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 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 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 . 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 , 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 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 and apart is the way in which Frenkel uses his personal later and . 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 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 + = 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 and people who understand some of the basics of first- . 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. He further says that b can be seen to Let us return to Frenkel’s book. Some detail about { p} Chapter 8, which is where the Langlands program arise from a two-dimensional representation of is introduced, may give the reader a feel for the the Galois group of an extension of the rationals. nature of the author’s writing style. It needs to Although harder mathematics is coming in later be said that the mathematical level of the topics chapters, this is the high point of the concrete jumps around. For example, the material on Galois mathematics in the book. groups and polynomial equations in Chapter 7 is In trying to summarize the foregoing, the author followed by half a page at the beginning of Chapter goes a little astray at this point and asserts that 8 about proofs by contradiction. Then Chapter 8 the correspondence of curves to forms such that the data sets a and b match is one-one. This states Fermat’s Last Theorem and explains briefly { p} { p} how a certain conjecture (Shimura-Taniyama-Weil) correspondence is not actually one-one, even if about cubic two-variable equations implies the we take into account isomorphisms among elliptic theorem. Frenkel does not say what cubics are curves. In fact, (1) and the curve allowed in the conjecture, but he indicates in the (2) y2 y x3 x2 10x 20 + = endnotes, without giving a definition, that the are nonisomorphic and correspond to the same allowable ones are those defining “elliptic curves.” modular form.1 This mistake is not fatal to the He soon gets concrete, working with the specific book, but it takes the reader’s focus offthe data elliptic curve sets and is a distraction. (1) y2 y x3 x2. + = L-Functions He introduces prime numbers and the finite field Traditionally the data sets are encoded into certain of integers modulo a prime p. Then he counts generating functions called L-functions, which are the number of finite solution pairs (x, y) of (1) functions of one complex variable. L-functions modulo p for some small primes p, observing that and the name for them go back at least to the number of solution pairs does not seem easily Dirichlet in the nineteenth century. Recall that predictable. He defines a by the condition that p conjectures about L-functions are at the heart the number of finite solution pairs is p a . Thus p of the Langlands program. One might think of he has attached a data set a to the elliptic curve, { p} L-functions as of two kinds, arithmetic/geometric running over the primes. Then he considers two p and analytic/automorphic. Prototypes for them examples of sequences definable in terms of a in the arithmetic/geometric case are the Artin generating function. The first, which is included L-functions of Galois representations and the just for practice with generating functions, is Hasse-Weil L-functions of elliptic curves; in the the Fibonacci sequence. The second, studied by analytic/automorphic case, prototypes are the M. Eichler in 1954, is the sequence b such that { p} Dirichlet L-functions and various L-functions of is the coefficient of p in bp q Hecke. Arithmetic/geometric L-functions contain 2 11 2 2 2 22 2 3 2 a great deal of algebraic information, much of q(1 q )(1 q ) (1 q ) (1 q ) (1 q ) it hidden, and analytic/automorphic L-functions (1 q33)2(1 q4)2(1 q44)2 . ⇥ ⇥··· tend to have nice properties. The key to unlocking Frenkel states that the sequences a and b the algebraic information is reciprocity laws, such { p} { p} match: a b for all primes p. Moreover, he says, as the Quadratic Reciprocity Law of Gauss and the p = p this is what the Shimura-Taniyama-Weil Conjecture of E. Artin, which say that says for the curve (1). There is no indication why certain arithmetic/geometric L-functions coincide 2 this equality holds, nor could there be in a book with analytic/automorphic L-functions. In some of this scope. Frenkel’s point is to get across further known cases, the above example of elliptic the beauty of the result. The seemingly random 1The two curves are isogenous but not isomorphic, accord- integers are thus seen to have a manageable ap ing to two of the lines under the heading “11” in the table pattern that one could not possibly have guessed. on page 90 of [2]. Their data sets a match by Theorem { p} The Shimura-Taniyama-Weil Conjecture is then 11.67 of [3], for example. tied to the Langlands program on pages 91–92, as 2The principle still holds if the two are equal except for an follows: Without saying much in the text about elementary factor.

1058 Notices of the AMS Volume 61, Number 9 curves being one of them, a similar reciprocity column, and the Langlands program already asso- formula holds, and equality reveals some of the ciates suitable automorphic functions to them. The hidden algebraic information. question is what to do with the third column (the For example, the Hasse-Weil L-function of an setting of Riemann surfaces). In endnote 21 for elliptic curve is a specific function of a complex Chapter 9, he gives a careful explanation of how variable s built out of the integers ap and defined the proper analog of the Galois group of a number 3 for Re s>2 . Because of the Shimura-Taniyama- field is the fundamental group of the Riemann Weil Conjecture, this L-function, call it L(s), equals surface. He can speak easily of fundamental groups a certain kind of L-function of Hecke, and such because of his detailed treatment of braid groups functions are known to extend analytically to in Chapter 5. He says that, for a proper analog of entire functions of s. Thus L(1) is well defined. The automorphic functions in the context of Riemann Birch and Swinnerton-Dyer Conjecture, a proof or surfaces, functions are inadequate. He proposes disproof of which is one of the Clay Millennium “sheaves” as a suitable generalization of functions, Prize Problems, is a precise statement of how the and “automorphic sheaves” will be the objects he behavior of L(s) near s 1 affects the nature of uses for harmonic analysis in this setting. = the rational solution pairs for the curve. Toward He does not introduce sheaves until Chapter rational settling the full conjecture, it is already 14, after spending several chapters on his further known that there are only finitely many solution personal history while touching very briefly on pairs if rational L(1) î 0 and there are infinitely mathematical notions like loop groups and Kac- many solution pairs if L(1) 0 and L0(1) î 0. = Moody algebras. A fond unstated hope of the Langlands program is that every algebraic/geometric L-function can Chapters 14–17 be seen to equal an automorphic -function except L Chapters 14–17 contain the remaining substantive possibly for an elementary factor. -functions do L comments on mathematics. They are tough slog- not appear in Love and Math. ging, and they are short on details. Chapter 14 is about sheaves. Some intuition is included, but Weil’s Rosetta Stone there is no definition in the text or the endnotes. Chapter 9 seeks to fit the Langlands program Nor did I find a single example. more fully into a framework first advanced by However, the endnotes for Chapter 14 contain André Weil, and then it looks at what is missing to a nice discussion of algebraic extensions of finite include the Langlands program in this framework.3 fields and the Frobenius element of the Galois In a 1940 letter [6] written from prison to his group of the algebraic closure, and the endnotes go sister, Weil proposed thinking about three areas of on to illustrate how to compute with the Frobenius mathematics as written on an imaginary Rosetta element. Chapter 15 is about his and Drinfeld’s stone, one column for each area. The varying efforts to merge Frenkel’s earlier work on Kac- columns represent what seems to be the same Moody algebras, which is not actually detailed in mathematics, but each is written in its own the book, with the theory of sheaves. The reader language. The three languages in Frenkel’s terms 4 5 may suppose that this merger can take place and are number fields, curves over finite fields, and 6 that the result completes the enlargement of Weil’s Riemann surfaces. Weil understood some of the entries in each column and sometimes knew how Rosetta stone. to translate part of one column into part of another. More than half of Chapter 15 is occupied with Weil sought to create a dictionary to translate each the screenplay of a conversation between Drinfeld language into the others. Frenkel wants to add and Frenkel. Part of the screenplay reads as follows: versions of the Langlands program to each column. DRINFELD writes the symbol LG on the Representations of Galois groups of number blackboard. fields fit tidily into the first column, and the Langlands program conjecturally associates suit- able automorphic functions to them. Similarly, as EDWARD Frenkel says, representations of Galois groups of Is the L for Langlands? curves over finite fields fit tidily into the second DRINFELD 3 Weil’s framework predates the Langlands program. (hint of a smile) 4Finite extensions of the rationals. 5 Finite algebraic extensions of fields F(x), where F is a finite Well, Langlands’ original motivation was to field. understand something called L-functions, 6 Finite extensions of C(x). so he called this group an L-group …

October 2014 Notices of the AMS 1059 The snide inclusion of this exchange is completely that I find merely irritating. Nevertheless, Frenkel’s uncalled for. We have seen that the term L-function book is a valiant effort at promoting widespread goes back to Dirichlet or earlier and that the name love of mathematics in a wide audience. It could L-group and the notation LG were introduced by well be that it is actually impossible to write a book Jacquet, not Langlands. Langlands himself had of the scope envisioned by Frenkel that achieves initially used the notation G to indicate what is this goal fully. If he had not been so ambitious, now known as the L-group. the result might have been better. For example, Chapters 16 and 17 are aboutb quantum duality, stopping after Chapter 9 and including a little , and superstring theory—all in an more detail might have enabled him to come closer effort to put them into the context of the Langlands to achieving what he wanted. program. In a sense this is ongoing research. To Three stars out of five. an extent it is also theoretical physics, which has to mesh eventually with the real universe in order Acknowledgment to be acceptable. Langlands put comments about I am grateful for helpful conversations with Robert this research on his website at [5] on April 6, 2014. Langlands and Martin Krieger. He said of Frenkel’s articles in this direction, These articles are impressive achievements References but often freewheeling, so that, although I [1] A. Borel, Formes automorphes et séries de have studied them with considerable care Dirichlet (d’apre`s Langlands), Séminaire Bour- and learned a great deal from them that baki, Exp. 466 (174/75), Lect. Notes in Math. 514 (1976), 183–222; Oeuvres, III, 374–398. I might never have learned from other [2] J. W. Cremona, Algorithms for Modular Elliptic Curves, sources, I find them in a number of respects Cambridge University Press, 1992. incomplete or unsatisfactory. … [3] A. W. Knapp, Elliptic Curves, Press, It [a certain duality that is involved] has 1992. [4] R. P. Langlands, “Functoriality,” http:// to be judged by different criteria. One is publications.ias.edu/rpl/section/21. The “edi- whether it is physically relevant. There torial comments” and “author’s comments” describe is, I believe a good deal of scepticism, early history of the Langlands program and the reasons which, if I am to believe my informants, for its introduction. Various papers of Langlands are is experimentally well founded. Although accessible from this Web location. the notions of functoriality and reciprocity [5] , “Beyond endoscopy,” http://publications. have, on the whole, been well received by ias.edu/rpl/section/25. Within the essay, see the section called “Message to ” and especially mathematicians, they have had to surmount the subsection “Author’s comments (Apr. 6, 2014).” some entrenched resistance, perhaps still [6] A. Weil, “Une lettre et un extrait de lettre à Simone latent. So I, at least, am uneasy about Weil,” March 26, 1940, Oeuvres Scientifiques, I, 244–255; associating them with vulnerable physical English translation, M. H. Krieger, Notices Amer. Math. notions. … Soc. 52 (2005), 334–341.

Chapter 18 Chapter 18, the last chapter, is completely different from the others. It is unrelated to the theme of the book and simply does not fit. The most charitable explanation that comes to mind for what happened is that inclusion of the chapter was a marketing decision. In any case, including it was a mistake, and I choose to disregard this chapter.

Summary Thus much of the book is a personal history about the author. This portion is well written and entertaining. Chapters 7–9 present some mathematics that is at once deep and beautiful, and they do so in a way that largely can be appreciated by many readers. The later chapters have nuggets of mathematics that are well done, but not enough to keep the attention of most readers. Most of those later chapters come dangerously close to the content-empty popular physics books

1060 Notices of the AMS Volume 61, Number 9