvature of continuously deformed algebraic varieties -- The geometry of the Riemann-Hilbert problem -- Chiral anomaly and Grassmannian boundary conditions -- The index of cone Mellin operators -- Part II. Eta-Invariants, Spectral Flows, and Seiberg-Witten Monopoles -- Symplectic functional analysis and spectral invariants -- 1. The Hörmander Index under Transversality Condition -- 2. The Maslov Index for Paths -- 3. The Fredholm Lagrangian Grassmannian -- 4. The Topology of the Fredholm Lagrangian Grassmannian -- 5. The Hörmander Index in Infinite Dimensions -- 6. An Example -- 7. General Boundary Data and Cauchy Data Spaces in Distribution Space -- 8. Generalized Yoshida-Nicolaescu Formula in H-½ (Σ) -- Appendix A. A Characterization of Lagrangian Fredholm Pairs -- References -- Eta invariants, spectral flows, and finite energy Seiberg-Witten monopoles -- Part III. Heat Kernels, Determinants, Torsion -- Heat asymptotics with spectral boundary conditions -- Heat content asymptotics -- Extremal Kähler metrics and Ray-Singer analytic torsion -- Noncommutative residues, Dixmier's trace, and heat trace expansions on manifolds with boundary -- Introduction -- 1. The Classical Results for Closed Manifolds -- 2. Boutet de Manvel's Calculus -- 3. The Noncommutative Residue for Manifolds with Boundary -- 4. Dixmier's Trace for Boundary Problems -- 5. Heat Trace Asymptotics -- 6. Remarks and References to Further Work -- References -- Part IV. Generalizations -- Spectral invariants and geometry of mapping spaces -- Spectral theory of boundary value problems for Dirac type operators -- Ellipticity and parametrices on manifolds with cuspidal edges -- Classical and quantum completeness for the Schrödinger operators on non-compact manifolds.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.