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Generalized Symplectic Geometries and the Index of Families of Elliptic Problems

Liviu I. Nicolaescu University of Michigan, Ann Arbor
Available Formats:
Electronic ISBN: 978-1-4704-0194-8
Product Code: MEMO/128/609.E
List Price: $45.00 MAA Member Price:$40.50
AMS Member Price: $27.00 Click above image for expanded view Generalized Symplectic Geometries and the Index of Families of Elliptic Problems Liviu I. Nicolaescu University of Michigan, Ann Arbor Available Formats:  Electronic ISBN: 978-1-4704-0194-8 Product Code: MEMO/128/609.E  List Price:$45.00 MAA Member Price: $40.50 AMS Member Price:$27.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1281997; 80 pp
MSC: Primary 58; 47; 55; Secondary 15; 16;

In 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 (eight symmetries in the real case and two in the complex case).

Readership

Graduate students and research mathematicians interested in index theory; topologists and gauge theorists.

• Table of Contents

• Chapters
• 1. Algebraic preliminaries
• 2. Topological preliminaries
• 3. $(p,q)$-lagrangians and classifying spaces for $K$-theory
• 4. Symplectic reductions
• 5. Clifford symmetric Fredholm operators
• 6. Families of boundary value problems for Dirac operators
• Appendix A. Gap convergence of linear operators
• Appendix B. Gap continuity of families of BVP’s for Dirac operators
• Appendix C. Pseudodifferential Grassmanians and BVP’s for Dirac operators
• Appendix D. The proof of Proposition 6.1
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1281997; 80 pp
MSC: Primary 58; 47; 55; Secondary 15; 16;

In 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 (eight symmetries in the real case and two in the complex case).

Readership

Graduate students and research mathematicians interested in index theory; topologists and gauge theorists.

• Chapters
• 1. Algebraic preliminaries
• 2. Topological preliminaries
• 3. $(p,q)$-lagrangians and classifying spaces for $K$-theory
• 4. Symplectic reductions
• 5. Clifford symmetric Fredholm operators
• 6. Families of boundary value problems for Dirac operators
• Appendix A. Gap convergence of linear operators
• Appendix B. Gap continuity of families of BVP’s for Dirac operators
• Appendix C. Pseudodifferential Grassmanians and BVP’s for Dirac operators
• Appendix D. The proof of Proposition 6.1
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.