In recent years, increasingly complex methods have been brought into play in the treatment of geometric and topological problems for partial differential operators on manifolds. This collection of papers, resulting from a Workshop on Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, provides a broad picture of these methods with new results. Subjects in the book cover a wide variety of topics, from recent advances in index theory and the more general theory of spectral invariants on closed manifolds and manifolds with boundary, to applications of those invariants in geometry, topology, and physics.Papers are grouped into four parts: Part I gives an overview of the subject from various points of view. Part II deals with spectral invariants, such as traces, indices, and determinants. Part III is concerned with general geometric and topological questions. Part IV deals specifically with problems on manifolds with singularities. The book is suitable for graduate students and researchers interested in spectral problems in geometry.
Part I. Basic material-Reviews: Spectral problems from quantum field theory by D. V. Vassilevich Euclidean quantum gravity in light of spectral geometry by G. Esposito Analysis of invariants associated with spectral boundary problems for elliptic operators by G. Grubb Part II. Spectral invariants and asymptotic expansions: A resolvent approach to traces and zeta Laurent expansions by G. Grubb Asymptotic expansion of the zeta-determinant of an invertible Laplacian on a stretched manifold by Y. Lee Agranovich-Dynin formula for the zeta-determinants of the Neumann and Dirichlet problems by J. Park and K. P. Wojciechowski Part III. Geometric and topological problems: The Calderon projector for the odd signature operator and spectral flow calculations in 3-dimensional topology by H. U. Boden, C. M. Herald, and P. Kirk Cut-and-paste on foliated bundles by E. Leichtnam and P. Piazza The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators by M. Lesch Variants of equivariant Seiberg-Witten Floer homology by M. Marcolli and B.-L. Wang Part IV. Manifolds with singularities: Dirac operators, boundary value problems, and the $b$-calculus by P. Loya Guillemin transform and Toeplitz representations for operators on singular manifolds by V. E. Nazaikinskii, G. Rozenblum, A. Yu. Savin, and B. Yu. Sternin Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains by V. Nistor.