One of the things that I find hardest about discussing mathematics with non-mathematicians, whether it is my students, my parents, or the english professors I find myself standing with at cocktail parties, is convincing them that mathematics is a living and breathing subject rather than a field that ended the day that the fundamental theorem of calculus was proved. We all know that there are many fascinating stories throughout the history of mathematics, but only a few of them — Galois and his duel, Newton-v.-Liebniz, Fermat's Last Theorem — have made their way into the popular imagination, and when they do they are often caricatures of what the life of a mathematician is like.
Mark Ronan has found another example of a compelling story, and he tells it in his new book, Symmetry and the Monster: One of the Greatest Quests in Mathematics. The story he tells is the classification of finite simple groups, and its culmination in the discovery of the Monster, a simple group of order
But before he gets there, Ronan has quite a bit of ground to cover. He starts by talking about what groups are, presenting them primarily as the symmetry groups of various shapes but hinting at deeper and more formal definitions. He then discusses the story of Galois, both the exciting personal history and the mathematical contributions that tie group theory in with the study of solving polynomial equations. While this is well-tread ground (covered in much more depth in the recent book by Mario Livio for example), it is necessary to set up the rest of the book, which is not.
The star of the next section of the book is Sophus Lie, a Norweigan mathematician who lived in the second half of the nineteenth century who studied the continuous symmetry groups which we now call Lie groups. Lie was quite a character, who was once imprisoned on suspicion of being a spy, and his contributions to mathematics were enormous. In particular, the finite Lie groups provided many new examples of simple groups.
After discussing the classification of finite Lie groups, Ronan moves through the contributions of Feit, Thompson, McKay, Leech, Ogg, and many other mathematicians over the course of nearly a century until 1982 when Robert Greiss announced the existence of the Monster Group. The book concludes with a look at John Conway (yet another interesting character in the world of mathematics) and the Moonshine conjectures, which give extremely surprising connections between the Monster and modular functions, which in turn lead to surprising connections with Lie algebras, string theory, and physics.
These characters and their stories are definitely the primary focus on the book, letting the mathematics itself take a back seat for the most part. This is not to say that Ronan doesn't describe the mathematics that his characters are working on — he does describe it, and for the most part describes it quite well (although I think most mathematicians would find it discomforting that he often uses non-standard terms, such as "atoms of symmetry" instead of "simple groups" and "cyclic arithmetic" instead of "modular arithmetic"). But most readers will get more out of Ronan's storytelling than his mathematical exposition. Despite the fact that Ronan devotes some pages to discussing topics such as character tables and j-functions, I don't think that any non-mathematician would walk away from this book truly understanding much more than what a group is and a handful of examples, but Ronan does a good job of describing the mathematics in broad strokes and giving a flavor of what is happening and — more importantly — why mathematicians get excited about these questions. Ironically, it is this lack of technical mathematics that I think will make this book more interesting to many mathematicians than some of the other pop-math books of the last decade: there is enough there to give a good sense of what is going on to those of us who don't work in group theory, but not so much as to lose the interest of anyone other than a handful of group theorists.
One of the nice touches is that throughout the book, Ronan discusses not only mathematics and mathematicians, but also mathematical culture. These explanations — of how papers get written and refereed, how conferences work, and who exactly this Bourbaki guy is, for example — do an excellent job of explaining to the reader the life of mathematicians. These descriptions, and the book as a whole, should explain to those lay readers what types of questions mathematicians are interested in and how we go about trying to solve them.
While Symmetry and the Monster is not a perfect book, it does a good job of serving multiple audiences and I think that both mathematicians and non-mathematicians will enjoy and get something out of this book. Mark Ronan found a good story and he tells it well, and that's about all one can ask for.
Darren Glass is an assistant professor at Gettysburg College. His research interests include number theory, Galois theory, and cryptography. He can be reached at firstname.lastname@example.org