1.1. From the Preface.

Gauge theories arise in the description of electromagnetic phenomena, including superconductivity. They also have become central to elementary particle physics, as the starting point for a quantum theory of the weak, strong and electromagnetic interactions. In the past few years classical gauge theories have been widely studied by topologists and geometers interested in their underlying structure. One outcome of the work of physicists and mathematicians has been the discovery of all "instantons", as well as a partial understanding of their role in quantum theory.

We present here another point of view. A combination of analytic and topological methods can give useful insights into classical gauge theories for which purely geometric methods have yet to be successful. In particular we use these methods to analyse general features of the Ginzburg-Landau equations for vortices and the Yang-Mills-Higgs equations for monopoles, for which solutions in closed form have not been found.

1.2. Review by: Masatsugu Minami.
Mathematical Reviews MR0614447 (82m:81051).

This is a book which is a quite up-to-date rigorous treatment of vortices and magnetic monopoles based upon the authors' recent results. ...
...

The highlight of the book is a careful construction of the existence theorem on the multimonopole solution to the first order Bogomol'nyi equation in the sense that the monopoles are spatially separated. Further properties of second order systems of equations on R3 outside the Bogomol'nyi-Prasad-Sommerfield limit are also discussed.

This book is designed to be readable by both physicists and mathematicians. For example, it starts with an explanation of what physicists call gauge theories, the Higgs mechanism and so on, and also includes a final chapter where some of the analytic tools used in this book are concisely explained.

2.1. From the Publisher.

This volume presents the extensive research of C Taubes. It gives the readers a clear and concise explanation of L2 Moduli Spaces on 4-Manifolds. This monograph discusses important results in the fields of differential geometry and topology. It sets the framework for development in these areas and should be a useful reference tool for both students and researchers in these fields.

2.2. Review by: Daniel Ruberman.
Mathematical Reviews MR1287854 (96b:58018).

The monograph under review represents an important strand in the ongoing effort to understand the powerful invariants of smooth 4-manifolds introduced by S Donaldson in the mid-1980s.
...

Recent events (in which Taubes has played a key role) have, in a sense, overtaken this massive enterprise, as the introduction of Seiberg-Witten invariants [E Witten, (1994)] has provided a technically simpler and more powerful approach to the main topological applications, such as the Thom conjecture. There remains, however, the fundamental task of understanding Donaldson theory in its most general setting as a theory of invariants on arbitrary 4-manifolds with boundary, complete with computable Floer groups and gluing laws. If such a program is ever carried out, the ideas in Taubes' book will play an essential role.

3.1. Review by: Dave Auckly.
Mathematical Reviews MR1331151 (96b:57039).

At the joint meetings in January 1995, Taubes gave a series of talks entitled "Mysteries in three and four dimensions". This is a video of the first two lectures. The lectures were very timely because they came just three months after a major breakthrough in gauge theory. In Taubes' usual style, the lectures are a mix of space alien, whale watching, Forrest Gump anecdotes and mathematical exposition. The first 50-minute lecture is a very complete, well-balanced review of Donaldson theory. It also contains an inside look at the recent history of gauge theory. The second lecture is an introduction to the Seiberg-Witten equations which is appropriate for advanced students of geometry. The author crams an amazing amount of detail into 100 minutes without losing the audience. I give this video four stars.

4.1. From the Publisher.

In this book, the author's goal is to provide an introduction to some of the analytic underpinnings for the geometry of anti-self duality in 4-dimensions. Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics.

The book reviews some basic geometry, but it is assumed that the reader has a general background in differential geometry (as would be obtained by reading a standard text on the subject). Some of the fundamental references include Atiyah, Hitchin and Singer, Freed and Uhlenbeck, Donaldson and Kronheimer, and Kronheimer and Mrowka. The last chapter contains open problems and conjectures.

4.2. From the Introduction.

My goal in these lectures is to provide an introduction to some of the analytic underpinnings for the geometry of anti-self duality in 4-dimensions. Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and also on curvatures of Riemannian metrics has resulted in some spectacular mathematics. In the ensuing lectures, I will review some of the basic geometry, but even so, I will assume that the reader has a generalist sort of background in differential geometry, such as one might obtain by reading a standard text on the subject (Kobayashi and Nomizu's books comes to mind). The final lecture consists of open problems and conjectures. (Actually, it consists of a series of questions that I would ask an alien spacefarer should one land in my back yard and profess a profound understanding of 4-dimensional geometry/topology.)

Before beginning, I should point out some of the fundamental references for the subject. The starting reference is the manuscript by Atiyah, Hitchin and Singer where the basic geometry of anti-self duality is presented. My presentation of the geometry borrows rather heavily from ... . Next comes the book by Freed and Uhlenheck which describes Simon Donaldson's remarkable first theorem which connects anti-self duality to geometric topology. Freed and Uhlenbeck's book also describes many of the underlying analytical issues. The next fundamental reference is the book by Donaldson and Kronheirner. This marvellous manuscript details almost all you have to know to understand the anti-self dual equations and their applications as of 1992. I also add as a basic reference, the recent paper by Kronheimer and Mrowka; these contain the latest breakthroughs on the subject. Finally, the well equipped gauge theory library should have a reference to Floer cohomology, so I suggest the forthcoming book by Donaldson, Fukaya and Kotschick.

Some of the subject matter in these lectures will overlap with various parts of the aforementioned references. However, as I remarked above, my goal here is to spotlight the analytic issues, and also to present the Riemannian metric analogy of the anti-self dual equations.

4.3. Review by: Daniel Pollack.
Mathematical Reviews MR1400226 (97m:53047).

This book provides an excellent introduction to the application of certain analytic techniques to problems in differential geometry. The conduit for this introduction is a two-part treatment of gluing and perturbation methods as applied to the study of compact Riemannian four-manifolds. The first part is a treatment of the existence and deformation theory for anti-self-dual connections on compact, oriented four-manifolds. The author reviews the basic notions of covariant derivatives and anti-self-duality by giving precise definitions as well as citing simple, concrete illustrative examples and raising numerous exercises which encourage readers to get their hands dirty with basic computations. The presentation is both casual and concrete. For example, when introducing the moduli space of anti-self-dual connections the author focuses from the start on the essential properties of a set of equations which lead to an interesting and useful moduli space. These are the Fredholm nature of the linearisation, the existence of a geometric compactification of the moduli space, and the algebraic constructions of special solutions. He then proceeds to explain in detail how the anti-self-dual equations exhibit each of these features.
...

The casual style in which this book is written together with the straightforward explanations of the key ideas underlying the theory makes it an excellent source for those wishing to learn about these basic techniques. As the book is certainly not a comprehensive treatment of the subject, ample references are given where necessary; in particular, a perfect balance seems to have been struck in the choice between what to include and what to refer the reader elsewhere for.

5.1. From the Publisher.

On March 28-30, 1996, International Press, the National Science Foundation, and the University of California at Irvine sponsored the First Annual International Press Lecture Series, held on the Irvine campus. The inaugural speaker for this event was Professor Clifford Henry Taubes of Harvard University who delivered three lectures on "Seiberg-Witten and Gromov Invariants." In addition, there were ten one-hour lectures delivered by some of the foremost researchers in the field of four dimensional smooth and symplectic topology. Volume I of these proceedings contains articles based on six of those lectures. The present volume consists of four papers by Taubes comprising the complete proof of his remarkable result relating the Seiberg-Witten and Gromov invariants of symplectic four manifolds. The first paper "SW$\implies$Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves" appeared in print in 1996 in the "Journal of the American Mathematical Society". The remaining three papers appeared in the "Journal of Differential Geometry".

5.2. Review by: Ignasi Mundet-Riera.
Mathematical Reviews MR1798809 (2002j:53115).

This book brings together the four papers of Taubes in which a proof is given of the equality between Seiberg-Witten invariants on a compact symplectic four-manifold and certain invariants obtained by counting pseudohomolomorphic curves (Gromov invariants).

In the first paper, the author recalls a certain perturbation (depending on a parameter r) of the Seiberg-Witten equations. ...

The second paper is devoted to giving a precise definition of the Gromov invariant which counts J-holomorphic curves. It should be noted that the invariant counts non-parametrised embedded curves, in contrast to the usual Gromov-Witten invariants, which count parametrised (and not necessarily embedded) curves. ...

In the next paper Taubes shows how to obtain solutions to the perturbed Seiberg-Witten equations from an embedded $J$-holomorphic curve. ...

Finally, the fourth paper of the book identifies the two different moduli spaces used to define Seiberg-Witten and Gromov invariants, thus proving that the invariants do coincide.
...

In the reviewer's opinion, it is a great idea to publish the four papers of Taubes in a single book. Not only is the result proved by Taubes extremely interesting per se, but also his great mastery of the several techniques used in gauge theory and the clarity and precision with which he explains their use make this book highly recommended reading.

6.1. From the Publisher.

For undergraduate courses in math modelling, and biology calculus courses in departments of math and life sciences.

Designed for students who understand the simple basics of calculus, this text focuses on the differential equations and related subjects that are commonly used today by working life scientists. It emphasises both the mathematics and how the mathematics is employed in order to introduce some potentially useful tools and modes of thought to future experimental biologists.

6.2. From the Preface.

This book is a compendium of chapters for a course on differential equations and their applications in the biological sciences that I developed at Harvard University. The book and the course roughly follow a wonderful book by Edward Beltrami called Mathematics for Dynamic Modeling published by Academic Press, which is a book written for students who already have a reasonably sophisticated mathematics background. This book covers many of the topics in Beltrami's book (and shamelessly borrows some of his examples), but it is designed for life science students who have had only the basics of calculus, which is to say that students should have a good intuitive feel for the meaning of differentiation and integration. and they should be at home integrating and differentiating sines, cosines, powers, and exponential functions, Note that the book is not really aimed at potential applied mathematicians; instead, my goal is to introduce to future experimental biologists some potentially useful tools and modes of thought.

The material here is organised into 28 chapters with accompanying articles from the current (circa late 1990s) biology research literature that illustrates the utility of the mathematics. (A few of the supporting articles come from the geology and earth science literature, geology being a "hobby" of mine.) I have supplied a paragraph or so of commentary about each of the illustrative articles. I don't require students to understand the biology in these articles (goodness knows how little I understand); rather I mean for the articles to make a convincing case that the mathematics from the course is re levant to specific areas of current biological research.

6.3. From the Introduction.

First, I freely admit to not being a biologist. In fact, until I started teaching the course on which this book was based, I knew very little of recent work in biology. I took biology in high school, dissected a worm and a frog, and happily found other interests. Subsequently, I kept minimally abreast of the subject by reading articles from popular science journals such as Science News and Scientific American. However, since I started teaching this course, I have endeavoured to educate myself about modern biology and have found it to be a glorious thing. In fact, I would be happy to argue the case that our understanding of biology now ranks as the (or at least one of the) crowning achievements of human knowledge.

My recent and ongoing education in biology has taught me the following lesson: With some notable exceptions, biology at the cusp of the twenty-first century is very much an experimentally driven science. Life is extremely complicated, and sorting out these complications is the task at hand. This is to say that the laboratory rules the field. At the risk of some exaggeration, one might say that it is somewhat premature in most fields of biology to spend too much energy with theory. In fact, I think that the following situation is common: You are trying to guess how a particular biological process works. You come up with a good proposal for the process. But, does nature use your proposal? You do some experiments and you find, lo and behold, that nature maybe uses your proposal, but probably not; and in any event, nature has found 20 completely different ways to work the process and is using all 20 simultaneously. (On the other hand. there are certain subfields of biology that could use, perhaps, more experimentation and less theory. Population biology is a particular example, for in this field, controlled experiments on macroscopic life forms are not easy to devise.)

With the preceding understood, where is the place for mathematics in biology? The answer to this question necessarily requires an understanding of what modern mathematics is. In this regard. I should say that term mathematics covers an extremely broad range of subjects. Even so, a unifying definition might be as follows:

Mathematics consists of the study and development of methods for prediction.

Meanwhile, a science such as biology has, roughly, the following objective:

To find useful and verifiable descriptions and explanations of phenomena in the natural world.

To be useful, a description need be nothing more than a catalogue or index. But an explanation is rarely useful without leading to verifiable predictions. It is here where mathematics can be a great help. In practice, working biologists use mathematics as a tool to facilitate the development of predictive explanations for observed phenomena. This is how mathematics will be viewed in this text. (The use of mathematics as a tool to make predictions of natural phenomena is called modelling and the resulting predictive explanation is often called a mathematical model.)

At this point, it is important to realise that a vast range of mathematics has found biological applications. Two, in particular, are differential equations and probability/ statistics. This text is concerned almost solely with differential equations; almost nothing will be said about probability and statistics. However, this is not to say that the latter are less important. In fact, they are extremely important, and you should take a course on probability and statistics (or experimental design) if you haven't done so already.

6.4. Review by George Oster.
Bulletin of the American Mathematical Society 39 (3) (2002), 431.

There is no shortage of introductory texts on differential equations, nor of introductory math books for biologists. But Taubes' book is a welcome addition. While the presentation of the mathematics is fairly conventional, the exposition is exceptionally clear, and each chapter is followed by a "Lessons" section summarising the key ideas covered. However, what distinguishes this text from its competitors are the "Readings" that follow: a collection of research papers the author has gleaned from the literature to complement each chapter. These cover a wide variety of topics, sometimes only peripherally related to the subject matter of the chapter, but never mind, for they constitute an interesting, if idiosyncratic, medley.

Beginning with simple exponential growth, the text covers the basics of ordinary differential equations, including phase plane analysis and stability of linear and nonlinear systems. Partial differential equations are introduced via diffusion and advection models. Separation of variables is introduced by a separate chapter that constructs a fisheries model, followed by a more detailed exposition in the following chapter. The chapter on pattern formation by Turing instabilities is rather cursory compared to the Readings, but the emphasis on diffusion-advection patterns is unusual and appropriate for ecological applications. I especially enjoyed the three chapters on fast and slow time scaling, estimating time scales in dynamic problems, and models for biological switches. The book concludes with short chapters on testing for periodicities in data and on chaotic systems.

One quibble I have is the small number of exercises at the end of each chapter, most of which do not address much of the content in the Readings. However, there is a collection of "extra exercises" at the end of the book, many of which include answers. I would hope that in the next edition, Taubes would try to integrate the exercises more closely with his selections from the literature.

Finally, it would not be fair to compare this book with the classic text by J D Murray, Mathematical biology, which is far more comprehensive and advanced, nor with Edelstein-Keshet's Introduction to mathematical biology, which is more intuitive and aimed at a less mathematical audience. Taubes' text is intended to be a pedagogical introduction to biological modelling for mathematics students, and in this it succeeds admirably, not least because of the exposure it provides to the current literature.

7.1. Review by: Hal Leslie Smith.
Mathematical Reviews MR2374282 (2008k:34001).

This book, well described by its title, stands out among the many similar books on the same subject matter by the inclusion, at the end of chapters, of quite a number of brief journal articles from the research literature which support and amplify the topics under discussion. ...

This second edition (2008) appears to be little changed from the original (2001). ...

The main mathematical topics included are ordinary differential equations of a single or a pair of unknown functions, phase plane analysis, equilibrium and stability analysis, elementary linear algebra, partial derivatives, the elementary partial differential equations of advection and diffusion and their special solutions, separation of variables solutions. The stability of steady states of scalar reaction diffusion equations on bounded intervals and the traveling waves for the Fisher equation are also considered. The book ends by returning to ordinary differential equations and periodic solutions (a simplified Poincaré-Bendixson theorem), fast-slow systems and chaos.

Biological applications treated by the author include population dynamics (exponential growth, logistic growth, predator-prey dynamics) simple epidemic models, and epidemic waves. The included research articles greatly expand the application areas.

The target audience for the book is "life science students who have had only the basics of calculus"; the book "is not really aimed at potential applied mathematicians; instead my goal is to introduce to future experimental biologists some potentially useful tools and modes of thought".

The above-mentioned mathematics is introduced as needed, in brief chapters (28 in all). A typical chapter begins by introducing a topic in a very few pages, followed by one or more research articles from Science or Nature, usually preceded by author commentary, and ending in a set of exercises. The mathematics is particularly well motivated, the style is informal, and the author is not afraid to simply introduce a solution.

I found myself reading several of the included research articles. However, they date to the 1990s, are sometimes poorly reproduced with the original colour graphics turned to shades of grey, and in some cases seem only marginally related to the topic at hand. There are some classics included but others are tough going. I wonder to what extent these are read/comprehended by the student. Overall, I think the exposition is excellent; a student can learn a great deal from this book.

8.1. From the Publisher.

- Introduction to many of the foundational concepts for modern mathematics, mathematical physics and theoretical physics in one volume.

- Unique focus on the foundational material to provide a concise, coherent introduction to the subject.

- Many of the classic examples in the subjects covered are fully worked out.

- Proofs of most of the background material from differential topology provided.

- The required linear algebra and complex function theory is presented in full.

- Inspired by Bott's famous Harvard course.

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects.

Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry.

Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.

8.2. From the Preface.

This book is meant to be an introduction to the subject of vector bundles, principal bundles, memes (Riemannian and otherwise), covariant derivatives, connections and curvature. I am imagining an audience of first-year graduate students or advanced undergraduate students who have some familiarity with the basics of linear algebra and with the notion of a smooth manifold. Even so, I start with a review of the latter subject. I have tried to make the presentation as much as possible self-contained with proofs of basic results presented in full. In particular, I have supplied proofs for almost all of the background material either in the text or in the chapter appendices. Even so, you will most likely have trouble if you are not accustomed to matrices with real and complex number entries, in particular the notions of an eigenvalue and eigenvector. You should also be comfortable working with multi-variable calculus. At the very end of each chapter is a very brief list of other book s with parts that cover some of the chapter's subject matter.

I have worked our many examples in the text, because truth be told, the subject is not interesting to me in the abstract. I for one need to feel the geometry to understand what is going on. In particular, I present in detail many of the foundational examples.

I learned much of the material that I present here from a true master, Raoul Bott. In particular, I put into this book the topics that I recall Raoul covering in his first-semester graduate differential geometry class. Although the choice of topics are those I recall Raoul covering, the often idiosyncratic points of view and the exposition are my own.

8.3. Review by: Josef Janyska.
Mathematical Reviews MR3135161.

The book is a very good introduction to differential geometry suitable for both advanced undergraduate students or first-year graduate students. The book consists of 19 chapters (each chapter contains a basic bibliography of the topic) and covers all the basic areas of differential geometry: smooth manifolds, Lie groups, vector bundles, metrics on vector bundles and Riemannian geometry, principal bundles, covariant derivatives and connections, holonomy, complex manifolds and holomorphic submanifolds. The book assumes only minimal prior knowledge and it is written in a very legible manner with many examples. I can highly recommend the book for a first introduction to differential geometry.

8.4. Review by: Peter Giblin.
The Mathematical Gazette 97 (539) (2013), 373.

The author of this book, C H Taubes, is a distinguished Harvard mathematician famous for his work in gauge theory, 4-manifolds (including the proof that 4-dimensional Euclidean space $\mathbb{R}^{4}$ has an uncountably infinite number of smooth structures) and 'Seiberg-Witten theory'. He says in the preface that he learnt the material of the book from a 'true master, Raoul Bott', but that the idiosyncratic point of view [charmingly misprinted as 'idionsyncratic'] is his own. The material, which is well described in the subtitle, is that of advanced real and complex differential geometry, suitable for well-prepared graduate students or professionals needing a good reference. For example, the Gauss-Bonnet theorem, long regarded as the high point of many undergraduate differential geometry courses, does not appear until page 227 and the proof uses almost complex structures. Similarly, there are only two diagrams, on pages 5 and 6, one of a torus and the other of a disk with two holes. So this is not a beginner's textbook, but it is unusual for the number of detailed proofs throughout, including in the early 'introductory' chapters. The pace of the book is fast, but the style is almost that of lecturing, with asides and comments and illuminating remarks made in the middle of arguments where they can be of most value, and with long and complicated arguments broken into smaller pieces for easier digestion. One thing I did not find so attractive is the typography: as someone used now to the elegant sophistication of TEX typesetting, the use of Roman type in formulas was rather a shock. But this should not detract from the attractiveness, for its intended audience, of the book