1.1. Review by: Krzysztof Galicki.
American Scientist 78 (6) (1990), 574.

The volume under review is based on lectures that Nigel Hitchin gave at the University of Montreal in 1985. Its purpose is to provide a unifying framework for a number of different nonlinear differential equations: from the four-dimensional self-dual Yang-Mills equations through the monopole and the vortex equations to harmonic maps of a 2-torus into the Lie group. All of these equations are of interest to those engaged in many areas of modern physics.

A single concept of a momentum mapping for the action of a group on a quaternionic vector space is shown to provide such a framework in a very natural way. In each case a solution is generated by an algebraic curve. Hitchin emphasises the common structure of all these equations and the role it plays in constructing the solutions. He very successfully accomplishes his goal. The material is chosen carefully and presented in a modern and original way. The splendid choice of examples makes the book easy to read. It is a welcome and valuable addition to the literature.

2.1. 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.

2.2. From the Preface.

In January 1987 I gave the Milton Brockett Porter Lectures at Rice University. This provided me with the opportunity of presenting, at some length, the results on magnetic monopoles which Nigel Hitchin and I have been investigating over the past few years. This book, written jointly, is an expanded version of the lectures and it contains a full and detailed treatment of the essentially new results. Although dependent on earlier work by many authors we have endeavoured to make it more self-contained by adding some introductory and background material.

Michael Atiyah

2.3. From the Introduction.

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 purely 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 localised in space and should be "conserved" in collisions.

2.4. 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 analyse these equations, subject to the following approximations: the Higgs self interaction is neglected, and the solution is assumed to vary slowly in time. Remarkably, this then amounts to understanding the geodesies of a certain solution of Einstein's vacuum equations in dimension $\Large\frac{k}{c}\normalsize$, where $k$ is the topological charge of the solution (roughly, the number of "lumps" at which the field is non trivial).

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.

2.5. 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 opposed to the one-dimensional space of solitons, and their interaction is correspondingly more complex.

The authors exploit an idea of Manton, which is that the time evolution should be described as geodesic motion on the space of static solutions. This space, in turn, has a quite beautiful and complete description, involving algebraic geometry and the method of inverse scattering. After giving some general results, the authors concentrate on the two-particle case, and reveal some quite intricate and surprising interactions. The book is well written and deserves a wide mathematical audience: it focuses techniques from a wide array of areas of mathematics on one concrete problem, and solves it.

2.6. Review by: R S Ward.
Bulletin of the American Mathematical Society 20 (2) (1989), 230-231.

The monopoles of this book are "lump-like" solutions of a nonlinear system of hyperbolic partial differential equations. A snapshot of such a solution, at any moment in time, reveals a finite number of lumps (or solitons), localised in space. As time evolves, these solitons interact with one another in a manner determined by the equations, while maintaining their separate identities.

This whole subject of solitons is truly interdisciplinary; it has caught the interest and imagination of analysts and geometers as well as of applied mathematicians, mathematical physicists, and theoretical physicists. Sometimes the word "soliton" is reserved for the solutions of the rather special class of equations described as completely integrable: in these cases, the scattering of solitons is essentially trivial (their velocities are unaffected by the collision, and no radiation is emitted). The best-known examples of these involve just one space dimension, and include such (sometimes bizarrely-named) systems as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and sine-Gordon equations. By contrast, monopoles live in three spatial dimensions, and scatter nontrivially; this book studies that scattering, in a slow-motion approximation. So, whereas for (say) the KdV equation one can write down an explicit solution representing a two-soliton collision, in this case one only has an approximate solution for small relative velocities. But the authors' point of view is that an equation such as KdV is in any event only an approximation to some more realistic physical model. As far as the physics is concerned, an approximate solution to an exact equation amounts to the same thing as an exact solution to an approximate equation.

But considerations of physics are, in any case, of only limited relevance. The main theme of the book is mathematical: the beautiful interplay between nonlinear systems and geometry.
...
This book is an expanded version of a series of lectures given by M F Atiyah at Rice University. It requires some background knowledge of algebraic and differential geometry, but not of the mathematical physics that originally inspired the problem. It should be read by any mathematician who wants to see something of the exciting connections between geometry and the nonlinear systems of mathematical physics.

With the introduction of general relativity, it became necessary to express the differential operators of mathematical physics in a coordinate-free form. This made it possible to define those operators on an arbitrary Riemannian manifold-the grads, divs, and curls got translated into the $d + d^{*}$ operator on the bundle of exterior forms. This particular operator found fruitful application in the theorem of Hodge which expressed the dimension of the null space {the space of harmonic forms) on a compact manifold in terms of topological invariants - the Betti numbers.

Another operator - the Dirac operator - made a later appearance in Riemannian geometry. It was used by Atiyah and Singer to explain the integrality of the $Â$-genus of a spin manifold, and then Lichnerowicz proved a strong vanishing theorem - if a spin manifold has positive scalar curvature, the null space of the Dirac operator (the space of harmonic spinors) is zero. Rearing in mind the formal similarity between the Dirac operator and the $d + d^{*}$ operator, one may ask if there is an analogue of Hodge's theorem - can we express the dimension of the null space in terms of topological invariants of the manifold? The main purpose of this paper is to show that this is impossible and in general the dimension of the space of harmonic spinors depends on the metric used to define the Dirac operator

This paper is based upon the author's doctoral thesis, supported by a United Kingdom SHC Research Studentship. Thanks are due especially to Professor Atiyah for his continuing help and encouragement.

We review the remarkably fruitful interactions between mathematics and quantum physics in the past decades, pointing out some general trends and highlighting several examples, such as the counting of curves in algebraic geometry, invariants of knots and four-dimensional topology.

4.1. Past history of physics.

The relation between mathematics and physics is one with a long tradition going back thousands of years and originating, to a great extent, in the great mystery of the cosmos as seen by shepherds on starry nights. Astronomy in the hands of Galileo ushered in the modern scientific era, and it was Galileo who said that the book of nature is written in the language of mathematics:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.

A giant step forwards in using the language of nature to describe physical phenomena was made by Isaac Newton, who developed and applied the calculus to the study of dynamics and whose universal law of gravitation explained everything from the fall of an apple to the orbits of the planets.

The nineteenth century witnessed the greater sophistication of Maxwell's equations to include the behaviour of electromagnet ism, and the twentieth century saw this process take a major step forwards with Einstein's theory of special relativity and then of general relativity. At this stage both gravitation and electromagnetism were formulated as field theories in four-dimensional space time, and this fusion of geometry and classical physics provided a strong stimulus to mathematicians in the field of differential geometry.

However, by this time, it had already been realised that atomic physics required an entirely new mathematical framework in the form of quantum mechanics, using radically new concepts, such as the linear superposition of states and the uncertainty principle, that no longer allowed the determination of both the position and momentum of a particle. Here the mathematical links were not with geometry, but with the analysis of linear operators and spectral theory. As experimental physics probed deep into the subatomic region, the quantum theories became increasingly complex and physics appeared to be diverging from classical mathematics, while at the same time dragging analysis along behind on a somewhat rocky journey. Indeed, in the 1950s under a barrage of newly discovered particles, the hope of capturing the fundamental physical laws in terms of deep and elegant mathematics seemed to evaporate rapidly. At the same time there was a strong inward movement in mathematics with a renewed focus on fundamental structures and rigour.

The picture began to change around 1955, ironically the year of Einstein's death, with the advent of the Yang-Mills equations, which showed that particle physics could be treated by the same kind of geometry as Maxwell's theory, but with quantum mechanics playing a dominant role. However, it took to the beginning of the 1970s before it became clear that these non-Abelian gauge theories are indeed at the heart of the standard model of particle physics, which describes the known particles and their interactions within the context of quantum field theory. It is a remarkable achievement that all the building blocks of this theory can be formulated in terms of geometrical concepts such as vector bundles, connections, curvatures, covariant derivatives and spinors.

This combination of geometrical field theory with quantum mechanics worked well for the structure of matter but seemed to face a brick wall when confronted with general relativity and gravitation. The search for a coherent framework to combine the physics of the very small (quantum mechanics) with the physics of the very large (general relativity) has, for several decades, been the 'Holy Grail' of fundamental physics. The most promising candidate for a solution to this problem is string theory, and the mathematical development of this theory, in its various forms, has been pursued with remarkable vigour and some success.

4.2. The impact of modern physics on mathematics.

What has been described so far is the familiar story of the advance of physics necessitating the use and development of increasingly sophisticated mathematics, to the mutual benefit of both fields. Mathematicians have been driven to investigate new areas; and physicists, to quote Eugene Wigner (1960), have been impressed by 'the unreasonable effectiveness of mathematics in physics' - the remarkable universal properties of mathematical structures.

But over the past 30 years a new type of interaction has taken place, probably unique, in which physicists, exploring their new and still speculative theories, have stumbled across a whole range of mathematical 'discoveries'. These are derived by physical intuition and heuristic arguments, which are beyond the reach, as yet, of mathematical rigour, but which have withstood the tests of time and alternative methods. There is great intellectual excitement in these mutual exchanges.

The impact of these discoveries on mathematics has been profound and widespread. Areas of mathematics such as topology and algebraic geometry, which lie at the heart of pure mathematics and appear very distant from the physics frontier, have been dramatically affected.

This development has led to many hybrid subjects, such as topological quantum field theory, quantum cohomology or quantum groups, which are now central to current research in both mathematics and physics. The meaning of all this is unclear and one may be tempted to invert Wigner's comment and marvel at 'the unreasonable effectiveness of physics in mathematics'.

The key role of physics in many of these areas is to produce an intuitive 'natural' context for various abstract mathematical constructions. Pure mathematics not only consists of theorems built step-by-step via logical deductions, but also has an intuitive side - the use of analogy and metaphor to jump from one context to another or to explore new areas to see what might be true. Geometrical intuition has often been an important part of this, treating functions as vectors in Hilbert spaces or dealing with the number theory of Fermat's last theorem with elliptic curves. In the same way classical mechanics has provided a rich setting for the principles of calculus, geometry and more recently dynamical systems. Remarkably, modern physical constructions such as quantum field theory and string theory, which are very far removed from everyday experience, have proven to be a similar fertile setting for mathematical problems. Indeed, in many ways quantum theory has turned out to be an even more effective framework for mathematics than classical physics. Particles and strings, fields and symmetries, they all have a natural role to play in mathematics.

Einstein once said that 'since the mathematicians have invaded the theory of relativity, I do not understand it myself' and there is a sense that the reverse invasion has wrong-footed a number of mathematicians. But the presence of physics in contemporary pure mathematics is undeniable and has presented the community with both opportunities and challenges.

This influence manifests itself in two ways. In some sense the easier one is for the mathematician to be presented with a clearly stated conjecture or problem. One then attempts a development of current conventional techniques to provide a rigorous solution. This situation requires a successive distillation of the problem from the physics to pure mathematics, and involves the active participation of committed individuals on the way. It may result in a seriously difficult task, but one that may perhaps be tackled with the mathematician's toolkit. The construction of instantons in the 1970s, to which we turn in more detail later, was an example of this: a problem derived from quantum physics but refined to one in conventional, but modern, differential geometry. The mathematics that developed from it, and its new viewpoints, was the Fields Medal-winning work of Simon Donaldson.

The second mode of influence is more direct. This is a more fundamental invasion of physical ideas based on the underlying structure and language of quantum field theory - thinking big in terms of path integrals on spaces of fields, for example. The challenge to mathematicians here is much more serious, since it involves developing a sense of intuitive understanding to parallel that of every physicist, for whom this has become second nature since graduate school. It has spurred new ways of thinking in pure mathematics. Clearly, this approach is a long-term programme with many ramifications, somewhat reminiscent of Hilbert's sixth problem: the axiomatisation of all branches of science, in which mathematics plays an important part.