The Spectral Flow, the Maslov Index and Decompositions of Manifolds
Author: Liviu I. Nicholaescu
Publisher:
Published: 1994
Total Pages: 196
ISBN-13:
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Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds
Author: Gerd Grubb
Publisher: American Mathematical Soc.
Published: 2005
Total Pages: 338
ISBN-13: 082183536X
DOWNLOAD EBOOKIn 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 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 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.
Spectral Flow
Author: Nora Doll
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2023-06-19
Total Pages: 460
ISBN-13: 3111172473
DOWNLOAD EBOOKThe Maslov Index in Symplectic Banach Spaces
Author: Bernhelm Booß-Bavnbek
Publisher: American Mathematical Soc.
Published: 2018-03-19
Total Pages: 134
ISBN-13: 1470428008
DOWNLOAD EBOOKThe authors consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying weak symplectic structures. Assuming vanishing index, they obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions the authors define the Maslov index of the curve by symplectic reduction to the classical finite-dimensional case. The authors prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction while recovering all the standard properties of the Maslov index. As an application, the authors consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, the authors derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting formula of the spectral flow on partitioned manifolds.
Geometric Aspects of Partial Differential Equations
Author: Krzysztof Wojciechowski
Publisher: American Mathematical Soc.
Published: 1999
Total Pages: 282
ISBN-13: 0821820613
DOWNLOAD EBOOKThis 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.
Aspects of Boundary Problems in Analysis and Geometry
Author: Juan Gil
Publisher: Birkhäuser
Published: 2012-12-06
Total Pages: 574
ISBN-13: 3034878508
DOWNLOAD EBOOKBoundary problems constitute an essential field of common mathematical interest, they lie in the center of research activities both in analysis and geometry. This book encompasses material from both disciplines, and focuses on their interactions which are particularly apparent in this field. Moreover, the survey style of the contributions makes the topics accessible to a broad audience with a background in analysis or geometry, and enables the reader to get a quick overview.
Topology, Geometry, and Algebra: Interactions and new directions
Author: Alejandro Adem
Publisher: American Mathematical Soc.
Published: 2001
Total Pages: 266
ISBN-13: 082182063X
DOWNLOAD EBOOKThis volume presents the proceedings from the conference on ``Topology, Geometry, and Algebra: Interactions and New Directions'' held in honor of R. James Milgram at Stanford University in August 1999. The meeting brought together distinguished researchers from a variety of areas related to algebraic topology and its applications. Papers in the book present a wide range of subjects, reflecting the nature of the conference. Topics include moduli spaces, configuration spaces, surgerytheory, homotopy theory, knot theory, group actions, and more. Particular emphasis was given to the breadth of interaction between the different areas.
Quantum Gravity and Spectral Geometry
Author: Giampiero Esposito
Publisher:
Published: 2002
Total Pages: 310
ISBN-13:
DOWNLOAD EBOOKGeneralized Symplectic Geometries and the Index of Families of Elliptic Problems
Author: Liviu I. Nicolaescu
Publisher: American Mathematical Soc.
Published: 1997
Total Pages: 98
ISBN-13: 0821806211
DOWNLOAD EBOOKIn this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family. All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eigh symmetries in the real case and two in the complex case). This text will also be of interest to those working in geometry and topology.
Index theory in nonlinear analysis
Author: Chungen Liu
Publisher: Springer
Published: 2019-05-22
Total Pages: 346
ISBN-13: 981137287X
DOWNLOAD EBOOKThis book provides detailed information on index theories and their applications, especially Maslov-type index theories and their iteration theories for non-periodic solutions of Hamiltonian systems. It focuses on two index theories: L-index theory (index theory for Lagrangian boundary conditions) and P-index theory (index theory for P-boundary conditions). In addition, the book introduces readers to recent advances in the study of index theories for symmetric periodic solutions of nonlinear Hamiltonian systems, and for selected boundary value problems involving partial differential equations.
