Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem.The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings.The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field.

Elliptic Operators, Topology and Asymptotic Methods (Pitman Research Notes in Mathematics)

Science & Math

John Roe

Mathematics

Longman Scientific & Technical

The Atiyah-Singer index theorem is one of the most powerful tools for relating geometry, analysis, and topology. In its original form, however, it applies only to compact manifolds. This book describes a version of index theory which works for noncompact spaces with appropriate control, such as complete Riemannian manifolds. The relevant ``control'' is provided by the large scale geometry of the space, and basic notions of large scale geometry are developed. Index theory for the signature operator is related to geometric topology via surgery theory. And, paralleling the analytic development, ``controlled'' surgery theories for noncompact spaces have been developed by topologists. This book explores the connections between these theories, producing a natural transformation from surgery to ``analytic surgery''. The analytic foundations of the work come from the theory of $C^*$-algebras, and the properties of the $C^*$-algebra of a coarse space are developed in detail. The book is based on lectures presented at a conference held in Boulder, Colorado, in August 1995 and includes the author's detailed notes and descriptions of some constructions that were finalized after the lectures were delivered. Also available from the AMS by John Roe is Lectures on Coarse Geometry.

Index Theory, Coarse Geometry, and Topology of Manifolds (CBMS Regional Conference Series in Mathematics, No. 90)

American Mathematical Society

<p>The winding number is one of the most basic invariants in topology. It measures the number of times a moving point $P$ goes around a fixed point $Q$, provided that $P$ travels on a path that never goes through $Q$ and that the final position of $P$ is the same as its starting position. This simple idea has far-reaching applications. The reader of this book will learn how the winding number can help us show that every polynomial equation has a root (the fundamental theorem of algebra), guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem), explain why every simple closed curve has an inside and an outside (the Jordan curve theorem), relate calculus to curvature and the singularities of vector fields (the Hopf index theorem), allow one to subtract infinity from infinity and get a finite answer (Toeplitz operators), generalize to give a fundamental and beautiful insight into the topology of matrix groups (the Bott periodicity theorem). All these subjects and more are developed starting only from mathematics that is common in final-year undergraduate courses. This book is published in cooperation with Mathematics Advanced Study Semesters.</p>

Winding Around: The Winding Number in Topology, Geometry, and Analysis (Student Mathematical Library) (Student Mathematical Library, 76)

Mathematics for Sustainability (Texts for Quantitative Critical Thinking)

Russ deForest

Sara Jamshidi

Springer

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Books by John Roe