Love and Math. The heart of hidden reality
Ever since he grew up as a boy in Kolomna (Russia), Frenkel was fascinated by elementary particles and quantum physics. It was pointed out to him that to understand these, he should start learning mathematics. So he started reading mathematics in his free time. An obvious choice would be to study at the department of Mechanics and Mathematics (Mekh-Mat) of the Moscow State University (MGU). However, back in 1984, his father being Jewish, this was impossible by the ruling anti-Semitism. His second choice was the Institute of Oil and Gas (Kerosinka), but he sneaked into the GMU to attend some courses and seminars by Gelfand. On the side he worked on a problem of braid groups proposed by D. Fuchs which resulted in his first paper published in Funct. Anal. Appl. at the age of 20. This brought him to study symmetry, (braid) groups and curves over finite fields. Further work brought him straight to the Langlands Program that was proposed by Robert Langland in 1967 and more formally in 1970. It is based on an earlier idea of André Weil who, while imprisoned in 1940 (having a disagreement with the French authorities), wrote a letter to his sister explaining the idea of a mathematical Rosetta Stone which would allow to translate results between three seemingly different fields in mathematics into each other: number theory, curves over finite fields, and Riemann surfaces. Exploring this connection has been shown successful by the proof of Fermat's Last Theorem. This connection is the mathematical analog to what the theoretical physicist call the Grand Unifying Theory in their study of quantum physics. The mathematical or physical aspects are just two different interpretation of the same theory. So quantum physics is like a fourth column to be added to Weil's Rosetta Stone. Frenkel's work with B. Feigin on Kac-Moody algebras came just in time because he got an invitation to spend a semester at Harvard in 1989 at the very time that perestroika was emerging. Because of the worsening situation in Russia with an unclear outcome, he decided after his 3 months stay, that it was better not to return home. So he stayed at Harvard where he got his PhD in 1991. Later he became professor of mathematics at UC Berkley. In 2003 he got directly involved in a multi-million DARPA grant to work out more elements of Weil's Rosetta Stone. Since then, his mathematical career is largely devoted to building the bits and pieces of this Grand Unifying Theory.
Frenkel makes it crystal clear that he is a passionate lover of mathematics and that his enthusiasm for the Langlands Program is immense. This love and passion is what he wants to convey to the reader. The math that most people learn in school is like learning to paint a fence in an art class, while true painting is about creating master pieces like Da Vinci or Picasso did. Mathematics is also a moral duty. Our world is ruled by mathematics that are hidden to most of us. The financial crisis in 2008 was caused by applying mathematics by people that were not controlled in a democratic way because our society does not care about mathematics and most people tend to stay away from it as far as possible. Mathematics should not be restricted to the "initiated few" but it should be shared by everybody. There is nothing more democratic than mathematics. There are no patents for formulas, its a universal language, and a correct formula can only represent truth, the universal truth.
With this conviction, Frenkel wants to transfer not only his love for mathematics but he also wants to show us the beauty of the mathematics that he is devoting most of his life to, and not just the "fence painting" bits. Of course reading this book will not make you a mathematician, but he succeeds by describing his life (at least the part related to his mathematical career) and gradually taking the reader along in his conquest of the mathematics he needed. So he explains symmetries, groups, finite fields, SU(3), manifolds, Galois groups, Lie algebras, sheaves, supersymmetry, strings, branes, etc. All things that are far beyond the low-fi kind of math that one usually finds in popular science books. Of course this is not easy, but I can imagine that his charismatic account will make some readers regret that they are not mathematicians, rather than the usual conviction that mathematics is a natural habitat where only nerds can survive. Many of the more technical details are removed from the main body as (sometimes quite extensive) notes that are collected at the end of the book. For a mathematical reader, they are of course useful, but others may want to skip them and still follow the essence of Frenkel's Conquest of Paradise.
But Frenkel is not only a mathematician. The last chapter of the book is still about mathematics and love, but now revealing the artistic talents of Frenkel. After a visit to Paris, he got the idea to make a film about math. With his neighbour, the author T. Farber, he wrote a screen-play called The two-body problem about two men in the South of France, one is a writer, the other a mathematician. They exchange their experiences, their passion for their profession and for women. It was published as a book in 2010. Before starting on the movie project, he wanted to get some cinematographic experience at a smaller scale and decided to produce a short movie. During another visit to France, he joined in with Reine Graves, a young film director. Inspired by a Japanese film of Y. Mishima Rites of Love and Death in which a lieutenant commits a ritual suicide together with his wife. Frenkel and Graves imitate the movie more or less. It shows a man (Frenkel) and a women (K.I. May) with in the back a poster with the text istina (Russian for truth). The man tattoos a mathematical formula (the formula of love) on the body of the women. The film is called Rites of Love and Math. It was well received, and you will find pictures on the Web of Frenkel teaching in Berkeley, but also where he shows up at the Cannes film festival. In fact by different media, Frenkel tries to transmit the same message: a mathematical formula or mathematics in general can be a thrilling thing of beauty, it can give you goose bumps, one may fall in love with it, it represents the ultimate truth, and it is worth committing your life to. The return you get from it is overwhelming.
One final remark. It is of course a side remark after Frenkel's plea for beauty, but I do not think that the cover design of the book is a success. It shows text in slightly tilted rectangles on a background image that is a detail of Van Gogh's The Starry Night painting. The symbolism is obviously well chosen, but it looks terribly chaotic, and I would have preferred a more stylish design representing the mathematical purity and beauty of its contents.