Being a fan of mathematical physics, I was very excited to be assigned this book to review. First and foremost, I was interested in what the author meant by physical mathematics. It turned out that this was not too hard to figure out. As the author mentions in his preface, over the course of the last few decades, mathematics and physics had many interactions which benefited both sides. Oftentimes physicists have reached into pure mathematics to pick up tools that could be useful in developing new physics; simultaneously, many new ideas from physics have had deep implications for mathematics. The latter is what the author calls physical mathematics: mathematics that’s inspired and motivated by physics.
When you skim through the table of contents of the book, you will see that the topics the author deems belong to physical mathematics are varied and diverse. In fact it is almost like a Who’s Who of modern mathematics: Kac-Moody algebras, Clifford algebras, Hopf algebras, quantum groups, homology and cohomology, 4-manifolds, symplectic manifolds, connections, characteristic classes, gauge theories, topological quantum field theories, moduli spaces, Chern-Simons, Yang-Mills, Atiyah-Segal, Atiyah-Jones, Seiberg-Witten, knot and link invariants, you can simply name your favorite mathematical notion or construct, and it is most likely here. The book covers a large subset of today’s mathematics, especially focusing on stuff that has been very active and in vogue for the last half century or so. As the author points out (page 385), of the twenty-eight Fields medalists from 1978 to 2006, fourteen have been awarded the medal for their work that lives in the boundaries of mathematics and physics. No matter how unfamiliar the phrase itself may be, physical mathematics is apparently very much alive and well, and is certainly getting much recognition.
The book begins with a chapter on algebra, followed by a chapter surveying topics in topology; the third chapter provides an introduction to differential geometry. The author depicts these first three chapters as mostly review material, mainly intended to make the book self-contained. The audience of the book, he expects, will consist of graduate students and researchers. However it is clear that the background he assumes is quite advanced and varied.
Even though the focus is mainly on the mathematics, the author often throws in sentences that make you realize that if you do not know much about modern physics, then perhaps you were not in the intended audience after all. Reading, I had to wonder who the real intended audience was and what they were supposed to gain from it, for it is clear to me that a typical graduate student, even one who has taken the standard graduate algebra, analysis and geometry sequences, will most likely not learn much from this book. It is true that the book is full of new information for someone with such a background, but the style is not one that seems too concerned about pedagogy. This is clearly not a textbook.
It is also not a typical monograph. The book does not contain many proofs, though the presentation style at first resembles a typical mathematics monograph. In the first three introductory review chapters there are a total of two proofs, and it is not totally clear why they are included and others are not. Furthermore the author sprinkles in many more personal anecdotes and opinions along the way than are found in most mathematical texts. I believe the author describes the book best in his own Epilogue (page 378):
[w]e might liken this book with a modern tour through many countries in a few days. When you return home you can look at the pictures, think of what parts you enjoyed and decide where you would like to spend more time.
The author wraps up his epilogue with:
I hope that readers found several parts enjoyable and perhaps some that they may want to explore further. The vast and exciting landscape of physical mathematics is open for exploration.
I certainly enjoyed it; all in all, reading this book was a fun, but challenging.
Algebra.- Topology.- Manifolds.- Bundles and Connections.- Characteristic Classes.- Theory of Fields, I: Classical.- Theory of Fields, II: Quantum and Topological.- Yang-Mills-Higgs Fields.- 4-Manifold Invariants.- 3-Manifold Invariants.- Knot and Link Invariants.- Dictionary of Terminology.- Historical and Biographical Notes.- Categories and Chain Complexes.- Operator Theory.