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## Comments & Reviews

This book … describes in impressive detail one of the greatest achievements of 20th-century mathematics, the Atiyah-Singer index theorem. As a bonus, the author includes an extensive and detailed discussion of some of the applications of the theorem and more recent developments in the direction of gauge theoretic physical models. For the student, the gold lining in the book is that all the background is there—starting at a level that is accessible to anyone with a reasonable mathematical education. There is also a bridge to the point of view from the physics side. The student can dip into the book at many points and use it to learn individual topics. The discussion covers a very broad sweep of mathematics which will be of interest to anyone aiming to understand geometric analysis, whether or not gauge theories are their primary focus. Two proofs of the Atiyah Singer theorem are given, one based on K-theory and the other on the heat kernel approach. … The authors have included many discussion sections that illuminate the thinking behind the more general theory. … There is much valuable historical context recorded here. Moreover, it captures the flavour of contemporary mathematics, with the interaction of geometric, topological, analytic, and physical perspectives and methods being in the forefront of the discussion.

Alan Carey

Mathematical Sciences Institute at Australian National University, Canberra

This book has something for almost everyone. An experienced index theorist may find in the juxtapositions of ideas, in the historical comments, or in the 458 references stimuli for new perspectives. An inexperienced index theorist will benefit from an overview of the literature with valuable heuristic comments.

Peter Haskell

AMS Mathematical Reviews

…The authors provide two proofs for the Index Theorem: one based on K-theory and the other following the heat kernel approach. All necessary background information is thoroughly discussed…This work can serve as a textbook for students of mathematics and physics in the upper-undergraduate and graduate levels. It is also a reference book, and good survey, written in a lively fashion.

László Kérchy

Acta Scientiarum Mathematicarum 80:3-4 (2014), Bolyai Institute, University of Szeged, Hungary

[The authors] give all the background information on such diverse topics as Fredholm operators, (elliptic) pseudo-differential operators, analysis on manifolds, principal bundles and curvature, and K-theory. Many applications of the [Atiyah-Singer index] theorem are given, with emphasis on low-dimensional topology and gauge theoretic particle physics.

Thus the book introduces various topics with basic examples and aspects in recent developments and includes many discussion sections to illuminate the thinking behind the more general theory. This monograph is informative and well written in a systematic way.

In-Sook Kim

Zentralblatt MATH

Those wishing to start their study of the theory of elliptic operators will appreciate the book’s use of rigorous exposition intermingled with intuition-building casual and historical discussion. Experts will find many features of the book useful and convenient.

Paul Kirk

Indiana University Bloomington

The real strength of the book is its detailedness, which makes it particularly attractive for the learning (graduate) student. But the senior researcher, likewise, will treasure this somewhat unconventional textbook as a very valuable source of information.

Matthias Lesch

Universität Bonn, Germany

Two or three famous Index Formulas discovered and proved in the course of middle decades of the last century are some of the highest peaks in a mountain country rising from the vast plains of functional analysis, theory of smooth manifolds, and homotopical topology.

This treatise, written with ambition, wit and (mathematical) eloquence, strives to combine the qualities of a guide-book, historical chronicles, and a hiking manual for enthusiastic travellers and budding future explorers of this vast territory.

Any reader possessing will and enthusiasm can profit from studying (parts of) this book and enjoy finding his or her own path through this land.

Yuri I. Manin

Max Planck Institute for Mathematics, Bonn, Germany

My struggles with this theorem are vivid in my mind. I still remember the sunny autumn afternoon when I picked from the library the first edition of this book. From the first moment I opened it I thought it spoke to me, the beginning graduate student with a limited mathematical experience. The proofs had the level of detail I needed at that stage in my life. What awed me most was the wealth of varied examples from exotic worlds I did not even suspect existed. Reading those examples and the carefully crafted proofs I began to get a glimpse at the wonderful edifice behind the index theorem.

The present edition has the same effect on the more mature me. And it does quite a bit more. Old examples are refined, and new ones added. Facts whose proofs were only sketched in the old edition are now given complete, or almost complete proofs. The chapter on gauge theory is completely and massively rewritten and it incorporates some of the spectacular developments in this area that took place in the intervening time. As importantly, throughout the book, the exposition is sprinkled with many mathematical “anectodes” which give the reader a glimpse into the minds of the pioneers of the subject.

I believe that any youngster who will pick this new edition will be as grateful as I am to the authors for the care and their concern for the reader. The teacher of this subject, as well, will have many things for which to be grateful: this is one of the few places containing such a wealth of examples and care for detail.

Liviu Nicolaescu

University of Notre Dame, Indiana

The present book is a landmark, being a complete introduction to practically all aspects of the Index Theorem, complete with basic examples and exercises, covering topological ideas: homotopy invariance and K-theory; analysis: elliptic operators and heat equations; geometry: principal bundles and curvature; physics: gauge theory and Seiberg-Witten theory. The first English version appeared in 1985, but the present book is a largely reworked and much expanded treatise; it is a highly informative presentation, based on the authors’ expert knowledge and pedagogical interests. Indeed, it seems to be presently the most complete single book on the many approaches to and applications of the Atiyah-Singer Index Theorem. Experts as well will find inspiration in this book.

Bent Ørsted

Aarhus University, Denmark

The student who wants to explore the whole shape of this huge and complex territory—as well as the usual climbing routes to the summit of the Index Theorem itself—can hardly do better than to enlist the genial and enthusiastic guide service of Bleecker and Booss-Bavnbek.

John Roe

Pennsylvania State University

Professors Bleecker and Booss-Bavnbek have followed … developments in index theory from the beginning, and made original contributions of their own… Assuming only basic analysis and algebra, [this book] gives detailed constructions and proofs for all the necessary concepts, along with illuminating digressions on the various paths through the rich territory of index theory.

Robert Seeley

Professor Emeritus, University of Massachusetts, Boston

The very appreciable feature of the book is its discussion of the applications of the theorem in low-dimensional topology and in gauge theories. For these reasons, the book will be a valuable reference for theoretical and mathematical physicists, especially those interested in the foundations of quantum gauge theories, at the basis of the standard model of elementary particle physics. Mathematically oriented graduate students will greatly profit from reading this excellent book.

Franco Strocchi

Scuola Normale Superiore, Pisa, Italy

Students of mathematics and physics will find this book to be an excellent resource for study of this vast subject. In particular, attention is given to exposition of the subject from different angles, which is very helpful as a bridge between physics and mathematics.

Cumrun Vafa

Donner Professor of Science, Physics Department, Harvard University

Readers from a wide range of backgrounds will find much to learn here.

Edward Witten

Institute for Advanced Studies, Princeton, New Jersey

[excerpt from a review published in the Journal of Geometry and Symmetry in Physics (http://www.emis.de/journals/JGSP), vol. 34 (2014), pp. 97–100.]

…The history and connections of and in index theory are covered by the book in an excellent way. The authors give many examples by which they help the reader to clarify his or her understanding of the subject and reveal the application as well. The examples are well elaborated and may serve as a way for understanding the theory. …the authors give definitions of notions which are basic in an appropriate field of mathematics or physics (range from homological groups to compact operators, manifolds and vector bundles), but the majority of these terms are not common for most experts working in the individual parts. Putting all these deep themes together without disrupting the integrity of the text, makes this book exceptional among existing text-books on this topic.

Svatopluk Krýsl

Faculty of Mathematics and Physics, Charles University in Prague

This titanic book (almost 800 pages) is something of a marvel ... [it] can be read with great profit by strong graduate students in differential geometry, global analysis, operator theory, and so on, and provides a wonderful resource for seasoned scholars in these fields given its encyclopedic scope. Additionally, there is a strong historical dimension to the book, which makes for a heightened experience and a deeper understanding...

Michael Berg

(For MAA Reviews) Loyola Marymount University, Los Angeles, California