Book Review Love and Math: The Heart of Hidden Reality Reviewed by Anthony W. Knapp My dream is that all of us will be able to Love and Math: The Heart of Hidden Reality see, appreciate, and marvel at the magic Edward Frenkel beauty and exquisite harmony of these Basic Books, 2013 ideas, formulas, and equations, for this will 292 pages, US$27.99 give so much more meaning to our love for ISBN-13: 978-0-465-05074-1 this world and for each other. Edward Frenkel is professor of mathematics at Frenkel’s Personal Story Berkeley, the 2012 AMS Colloquium Lecturer, and Frenkel is a skilled storyteller, and his account a 1989 émigré from the former Soviet Union. of his own experience in the Soviet Union, where He is also the protagonist Edik in the splendid he was labeled as of “Jewish nationality” and November 1999 Notices article by Mark Saul entitled consequently made to suffer, is gripping. It keeps “Kerosinka: An Episode in the History of Soviet one’s attention and it keeps one wanting to read Mathematics.” Frenkel’s book intends to teach more. After his failed experience trying to be appreciation of portions of mathematics to a admitted to Moscow State University, he went to general audience, and the unifying theme of his Kerosinka. There he read extensively, learned from topics is his own mathematical education. first-rate teachers, did mathematics research at a Except for the last of the 18 chapters, a more high level, and managed to get some of his work accurate title for the book would be “Love of Math.” smuggled outside the Soviet Union. The result was The last chapter is more about love than math, and that he finished undergraduate work at Kerosinka we discuss it separately later in this review. and was straightway offered a visiting position at Raoul Bott once gave a lecture called “Sex and Harvard. He tells this story in such an engaging Partial Differential Equations.” Come for the sex. way that one is always rooting for his success. He Stay for the partial differential equations. The title shows a reverence for various giants who were in “Love and Math” is the same idea. the Soviet Union at the time, including I. M. Gelfand Much of the book is a narrative about Frenkel’s and D. B. Fuchs. The account of how he obtained an own personal experience. If this were all there exit visa is particularly compelling. Once he is in the were to the book, it would be nice, but it might not United States, readers get to see his awe at meeting justify a review in the Notices. What sets this book mathematical giants such as Vladimir Drinfeld and apart is the way in which Frenkel uses his personal later Edward Witten and Robert Langlands. The story to encourage the reader to see the beauty reader gets to witness in Chapter 14 the 1990 of some of the mathematics that he has learned. unmasking of the rector (president) of Moscow In a seven-page preface, Frenkel says “This book University, who had just given a public lecture in is an invitation to this rich and dazzling world. I Massachusetts and whitewashed that university’s wrote it for readers without any background in discriminatory admissions policies. From there mathematics.” He concludes the preface with this one gets to follow Frenkel’s progress through sentence: more recent joint work with Witten and up to a Anthony W. Knapp is professor emeritus at Stony Brook Uni- collaboration with Langlands and Ngô Bao Châu. versity, State University of New York. His email address is I have to admit that I was dubious when I read [email protected]. the publisher’s advertising about the educational DOI: http://dx.doi.org/10.1090/noti1152 aspect of the book. I have seen many lectures and 1056 Notices of the AMS Volume 61, Number 9 books by people with physics backgrounds that 6 touches on mathematics by alluding to Betti contain no mathematics at all—no formulas, no numbers and spectral sequences without really equations, not even any precise statements. Such discussing them. It also gives the Fermat equation books tend to suggest that modern physics is really xn yn zn but does not dwell on it. + = just one great thought experiment, an extension of Chapters 7–9 contain some serious quantitative Einstein’s way of thinking about special relativity. mathematics, and then there is a break for some Love and Math at first sounded to me exactly like narrative. The mathematics resumes in Chapter that kind of book. I was relieved when I opened Love 14. The topics in Chapters 7–9 quickly advance and Math and found that Frenkel was trying not to in level. The mathematical goals of the chapters treat his subject matter this way. He has equations are respectively to introduce Galois groups and to and other mathematical displays, and when his say something precise yet introductory about the descriptions are more qualitative than that, those Langlands Program as it was originally conceived descriptions usually are still concrete. The first [4]. Giving some details but not all, Chapter 7 equation concerns clock arithmetic and appears speaks of number systems—the positive integers, on page 18, and there are many more equations the integers, the rationals, certain algebraic exten- and mathematical displays starting in Chapter 6. sions of the rationals, Galois groups, and solutions As a kind of compensation for formula-averse of polynomial equations by radicals. Frenkel con- readers, he includes a great many pictures and cludes by saying that the Langlands program “ties diagrams and tells the reader to “feel free to skip together the theory of Galois groups and another [any formulas] if so desired.” area of mathematics called harmonic analysis.” Audience Nature of the Langlands Program The equations being as they are, to say that It is time in this review to interpolate some remarks his audience is everyone is an exaggeration. My about the nature of the Langlands program. The experience is that the average person in the United term “Langlands program” had one meaning until States is well below competency at traditional first- roughly 1979 and acquired a much enlarged year high-school algebra, even though that person meaning after that date. The term came into use may once have had to pass such a course to get a about the time of A. Borel’s Séminaire Bourbaki high-school diploma. For one example, I remember talk [1] in 1974/75. In the introduction Borel wrote a botched discussion on a local television newscast (my translation): of the meaning of the equation xn yn zn after This lecture tries to give a glimpse of the set + = Andrew Wiles announced his breakthrough on of results, problems and conjectures that Fermat’s Last Theorem. As a book that does more establish, whether actually or conjecturally, than tell Frenkel’s own personal story, Love and some strong ties between automorphic Math is not for someone whose mathematical forms on reductive groups, or representa- ability is at the level of those newscasters. tions of such groups, and a general class of Frenkel really aims at two audiences, one wider Euler products containing many of those than the other. The wider audience consists of that one encounters in number theory and people who understand some of the basics of first- algebraic geometry. year high-school algebra. The narrower audience At the present time, several of these consists of people with more facility at algebra conjectures or “questions” appear quite who are willing to consider a certain amount of inaccessible in their general form. Rather abstraction. To write for both audiences at the they define a vast program, elaborated by same time, Frenkel uses the device of lengthy R. P. Langlands since about 1967, often endnotes, encouraging only the interested readers called the “Langlands philosophy” and al- to look at the endnotes. Perhaps he should also ready illustrated in a very striking way have advised the reader that the same zippy pace by the classical or recent results that are one might use for reading the narrative parts of behind it, and those that have been obtained the book is not always appropriate for reading the since. … mathematical parts. Anyway, the endnotes occupy Finally sections 7 and 8 are devoted to the 35 pages at the end of the book. general case. The essential new point is the introduction, by Langlands [in three cited Initial Mathematics papers] of a group associated to a connected The mathematics begins gently enough with a reductive group over a local field, on which discussion of symmetry and finite-dimensional are defined L factors of representations of representations in Chapter 2. No more mathematics G; also, following a suggestion of H. Jacquet, really occurs until Chapter 5, when braid groups we shall call it the L-group of G and denote are introduced with some degree of detail. Chapter it LG.… October 2014 Notices of the AMS 1057 After 1979 Langlands and others worked on what modular forms are, Frenkel explains how the related matters that expanded the scope of the term Eichler sequence can be interpreted in terms of “Langlands program.” More detail about the period modular forms of a certain kind. The endnotes before 1979 and the reasons for the investigation come close to explaining this statement completely. appear in Langlands’s Web pages, particularly [4]. In addition, he says also that the statement of the Shimura-Taniyama-Weil Conjecture is that one can Nature of Author’s Expository Style find a modular form of this kind for any elliptic curve.