Index Theory For Invariant Elliptic Operators On Manifolds With Proper Cocompact Group Actions
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Author | : Gong Cheng (Mathematician) |
Publisher | : |
Total Pages | : 54 |
Release | : 2018 |
Genre | : Electronic dissertations |
ISBN | : |
In this thesis, we study G-invariant elliptic operators, and in particular Dirac operators, on the space of invariant sections of a Hermitian bundle over a (non-compact) manifold with a proper and cocompact Lie group action. We provide a canonical way to define the Hilbert space of invariant sections for proper and cocompact actions and prove that the G-invariant Dirac operators, and more generally, elliptic operators, are Fredholm for the Hilbert space we constructed. Using the framework developed in this thesis, we give a new proof of a generalized Lichnerowicz Vanishing Theorem for proper cocompact group actions as an application.
Author | : Jerome Kaminker |
Publisher | : American Mathematical Soc. |
Total Pages | : 334 |
Release | : 1988 |
Genre | : Mathematics |
ISBN | : 0821850776 |
Combining analysis, geometry, and topology, this volume provides an introduction to current ideas involving the application of $K$-theory of operator algebras to index theory and geometry. In particular, the articles follow two main themes: the use of operator algebras to reflect properties of geometric objects and the application of index theory in settings where the relevant elliptic operators are invertible modulo a $C^*$-algebra other than that of the compact operators. The papers in this collection are the proceedings of the special sessions held at two AMS meetings: the Annual meeting in New Orleans in January 1986, and the Central Section meeting in April 1986. Jonathan Rosenberg's exposition supplies the best available introduction to Kasparov's $KK$-theory and its applications to representation theory and geometry. A striking application of these ideas is found in Thierry Fack's paper, which provides a complete and detailed proof of the Novikov Conjecture for fundamental groups of manifolds of non-positive curvature. Some of the papers involve Connes' foliation algebra and its $K$-theory, while others examine $C^*$-algebras associated to groups and group actions on spaces.
Author | : Vladimir E. Nazaykinskiy |
Publisher | : Springer Science & Business Media |
Total Pages | : 224 |
Release | : 2008-06-30 |
Genre | : Mathematics |
ISBN | : 3764387750 |
This comprehensive yet concise book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. This is the first book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. To make the book self-contained, the authors have included necessary geometric material.
Author | : A. Connes |
Publisher | : American Mathematical Society |
Total Pages | : 592 |
Release | : 2023-02-23 |
Genre | : Mathematics |
ISBN | : 1470469774 |
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme. The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.
Author | : Peter B. Gilkey |
Publisher | : CRC Press |
Total Pages | : 534 |
Release | : 1994-12-22 |
Genre | : Mathematics |
ISBN | : 9780849378744 |
This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
Author | : Bogdan Bojarski |
Publisher | : Springer Science & Business Media |
Total Pages | : 332 |
Release | : 2006-11-09 |
Genre | : Mathematics |
ISBN | : 3764376872 |
This book consists of reviewed original research papers and expository articles in index theory (especially on singular manifolds), topology of manifolds, operator and equivariant K-theory, Hopf cyclic cohomology, geometry of foliations, residue theory, Fredholm pairs and others, and applications in mathematical physics. The wide spectrum of subjects reflects the diverse directions of research for which the starting point was the Atiyah-Singer index theorem.
Author | : John Roe |
Publisher | : Longman Scientific and Technical |
Total Pages | : 208 |
Release | : 1988 |
Genre | : Mathematics |
ISBN | : |
Author | : Thomas Donaldson |
Publisher | : |
Total Pages | : 117 |
Release | : 1978 |
Genre | : |
ISBN | : |
Author | : Amiya Mukherjee |
Publisher | : Springer |
Total Pages | : 280 |
Release | : 2013-10-30 |
Genre | : Mathematics |
ISBN | : 9386279606 |
This monograph is a thorough introduction to the Atiyah-Singer index theorem for elliptic operators on compact manifolds without boundary. The main theme is only the classical index theorem and some of its applications, but not the subsequent developments and simplifications of the theory. The book is designed for a complete proof of the K -theoretic index theorem and its representation in terms of cohomological characteristic classes. In an effort to make the demands on the reader's knowledge of background materials as modest as possible, the author supplies the proofs of almost every result. The applications include Hirzebruch signature theorem, Riemann-Roch-Hirzebruch theorem, and the Atiyah-Segal-Singer fixed point theorem, etc.
Author | : Michael Atiyah |
Publisher | : Oxford University Press |
Total Pages | : 632 |
Release | : 1988-04-28 |
Genre | : Biography & Autobiography |
ISBN | : 9780198532774 |
This is a collection of the works of Michael Atiyah, a well-established mathematician and winner of the Fields Medal. It is thematically divided into volumes; this one discusses index theory.