1. K-theory (1967), by Michael Atiyah.
These 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 made 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 real K-theory and so fills an obvious gap in the lecture notes.
1.2. Review by: Friedrich Hirzebruch.
American Scientist 57 (2) (1969), 153A-154A.
Based on Atiyah's Harvard course of 1964 this is the first rather comprehensive account of K-theory. ... Stemming from linear algebra and topology, K-theory has had most important applications in linear analysis. It is the right tool to handle elliptic operators on manifolds (Atiyah, Bott, Singor). This development is already present in the book under review.
This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of providing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts on commutative algebra such as those of Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott's Ideal theory (1953) and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization. ... The lecture-note origin of this book accounts for the rather terse style, with little general padding, and for the condensed account of many proofs. We have resisted the temptation to expand it in the hope that the brevity of our presentation will make clearer the mathematical structure of what is by now an elegant and attractive theory. Our philosophy has been to build up to the main theorems in a succession of simple steps and to omit routine verifications. The authors have used the methods of homological algebra but do not pursue the subject to any great depth. ... Anyone writing now on commutative algebra faces a dilemma in connection with homological algebra, which plays such an important part in modern developments. A proper treatment of homological algebra is impossible within the confines of a small book: on the other hand, it is hardly sensible to ignore it completely. The compromise we have adopted is to use elementary homological methods - exact sequences, diagrams, etc. - but to stop short of any results requiring a deep study of homology.
2.2. Review by: Donald John Lewis and Bernard Robert McDonald.
Amer. Math. Monthly 77 (7) (1970), 783-784.
This is a short 126 page introduction to commutative ring theory. It assumes the student has had a very solid first course (graduate level in North America, Part II level in England) in general algebra. Few specific results are assumed, but the maturity and sophistication that comes from such a course are assumed. The material is approximately what one might cover in a short course on ideal theory in commutative rings. However, here the emphasis is more on modules and localization than on classical ideal theory. ... The authors' main motivation is the preparation of the student for a systematic study of homological algebra, the first step for serious study of modern algebraic geometry and class field theory. ... Here, certain elementary homological methods are introduced but the heavier parts of the machinery are left for later courses. The text is very tersely written, examples are a bit scarce and proofs are condensed. This reviewer doubts that many students can profitably read it unassisted. Each chapter has a motivational introduction which is excellent for the initiated but may be obscure to others. The highlight of the text is the very excellent set of problems which constitute one-third of the text.
2.3. Review by: Johnny Albert Johnson.
Mathematical Reviews, MR0242802 (39 #4129).
The intent of this book is to provide a rather quick introduction to the theory of commutative algebra. ... The general style of the book is concise and to the point. ... The authors have used the methods of homological algebra but do not pursue the subject to any great depth. ... A substantial number of exercises have been provided at the end of each chapter. Some of them are simple and others are rather difficult. Some hints and some complete solutions are provided for the more difficult problems.
Lecture 1: Transversally elliptic operators.- In this lecture we review the basic properties of pseudo-differential operators and introduce the subclass of transversally elliptic operators.
Lecture 2: The index of transversally elliptic operators.
Lecture 3: The excision and multiplicative properties. - In this lecture we prove that an operator in X, transversally elliptic relative to a free G-action, can be reduced to an elliptic operator on the orbit manifold X/G. We then turn to the multiplicative properties of the index and the excision theorem ...
Lecture 4: The naturality of the index and the localization theorem. - We prove, in this lecture, that the index is natural with respect to equivariant embeddings and enlargement of the group. We then show that the problem of computing the index of any transversally elliptic operator can be reduced to the calculation of the index for an operator on Euclidean space transversally elliptic relative to a toral action.
Lecture 5: The index homomorphism for G = S1.
Lecture 6: The operators Δ ± for G = S1.
Lecture 7: Toral actions with finite isotropy groups.
Lecture 8: The index homomorphism for G = Tn.
Lecture 9: The cohomology formula.
Lecture 10: Applications.
4.2. Review by: Robert Evert Stong.
Mathematical Reviews, MR0482866 (58 #2910).
This volume consists of lecture notes describing an extension of the index theory of elliptic operators. Given a compact Lie group G acting on a compact manifold X and a pseudodifferential operator P on X which is G-invariant and elliptic in the directions transverse to the orbits of G, one can associate to P an index which is a distribution on G.
These Lectures Notes are an expanded version of the Fermi Lectures which I gave at the Scuola Normale in Pisa in June 1978. They also cover material presented in the spring of 1978 in the Loeb Lectures at Harvard and the Whittemore Lectures at Yale. In all cases I was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. In writing up the lectures I have tried as far as possible to keep this dual audience in mind, and the early chapters in particular attempt to bridge the gap between the two points of view. In the later chapters, where the material becomes more technical, there is a danger of falling between two stools. On the one hand the mathematical jargon may be unintelligible to the physicist, while the presentation may, by mathematical standards, be lacking in rigour. This is a risk I have deliberately taken. The initiated mathematician should be able to fill in most of the gaps by himself or by referring to other published papers. Physicists who have survived the early chapters may derive some benefit by being exposed to new mathematical techniques, applied to problems they are familiar with. With this aim in mind I have throughout presented the mathematical material in a somewhat unorthodox order, following a pattern which I felt would relate the new techniques to familiar ground for physicists. The main new results presented in the lectures, namely the construction of all multi-instanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathematicians.
The purpose of this book is to apply geometrical methods to investigate solutions of the non-linear system of hyperbolic equations which describe the time evolution of non-abelian magnetic monopoles. The problem we study is, in various respects, a somewhat simplified model but it retains sufficient features to be physically interesting. It gives information about the low-energy scattering of monopoles and it exhibits some new and significant phenomena. From a mathematical point of view our investigation should be seen as a contribution to the area of "soliton" theory. In general a soliton is a solution of some non-linear differential equation which behaves in certain respects like a particle: it should be approximately localized in space and should be "conserved" in collisions.
6.2. From the Publisher.
Systems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly moving monopoles. This picture clarifies the important notion of solitons, which has attracted much attention in recent years. The soliton idea bridges the gap between the concepts of "fields" and "particles," and is here explored in a fully three-dimensional context. While the background and motivation for the work comes from physics, the presentation is mathematical. This book is interdisciplinary and addresses concerns of theoretical physicists interested in elementary particles or general relativity and mathematicians working in analysis or geometry. The interaction between geometry and physics through non-linear partial differential equations is now at a very exciting stage, and the book is a contribution to this activity.
6.3. Review by: Jacques Hurtubise.
American Scientist 77 (3) (1989), 296-297.
Nonlinear differential equations govern much of the world we live in, and many of the most intriguing phenomena which arise are due, in an essential way, to this nonlinearity. Solitons, which are particle like solutions to some of these equations, are one well-studied example. This beautiful book is devoted to the study of another example, that of the time evolution of magnetic monopoles. They share certain characteristics of solitons: for one, monopoles also exhibit particle-like behaviour. On the other hand, monopoles evolve in three-dimensional space, as op posed to the one-dimensional space of solitons, and their interaction is correspondingly more complex.
6.4. Review by: Claude LeBrun.
American Scientist 78 (1) (1990), 70-71.
The Yang-Mills-Higgs equations are the cornerstone of the modern physical theory of nongravitational fundamental forces. This book brings together an array of clever mathematical techniques from differential geometry, complex analysis, algebraic geometry, and mechanics to analyze these equations, subject to the following approximations: the Higgs self interaction is neglected, and the solution is assumed to vary slowly in time. ... The book has a wonderfully interdisciplinary flavour, tracing the problem from its physical motivations, through numerous beautiful areas of current mathematics, and on to the details of the solution, described both qualitatively and in terms of precise formulae. It manages simultaneously to be a chatty exposition and a research monograph, and by so doing contributes admirably to that active dialogue between mathematics and physics for which the current epoch seems so notable.
In recent years there has been a remarkable renaissance in the interaction between geometry and physics. After a long fallow period in which mathematicians and physicists pursued apparently independent paths their interests have now converged in a striking manner. However, it appears that parallel problems were being investigated in the past but a common language and framework were missing. This has now been rectified with gauge theory (alias the theory of connections) providing the common ground. In earlier periods geometry and physics interacted at the classical level, as in Einstein's theory of general relativity, with gravitational force being interpreted in terms of curvature. The new feature of the present interaction is that quantum theory is now involved and it turns out to have significant relations with topology. Thus geometry is involved in a global and not purely local way. A somewhat surprising feature of the new developments is that quantum field theory seems to tie up with deep properties of low-dimensional geometry ...
7.2. From the Publisher.
These notes arise from lectures presented in Florence under the auspices of the Accadamia dei Lincee and deal with an area that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah here presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.
7.3. Review by: Jim Stasheff.
American Scientist 79 (6) (1991), 568.
A slim volume of this size is more usually associated with a work of poetry. Indeed the comparison is apt, given the finely crafted nature of Sir Michael Atiyah's exposition. In eight swift chapters, Atiyah presents a panorama of some of the most exciting developments in contemporary mathematical physics (or physical mathematics). The subject of these lecture notes (expanded from lectures Atiyah gave for the Academia dei Lincei in 1988) is much more than is indicated in the title; knots play a small but significant role. The lectures successfully attempt to introduce the reader to a field that is in its infancy, but is growing at an astonishing rate. Thus the material cannot be considered definitive, but rather is full of ideas that are undergoing dynamic evolution. The current text provides an invitation to behold the early stages of a new paradigm for the interaction of mathematics and physics, Quoting Atiyah: "After a long fallow period in which mathematicians and physicists pursued apparently independent paths their interests have now converged in a striking manner."
7.4. Review by: Nick Lord.
The Mathematical Gazette 75 (472) (1991), 261.
Anything published by Sir Michael Atiyah, this country's most eminent mathematician (and an ex-president of the Mathematical Association), is worthy of note. This slim volume, arising from lectures given at the University of Florence in 1988, provides an entree into a new and rapidly developing area that lies at the crossroads of mathematics and physics. Emphasising motivational ideas rather than technicalities and a formal theorem-proof-corollary style, Sir Michael presents an introduction (from the mathematician's viewpoint) to the exciting recent work of Ed Witten relating topological quantum field theory to the new polynomial invariants of knots discovered by Vaughan Jones in the early 1980's. As ever, the author writes with seductive charm and clarity and the publishers are certainly not guilty of hyperbole when they assert that the book will be essential reading for all geometers and gauge theorists.
8.1. From the Publisher.
From the earliest times, the geometry of space has been intimately involved with physics. As science has evolved and our understanding has deepened, the relations between geometry and physics have become subtler and more complex. At the time of this lecture, fundamentally new ideas from both areas were dramatically altering conceptions about the nature of the universe. In this presentation, Sir Michael Atiyah, one of the foremost mathematicians of the 20th century, discusses some of the deep connections that have been discovered between mathematics and quantum physics. Starting with the viewpoints of Euclid and Newton, Atiyah moves on to ideas growing out of Jones' work on knots in 3-space and Donaldson's work on 4-manifolds. In describing how Witten has brought these developments into contact with quantum field theory, Atiyah shows how quantum field theory is in itself an effort to understand the structure of a vacuum. A witty, engaging, and clear-sighted lecturer, Atiyah makes this fascinating topic accessible to audiences with a general scientific background.
8.2. Review by: Bernd Wegner.
Zentralblatt MATH (Zbl 00424114).
This video tape provides a good opportunity to enjoy the vivid lecturing style of Sir Michael Atiyah. On occasion of the 1991 Gibbs Lecture he provides a survey on the relations between geometry and physics. Starting with the viewpoints of Euclid and Newton the lecture culminates in the applications of topology to quantum physics, growing out of Jones' work on knots in 3-space and Donaldson's work on 4-manifolds.
9.1. From the Publisher.
Michael Atiyah tells us about the craft of the mathematician, the mechanisms of the brain that regulate our mathematical reasoning and the importance of this science to other disciplines, without ever forgetting its intrinsic beauty.
These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may stimulate readers to investigate them.