Love and Math

by Frenkel, Edward

  The research on the interface between the Langlands Program and electromagnetic duality quickly became a hot topic that developed into a vibrant area of research. An important role in this process was played by the annual conferences we organized at the Kavli Institue for Theoretical Physics in Santa Barbara. The Institute director, Nobel Prize–winner David Gross, was our big supporter.

  In June 2009, I was asked to speak about these new developments at Séminaire Bourbaki. One of the longest running mathematics seminars in the world, it is revered in the mathematical community. Scores of mathematicians are drawn to its meetings at the Henri Poincaré Institute in Paris, which take up a weekend three times a year. The seminar was created shortly after World War II by a group of young and ambitious mathematicians who called themselves – using an assumed name – Association des collaborateurs de Nicolas Bourbaki. Their idea was to rewrite the foundations of mathematics using a new standard of rigor based on the set theory initiated by Georg Cantor in the late nineteenth century. They succeeded only partially, but their influence on mathematics has been enormous. André Weil was one of the founding members, and Alexander Grothendieck played a big role later on.

  The purpose of Séminaire Bourbaki is to report on the most exciting developments in mathematics. A secret committee that chooses the topics and the speakers has since its inception followed the rule that its members must be under fifty years of age. The founders of the Bourbaki movement apparently believed that it constantly needed fresh blood, and this has served them well. The committee invites the speakers and makes sure that they write up their lectures in advance. Copies are distributed to the audience at the seminar. As it is considered an honor to present a talk at the seminar, the speakers comply with the request.

  The title of my seminar was “Gauge Theory and Langlands Duality.”19 Though my talk was more technical and involved more formulas and mathematical terminology, I basically followed the story that I have recounted in this book. I started with André Weil’s Rosetta stone, giving a brief tour of each of its three columns, just as I did here. Because Weil was one of the founders of the Bourbaki group, I thought it was especially fitting to talk about his ideas at the seminar. I then focused on the newest developments, linking the Langlands Program and the electromagnetic duality.

  The talk was received well. I was pleased to see in the front row another key member of the Bourbaki, Jean-Pierre Serre, a legend in his own right. At the end of my talk he came up to me. After asking a few pointed technical questions, he made an observation.

  “I found it interesting that you think of quantum physics as the fourth column in Weil’s Rosetta stone,” he said. “You know, André Weil wasn’t particularly fond of physics. But I think that if he were here today, he would agree that quantum physics has an important role to play in this story.”

  This was the best compliment one could possibly get.

  In the last few years, a lot of progress has been made in the Langlands Program, across all columns of Weil’s Rosetta stone. We are still far from fully understanding the deepest mysteries of the Langlands Program, but one thing is clear: it has passed the test of time. We see more clearly now that it has led us to some of the most fundamental questions in math and physics.

  These ideas are as vital today as they were when Langlands wrote his letter to André Weil, almost fifty years ago. I don’t know whether we can find all the answers in the next fifty years, but there is no doubt that the next fifty years will be at least as exciting as the past fifty years have been. And perhaps some of the readers of this book will have the opportunity to contribute to this fascinating project.

  The Langlands Program has been the focus of this book. I think it provides a good panoramic view of modern mathematics: its deep conceptual structure, groundbreaking insights, tantalizing conjectures, profound theorems, and unexpected connections between different fields. It also illustrates the intricate links between math and physics and the mutually enriching dialogue between these two subjects. Thus, the Langlands Program exemplifies the four qualities of mathematical theories that we discussed in Chapter 2: universality, objectivity, endurance, and relevance to the physical world.

  Of course, there are many other fascinating areas of math. Some have been exposed in the literature for non-specialists and some have not. As Henry David Thoreau wrote,20 “We have heard about the poetry of mathematics, but very little of it has yet been sung.” Alas, his words still ring true today, more than 150 years after he wrote them, which is to say that we, mathematicians, need to do a better job of unlocking the power and beauty of our subject to a wider audience. At the same time, I hope that the story of the Langlands Program will inspire readers’ curiosity about mathematics and motivate the desire to learn more.

  *In this regard, Hitchin quotes the great German poet Goethe: “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.”

  Chapter 18

  Searching for the Formula of Love

  In 2008, I was invited to do research and lecture about my work in Paris as the recipient of a newly created Chaire d’Excellence awarded by Fondation Sciences Mathématiques de Paris.

  Paris is one of the world’s centers of mathematics; it is also a capital of cinema. Being there, I felt inspired to make a movie about math. In popular films, mathematicians are usually portrayed as weirdos and social misfits on the verge of mental illness, reinforcing the stereotype of mathematics as a boring and cold subject, far removed from reality. Who could want such a life for themselves, doing work that supposedly had nothing to do with anything?

  When I came back to Berkeley in December 2008, I felt the urge to channel my artistic energy. My neighbor, Thomas Farber, is a wonderful author, and he teaches creative writing at UC Berkeley. I asked him, “How about writing a screenplay together about a writer and a mathematician?” Tom liked the idea and suggested that we set it on a beach in the south of France. We decided that the film would start like this: a writer and a mathematician, on a beautiful sunny day, sitting at adjacent tables in an open-air café on the beach. They savor the beauty surrounding them, look at each other, and start talking. What happens next?

  We began writing. The process was similar to the way I collaborate with mathematicians and physicists. But it was also different: finding the right words to describe the characters’ feelings and emotions, getting to the heart of a story. The framework was much more fluid and unrestricted than I had been used to. And there I was, going toe-to-toe with a great writer for whom I had so much respect and admiration. Luckily for me, Tom did not try to impose his will but treated me as an equal, gently letting me develop my abilities as a writer. Like the mentors who guided me into the world of mathematics, Tom helped me to enter the world of writing, for which I will always be grateful.

  In one of the conversations, the mathematician tells the writer about the “two-body problem.” It refers to two objects (bodies) that interact only with each other, such as a star and a planet (we ignore all other forces acting on them). There is a simple mathematical formula that accurately predicts their trajectories all the way into the future once we know the force of attraction between them. How different, however, from the interaction of two human bodies – two lovers, or two friends. Here, even if the two-body problem has a solution, it’s not unique.

  Our screenplay is about the collision between the real world and the world of abstraction: for Richard, the writer, it’s the world of literature and art; for Phillip, the mathematician, the world of science and mathematics. Each man is fluent in his respective abstract domain, but in what way does this affect behavior in the real world? Phillip is trying to come to terms with a dichotomy between mathematical truth, where he is expert, and human truth, where he is not. He learns that approaching life’s problems in the same way as mathematical problems does not always help.

  Tom and I also asked: can we see the differences and similari
ties between art and science – the “two cultures,” as C.P. Snow called them1 – through the narratives of these two men? In fact, one can read the film as metaphorical, about two sides of the same character: the left half of the brain and the right half, if you will. They are in constant competition but also inform each other – two cultures co-existing in one mind.

  In our screenplay, the characters trade stories of their past relationships, love found and lost, heartbreaks. And they meet several women in the course of the day, so we can see the two men use their passion for their professions as a means of seduction. There is a lot of mutual interest between them as well, but at the same time a conflict is brewing, which reaches an unexpected conclusion at the end.

  We called our screenplay The Two-Body Problem and published it as a book.2 Its play version has been performed in a Berkeley theater, directed by award-winning director Barbara Oliver. This was the first time I had ventured into the arts, and I was both surprised and amused by the audience reaction. For example, most people took everything that happened to the mathematician in the screenplay as my autobiography. Of course, many of my real-life experiences contributed to the writing of The Two-Body Problem. For instance, I did have a Russian girlfriend in Paris, and some of the remarkable qualities of Natalia, Phillip’s girlfriend in the screenplay, were inspired by her. Some scenes in the screenplay were drawn from my experience, some from Tom’s. But as a writer, you are most driven by the desire to create compelling characters and an engaging story. Once Tom and I decided what we wanted to convey, we had to shape the characters in a certain way. Those real-life experiences got so embellished and distorted that they were no longer ours. The protagonists of The Two-Body Problem became their own men, as they had to, to be art.

  As we began looking for a producer to help us make The Two-Body Problem into a full-length feature film, I thought it would be worthwhile to do a cinematic project on a smaller scale. When I returned to Paris to continue my Chaire d’Excellence in April 2009, a friend, mathematician Pierre Schapira, introduced me to a young, talented film director, Reine Graves. A former fashion model, she had previously directed several original, bold short films (one of which won the Pasolini Prize at the Festival of Censored Films in Paris). At a lunch meeting arranged by Pierre, she and I hit it off right away. I suggested we work together on a short film about math, and Reine liked the idea. Months later, when asked about this, she said that she felt mathematics was one of the last remaining areas where there was genuine passion.3

  As we started throwing ideas around, I showed Reine a couple of photographs I had made previously, in which I painted (digitally) tattoos of mathematical formulas on human bodies. Reine liked them, and we decided that we would try to make a film involving the tattoo of a formula.

  Tattoo, as an art form, originated in Japan. I have visited Japan a dozen times (to work with Feigin, who had been spending his summers at Kyoto University) and am fascinated by Japanese culture. Not surprisingly, Reine and I turned to the Japanese cinema for inspiration. One film was Rite of Love and Death by the great Japanese writer Yukio Mishima, based on his short story. Mishima himself directed and starred in it.

  The film is shot in black-and-white, and it unfolds on the austere, stylized stage typical of Japanese Noh theater. The film has no dialogue, but there is music from Wagner’s opera Tristan and Isolde playing in the background. There are two characters: a young officer of the Imperial Guard, Lieutenant Takeyama, and his wife, Reiko. The officer’s friends stage an unsuccessful coup d’état (here the film refers to actual events of February 1936, which Mishima thought had a dramatic effect on Japanese history). The lieutenant is given the order to execute the perpetrators of the coup, which he cannot do – they are close friends. But neither can he disobey the order of the Emperor. The only way out is ritual suicide, seppuku (or harakiri).

  Although it is only twenty-nine minutes long, the film touched me deeply. I could sense the vigor and clarity of Mishima’s vision. His presentation was forceful, raw, unapologetic. One may disagree with his ideas (and in fact his vision of the intimate link between love and death does not appeal to me), but I have a tremendous respect for the author for being so strong and uncompromising.

  Mishima’s film went against the usual conventions of cinema: it was silent, with written text between the “chapters” of the movie to explain what’s going to happen next. It was theatrical; scenes carefully staged, with little movement. But I was captivated by the undercurrent of emotion. (I did not know yet the eerie resemblance of Mishima’s own death to what happened in his film.)

  Perhaps the film resonated with me so much because Reine and I were also trying to create an unconventional film, to talk about mathematics the way no one had talked about it before. I felt that Mishima had created the aesthetic framework and language we were looking for. I called Reine.

  “I have watched Mishima’s film,” I said, “and it’s amazing. We should make a film like this.”

  “OK,” she said, “but what will it be about?”

  Suddenly, words started coming out of my mouth. Everything was crystal clear.

  “A mathematician creates a formula of love,” I said, “but then discovers the flip side of the formula: it can be used for evil as well as for good. He realizes he has to hide the formula to protect it from falling into the wrong hands. And he decides to tattoo it on the body of the woman he loves.”

  “Sounds good. What do you think we should call it?”

  “Hmmm... How about this: Rites of Love and Math.”

  And just like this, the idea of the film was born.

  We envisioned it as an allegory, showing that a mathematical formula can be beautiful, like a poem, a painting, or a piece of music. The idea was to appeal not to the cerebral but rather to the intuitive and visceral. Let the viewers first feel rather than understand it. We thought emphasizing the human and spiritual elements of mathematics would help inspire viewer’s curiosity.

  Mathematics and science in general are often presented as cold and sterile. In truth, the process of creating new mathematics is a passionate pursuit, a deeply personal experience, just like creating art and music. It requires love and dedication, a struggle with the unknown and with oneself, which elicits strong emotions. And the formulas you discover really do get under your skin, just like the tattooing in the film.

  In our film, a mathematician discovers a “formula of love.” Of course, this is a metaphor: we are always trying to achieve complete understanding, ultimate clarity, to know everything. In the real world, we have to settle for partial knowledge and understanding. But what if someone were able to find the ultimate Truth; what if it could be expressed by a mathematical formula? This would be the formula of love.

  Henry David Thoreau put this eloquently:4

  The most distinct and beautiful statement of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.

  Even if a single formula can’t be powerful enough to explain everything, mathematical formulas are some of the purest, most versatile, and most economical expressions of truth known to mankind. They convey timeless and precious knowledge, unaffected by fads and fashion, and impart the same meaning to anyone who comes in contact with them. The truths they express are the necessary truths, steadfast beacons of reality guiding humanity through time and space.

  Heinrich Hertz, who proved the existence of electromagnetic waves and whose name is now used as the unit of frequency, expressed his awe this way:5 “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.”

  Hertz is not alone in his sentiment. Most math practitioners believe that mathematical formulas and ideas inhabit a separate world. Robert Langlands writes that mathematics “often comes in the form of intimations, a word that suggests that mat
hematics, and not only its basic concepts, exists independently of us. This is a notion that is hard to credit, but hard for a professional mathematician to do without.”6 This is echoed by another eminent mathematician, Yuri Manin (the advisor of Drinfeld), who talks about a “vision of the great Castle of Mathematics, towering somewhere in the Platonic World of Ideas, which [mathematicians] humbly and devotedly discover (rather than invent).”7

  From this point of view, Galois groups were discovered by the French prodigy, not invented by him. Until he did so, this concept lived somewhere in the enchanted gardens of the ideal world of mathematics, waiting to be found. Even if Galois’ papers had been lost and he had not been given the rightfully deserved credit for his discovery, exactly the same groups would have been discovered by someone else.

  Contrast this with discoveries in other areas of human endeavor: if Steve Jobs had not come back to Apple, we may have never known iPods, iPhones, and iPads. Other technological innovations would have been made, but there is no reason to expect that the same elements would have been found by others. In contrast, mathematical truths are inevitable.

  The world inhabited by mathematical concepts and ideas is often referred to as the Platonic world of mathematics, after the Greek philosopher Plato, who was first to argue that mathematical entities are independent of our rational activities.8 In his book The Road to Reality, acclaimed mathematical physicist Roger Penrose writes that the mathematical assertions that belong to the Platonic world of mathematics “are precisely those that are objectively true. To say that some mathematical assertion has a Platonic existence is merely to say that it is true in an objective sense.” Similarly, mathematical notions “have a Platonic existence because they are objective notions.”9