The Atiyah-Singer Index Theorem
Author: P. Shanahan
Publisher: Springer
Published: 2006-11-15
Total Pages: 230
ISBN-13: 3540359214
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Invariance Theory
Author: Peter B. Gilkey
Publisher: CRC Press
Published: 1994-12-22
Total Pages: 534
ISBN-13: 9780849378744
DOWNLOAD EBOOKThis 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.
The Atiyah-Patodi-Singer Index Theorem
Author: Richard Melrose
Publisher: CRC Press
Published: 1993-03-31
Total Pages: 392
ISBN-13: 1439864608
DOWNLOAD EBOOKBased on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.
Seminar on the Atiyah-Singer Index Theorem
Author: Michael Francis Atiyah
Publisher: Princeton University Press
Published: 1965-09-21
Total Pages: 384
ISBN-13: 9780691080314
DOWNLOAD EBOOKA classic treatment of the Atiyah-Singer index theorem from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.
Topology and Analysis
Author: D.D. Bleecker
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 467
ISBN-13: 1468406272
DOWNLOAD EBOOKThe Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe matical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ ent mathematical approaches, and must have experience with their inter connections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics", "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics": In spi te of i ts difficulty and immensely rich interrela tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many facetted and always new presentations of the material by M. F.
Atiyah-Singer Index Theorem - An Introduction
Author: Amiya Mukherjee
Publisher: Hindustan Book Agency
Published: 2013-10-30
Total Pages: 0
ISBN-13: 9789380250540
DOWNLOAD EBOOKOffers a thorough introduction to the Atiyah-Singer index theorem for elliptic operators on compact manifolds without boundary. The main theme is 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.
Heat Kernels and Dirac Operators
Author: Nicole Berline
Publisher: Springer Science & Business Media
Published: 2003-12-08
Total Pages: 384
ISBN-13: 9783540200628
DOWNLOAD EBOOKIn the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback.
K-theory
Author: Michael Atiyah
Publisher: CRC Press
Published: 2018-03-05
Total Pages: 181
ISBN-13: 0429973179
DOWNLOAD EBOOKThese notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.
Index Theorem. 1
Author: M. Furuta
Publisher: American Mathematical Soc.
Published: 2007
Total Pages: 230
ISBN-13: 9780821820971
DOWNLOAD EBOOKThe Atiyah-Singer index theorem is a remarkable result that allows one to compute the space of solutions of a linear elliptic partial differential operator on a manifold in terms of purely topological data related to the manifold and the symbol of the operator. First proved by Atiyah and Singer in 1963, it marked the beginning of a completely new direction of research in mathematics with relations to differential geometry, partial differential equations, differential topology, K-theory, physics, and other areas.
The Index Theorem And The Heat Equation Method
Author: Yanlin Yu
Publisher: World Scientific
Published: 2001-07-02
Total Pages: 309
ISBN-13: 981449111X
DOWNLOAD EBOOKThis book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems, and it seems to be as efficient as other methods.
