The Monodromy of Love
LOVE AND MATH: The Heart of Hidden Reality. Edward Frenkel. 286 pp. Basic Books, 2013. $27.99.
Last April the distinguished biologist Edward O. Wilson published an essay in the Wall Street Journal arguing that knowledge of mathematics is not necessary or even particularly helpful in most of the sciences. The champion who stepped forward from the ranks of mathematicians to contest this claim was not an elder statesman like Wilson but a Russian American mathematician in his 40s: Edward Frenkel of the University of California, Berkeley. Frenkel countered that math is indeed a useful tool, and more. “It’s about concepts and ideas that empower us to describe reality and figure out how the world really works,” he wrote. (The Wilson and the Frenkel essays were reprinted in Notices of the American Mathematical Society and are available online: http://www.ams.org/notices/201307/rnoti-p837.pdf.) He expands on this theme in Love and Math, a book that blends personal memoir with a tutorial introduction to some important ideas on the frontiers of modern mathematics.
First the memoir part. Frenkel grew up in an industrial city two hours from Moscow where a family friend—a teacher at a local college—encouraged and coached him in mathematics. In 1984, at age 16, he applied to the mathematics program at Moscow State University (known as MGU), even though he had been warned he would not be admitted because of his “nationality”: Jewish. The Soviet brand of anti-Semitism in that era had an especially cruel twist to it. Officially, discrimination didn’t exist; all students were eligible to apply. But those with Jewish names or heritage were turned away through a system of rigged oral exams. Frenkel was given a geometry problem about a circle inscribed in a triangle. As soon as he began to answer, his inquisitor interrupted to demand that he define a circle. “A circle is the set of points on the plane equidistant from a given point,” Frenkel said. Wrong. “It’s the set of all points on the plane equidistant from a given point.” Next he had to define a triangle and then a line. Of course none of the answers were ever quite satisfactory.
Frenkel ultimately enrolled at a less exalted school, the Institute of Oil and Gas, but he found ways to attend lectures at MGU. (The ways included scaling fences.) Within a few years he was collaborating with mathematicians in the same department that had rejected him, and publishing original research. Nevertheless, the unwritten rules of “nationality” would have blocked his path to an academic position in Russia. In the end it was MGU’s loss. Frenkel was offered a fellowship at Harvard, where he became a visiting professor even though he was just finishing his bachelor’s degree. He never went back.
Alongside the story of his life, Frenkel tells the story of his mathematics. His focus is a body of work called the Langlands Program, after Robert Langlands of the Institute for Advanced Study (although dozens of other mathematicians have contributed to it over the past 40 years). “I like to think of it as a Grand Unified Theory of Mathematics,” Frenkel writes, “because it uncovers and brings into focus mysterious patterns shared by different areas of math and thus points to deep, unexpected connections between them.” The most famous example of such a connection emerged 20 years ago with the proof of Fermat’s Last Theorem. The theorem is a statement in number theory. It says there are no positive integers x, y, and z that solve the equation xn + yn = zn when n is any integer greater than 2. The proof of this proposition, which was 350 years in coming, relied on ideas from two distant branches of mathematics: modular forms and the study of rational points on elliptic curves. I will not try to define those terms here, but Frenkel most certainly does make the attempt. He is never content merely to report what he sees from the mountaintops of mathematics; he wants you to climb up there with him and have a look around for yourself.
I admire this intrepid approach, but I also worry that some readers will not be prepared for the rigors of the journey. In his preface Frenkel writes:
Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars. This book is an invitation to this rich and dazzling world. I wrote it for readers without any background in mathematics. If you think that math is hard, that you won’t get it, if you are terrified by math, but at the same time curious whether there is something there worth knowing—then this book is for you.
Regrettably, I can’t fully endorse this invitation to neophytes and mathophobes. Those who are frightened of mathematics or hostile to it are not the ideal readers for Love and Math. Although Frenkel is a gifted tour guide and explainer, he knows the territory so well that he often misjudges how easily newcomers can get lost in the wilderness. Nonetheless, readers who come equipped with some level of interest and enthusiasm—and maybe their own compass—will have a splendid time.
Frenkel is passionate about his subject. In 2009 he wrote and acted in an erotic film in which a mathematician tattoos “a formula of love” on the body of his beloved. This is not the only occasion when the two nouns in his title come into close conjunction. I was even more taken with a metaphor he uses in the book to explain the concept of monodromy. A conventional definition of this term begins with the idea of a mathematical function that takes on some definite value for every possible position in the plane. Monodromy arises if you traverse a closed path in the plane and on returning to your starting point you discover that the value of the function has changed. Frenkel introduces this notion with the story of Rick and Ilsa, who meet at a party and fall in love. Then they go on a trip around the world. “As they were traveling, their relationship was evolving, so when they came back to the same point in space—their hometown—their love for each other may have changed.” This is “the monodromy of love.”
Brian Hayes is senior writer for American Scientist. He is the author of Group Theory in the Bedroom, and Other Mathematical Diversions (Hill and Wang, 2008) and Infrastructure: A Field Guide to the Industrial Landscape (W. W. Norton, 2005).