Book Review

Love and Math: TheHeart of Hidden RealityReviewed by Anthony W. Knapp

Love and Math: The Heart of Hidden RealityEdward FrenkelBasic Books, 2013292 pages, US$27.99ISBN-13: 978-0-465-05074-1

Edward Frenkel is professor of mathematics atBerkeley, the 2012 AMS Colloquium Lecturer, anda 1989 émigré from the former Soviet Union.He is also the protagonist Edik in the splendidNovember 1999 Notices article by Mark Saul entitled“Kerosinka: An Episode in the History of SovietMathematics.” Frenkel’s book intends to teachappreciation of portions of mathematics to ageneral audience, and the unifying theme of histopics is his own mathematical education.

Except for the last of the 18 chapters, a moreaccurate title for the book would be “Love of Math.”The last chapter is more about love than math, andwe discuss it separately later in this review.

Raoul Bott once gave a lecture called “Sex andPartial Differential Equations.” Come for the sex.Stay for the partial differential equations. The title“Love and Math” is the same idea.

Much of the book is a narrative about Frenkel’sown personal experience. If this were all therewere to the book, it would be nice, but it might notjustify a review in the Notices. What sets this bookapart is the way in which Frenkel uses his personalstory to encourage the reader to see the beautyof some of the mathematics that he has learned.In a seven-page preface, Frenkel says “This bookis an invitation to this rich and dazzling world. Iwrote it for readers without any background inmathematics.” He concludes the preface with thissentence:

Anthony W. Knapp is professor emeritus at Stony Brook Uni-versity, State University of New York. His email address isaknapp@math.sunysb.edu.

DOI: http://dx.doi.org/10.1090/noti1152

My dream is that all of us will be able tosee, appreciate, and marvel at the magicbeauty and exquisite harmony of theseideas, formulas, and equations, for this willgive so much more meaning to our love forthis world and for each other.

Frenkel’s Personal StoryFrenkel is a skilled storyteller, and his accountof his own experience in the Soviet Union, wherehe was labeled as of “Jewish nationality” andconsequently made to suffer, is gripping. It keepsone’s attention and it keeps one wanting to readmore. After his failed experience trying to beadmitted to Moscow State University, he went toKerosinka. There he read extensively, learned fromfirst-rate teachers, did mathematics research at ahigh level, and managed to get some of his worksmuggled outside the Soviet Union. The result wasthat he finished undergraduate work at Kerosinkaand was straightway offered a visiting position atHarvard. He tells this story in such an engagingway that one is always rooting for his success. Heshows a reverence for various giants who were inthe Soviet Union at the time, including I. M. Gelfandand D. B. Fuchs. The account of how he obtained anexit visa is particularly compelling. Once he is in theUnited States, readers get to see his awe at meetingmathematical giants such as Vladimir Drinfeld andlater Edward Witten and Robert Langlands. Thereader gets to witness in Chapter 14 the 1990unmasking of the rector (president) of MoscowUniversity, who had just given a public lecture inMassachusetts and whitewashed that university’sdiscriminatory admissions policies. From thereone gets to follow Frenkel’s progress throughmore recent joint work with Witten and up to acollaboration with Langlands and Ngô Bao Châu.

I have to admit that I was dubious when I readthe publisher’s advertising about the educationalaspect of the book. I have seen many lectures and

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books by people with physics backgrounds thatcontain no mathematics at all—no formulas, noequations, not even any precise statements. Suchbooks tend to suggest that modern physics is reallyjust one great thought experiment, an extension ofEinstein’s way of thinking about special relativity.Love and Math at first sounded to me exactly likethat kind of book. I was relieved when I opened Loveand Math and found that Frenkel was trying not totreat his subject matter this way. He has equationsand other mathematical displays, and when hisdescriptions are more qualitative than that, thosedescriptions usually are still concrete. The firstequation concerns clock arithmetic and appearson page 18, and there are many more equationsand mathematical displays starting in Chapter 6.As a kind of compensation for formula-aversereaders, he includes a great many pictures anddiagrams and tells the reader to “feel free to skip[any formulas] if so desired.”

AudienceThe equations being as they are, to say thathis audience is everyone is an exaggeration. Myexperience is that the average person in the UnitedStates is well below competency at traditional first-year high-school algebra, even though that personmay once have had to pass such a course to get ahigh-school diploma. For one example, I remembera botched discussion on a local television newscastof the meaning of the equation xn + yn = zn afterAndrew Wiles announced his breakthrough onFermat’s Last Theorem. As a book that does morethan tell Frenkel’s own personal story, Love andMath is not for someone whose mathematicalability is at the level of those newscasters.

Frenkel really aims at two audiences, one widerthan the other. The wider audience consists ofpeople who understand some of the basics of first-year high-school algebra. The narrower audienceconsists of people with more facility at algebrawho are willing to consider a certain amount ofabstraction. To write for both audiences at thesame time, Frenkel uses the device of lengthyendnotes, encouraging only the interested readersto look at the endnotes. Perhaps he should alsohave advised the reader that the same zippy paceone might use for reading the narrative parts ofthe book is not always appropriate for reading themathematical parts. Anyway, the endnotes occupy35 pages at the end of the book.

Initial MathematicsThe mathematics begins gently enough with adiscussion of symmetry and finite-dimensionalrepresentations in Chapter 2. No more mathematicsreally occurs until Chapter 5, when braid groupsare introduced with some degree of detail. Chapter

6 touches on mathematics by alluding to Bettinumbers and spectral sequences without reallydiscussing them. It also gives the Fermat equationxn + yn = zn but does not dwell on it.

Chapters 7–9 contain some serious quantitativemathematics, and then there is a break for somenarrative. The mathematics resumes in Chapter14. The topics in Chapters 7–9 quickly advancein level. The mathematical goals of the chaptersare respectively to introduce Galois groups and tosay something precise yet introductory about theLanglands Program as it was originally conceived[4]. Giving some details but not all, Chapter 7speaks of number systems—the positive integers,the integers, the rationals, certain algebraic exten-sions of the rationals, Galois groups, and solutionsof polynomial equations by radicals. Frenkel con-cludes by saying that the Langlands program “tiestogether the theory of Galois groups and anotherarea of mathematics called harmonic analysis.”

Nature of the Langlands ProgramIt is time in this review to interpolate some remarksabout the nature of the Langlands program. Theterm “Langlands program” had one meaning untilroughly 1979 and acquired a much enlargedmeaning after that date. The term came into useabout the time of A. Borel’s Séminaire Bourbakitalk [1] in 1974/75. In the introduction Borel wrote(my translation):

This lecture tries to give a glimpse of the setof results, problems and conjectures thatestablish, whether actually or conjecturally,some strong ties between automorphicforms on reductive groups, or representa-tions of such groups, and a general class ofEuler products containing many of thosethat one encounters in number theory andalgebraic geometry.

At the present time, several of theseconjectures or “questions” appear quiteinaccessible in their general form. Ratherthey define a vast program, elaborated byR. P. Langlands since about 1967, oftencalled the “Langlands philosophy” and al-ready illustrated in a very striking wayby the classical or recent results that arebehind it, and those that have been obtainedsince. …

Finally sections 7 and 8 are devoted to thegeneral case. The essential new point is theintroduction, by Langlands [in three citedpapers] of a group associated to a connectedreductive group over a local field, on whichare defined L factors of representations ofG; also, following a suggestion of H. Jacquet,we shall call it the L-group of G and denoteit LG. …

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After 1979 Langlands and others worked onrelated matters that expanded the scope of the term“Langlands program.” More detail about the periodbefore 1979 and the reasons for the investigationappear in Langlands’s Web pages, particularly [4].

Nature of Author’s Expository StyleLet us return to Frenkel’s book. Some detail aboutChapter 8, which is where the Langlands programis introduced, may give the reader a feel for thenature of the author’s writing style. It needs tobe said that the mathematical level of the topicsjumps around. For example, the material on Galoisgroups and polynomial equations in Chapter 7 isfollowed by half a page at the beginning of Chapter8 about proofs by contradiction. Then Chapter 8states Fermat’s Last Theorem and explains brieflyhow a certain conjecture (Shimura-Taniyama-Weil)about cubic two-variable equations implies thetheorem. Frenkel does not say what cubics areallowed in the conjecture, but he indicates in theendnotes, without giving a definition, that theallowable ones are those defining “elliptic curves.”He soon gets concrete, working with the specificelliptic curve

(1) y2 + y = x3 − x2.

He introduces prime numbers and the finite fieldof integers modulo a prime p. Then he countsthe number of finite solution pairs (x, y) of (1)modulo p for some small primes p, observing thatthe number of solution pairs does not seem easilypredictable. He defines ap by the condition thatthe number of finite solution pairs is p − ap. Thushe has attached a data set {ap} to the elliptic curve,p running over the primes. Then he considers twoexamples of sequences definable in terms of agenerating function. The first, which is includedjust for practice with generating functions, isthe Fibonacci sequence. The second, studied byM. Eichler in 1954, is the sequence {bp} such thatbp is the coefficient of qp in

q(1− q2)(1− q11)2(1− q2)2(1− q22)2(1− q3)2

× (1− q33)2(1− q4)2(1− q44)2 × · · · .Frenkel states that the sequences {ap} and {bp}match: ap = bp for all primes p. Moreover, he says,this is what the Shimura-Taniyama-Weil Conjecturesays for the curve (1). There is no indication whythis equality holds, nor could there be in a bookof this scope. Frenkel’s point is to get acrossthe beauty of the result. The seemingly randomintegers ap are thus seen to have a manageablepattern that one could not possibly have guessed.

The Shimura-Taniyama-Weil Conjecture is thentied to the Langlands program on pages 91–92, asfollows: Without saying much in the text about

what modular forms are, Frenkel explains how theEichler sequence can be interpreted in terms ofmodular forms of a certain kind. The endnotescome close to explaining this statement completely.In addition, he says also that the statement of theShimura-Taniyama-Weil Conjecture is that one canfind a modular form of this kind for any ellipticcurve. He further says that {bp} can be seen toarise from a two-dimensional representation ofthe Galois group of an extension of the rationals.Although harder mathematics is coming in laterchapters, this is the high point of the concretemathematics in the book.

In trying to summarize the foregoing, the authorgoes a little astray at this point and asserts thatthe correspondence of curves to forms such thatthe data sets {ap} and {bp}match is one-one. Thiscorrespondence is not actually one-one, even ifwe take into account isomorphisms among ellipticcurves. In fact, (1) and the curve

(2) y2 + y = x3 − x2 − 10x− 20

are nonisomorphic and correspond to the samemodular form.1 This mistake is not fatal to thebook, but it takes the reader’s focus off the datasets and is a distraction.

L-FunctionsTraditionally the data sets are encoded into certaingenerating functions called L-functions, which arefunctions of one complex variable. L-functionsand the name for them go back at least toDirichlet in the nineteenth century. Recall thatconjectures about L-functions are at the heartof the Langlands program. One might think ofL-functions as of two kinds, arithmetic/geometricand analytic/automorphic. Prototypes for themin the arithmetic/geometric case are the ArtinL-functions of Galois representations and theHasse-Weil L-functions of elliptic curves; in theanalytic/automorphic case, prototypes are theDirichlet L-functions and various L-functions ofHecke. Arithmetic/geometric L-functions containa great deal of algebraic information, much ofit hidden, and analytic/automorphic L-functionstend to have nice properties. The key to unlockingthe algebraic information is reciprocity laws, suchas the Quadratic Reciprocity Law of Gauss and theArtin Reciprocity Law of E. Artin, which say thatcertain arithmetic/geometric L-functions coincidewith analytic/automorphic L-functions.2 In somefurther known cases, the above example of elliptic

1The two curves are isogenous but not isomorphic, accord-ing to two of the lines under the heading “11” in the tableon page 90 of [2]. Their data sets {ap}match by Theorem11.67 of [3], for example.2The principle still holds if the two are equal except for anelementary factor.

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curves being one of them, a similar reciprocityformula holds, and equality reveals some of thehidden algebraic information.

For example, the Hasse-Weil L-function of anelliptic curve is a specific function of a complexvariable s built out of the integers ap and definedfor Re s > 3

2 . Because of the Shimura-Taniyama-Weil Conjecture, this L-function, call it L(s), equalsa certain kind of L-function of Hecke, and suchfunctions are known to extend analytically toentire functions of s. Thus L(1) is well defined. TheBirch and Swinnerton-Dyer Conjecture, a proof ordisproof of which is one of the Clay MillenniumPrize Problems, is a precise statement of how thebehavior of L(s) near s = 1 affects the nature ofthe rational solution pairs for the curve. Towardrational settling the full conjecture, it is alreadyknown that there are only finitely many solutionpairs if rational L(1) ≠ 0 and there are infinitelymany solution pairs if L(1) = 0 and L′(1) ≠ 0.

A fond unstated hope of the Langlands programis that every algebraic/geometric L-function canbe seen to equal an automorphic L-function exceptpossibly for an elementary factor. L-functions donot appear in Love and Math.

Weil’s Rosetta StoneChapter 9 seeks to fit the Langlands programmore fully into a framework first advanced byAndré Weil, and then it looks at what is missing toinclude the Langlands program in this framework.3

In a 1940 letter [6] written from prison to hissister, Weil proposed thinking about three areas ofmathematics as written on an imaginary Rosettastone, one column for each area. The varyingcolumns represent what seems to be the samemathematics, but each is written in its ownlanguage. The three languages in Frenkel’s termsare number fields,4 curves over finite fields,5 andRiemann surfaces.6 Weil understood some of theentries in each column and sometimes knew howto translate part of one column into part of another.Weil sought to create a dictionary to translate eachlanguage into the others. Frenkel wants to addversions of the Langlands program to each column.

Representations of Galois groups of numberfields fit tidily into the first column, and theLanglands program conjecturally associates suit-able automorphic functions to them. Similarly, asFrenkel says, representations of Galois groups ofcurves over finite fields fit tidily into the second

3Weil’s framework predates the Langlands program.4Finite extensions of the rationals.5Finite algebraic extensions of fields F(x), where F is a finitefield.6Finite extensions of C(x).

column, and the Langlands program already asso-ciates suitable automorphic functions to them. Thequestion is what to do with the third column (thesetting of Riemann surfaces). In endnote 21 forChapter 9, he gives a careful explanation of howthe proper analog of the Galois group of a numberfield is the fundamental group of the Riemannsurface. He can speak easily of fundamental groupsbecause of his detailed treatment of braid groupsin Chapter 5. He says that, for a proper analog ofautomorphic functions in the context of Riemannsurfaces, functions are inadequate. He proposes“sheaves” as a suitable generalization of functions,and “automorphic sheaves” will be the objects heuses for harmonic analysis in this setting.

He does not introduce sheaves until Chapter14, after spending several chapters on his furtherpersonal history while touching very briefly onmathematical notions like loop groups and Kac-Moody algebras.

Chapters 14–17Chapters 14–17 contain the remaining substantivecomments on mathematics. They are tough slog-ging, and they are short on details. Chapter 14is about sheaves. Some intuition is included, butthere is no definition in the text or the endnotes.Nor did I find a single example.

However, the endnotes for Chapter 14 containa nice discussion of algebraic extensions of finitefields and the Frobenius element of the Galoisgroup of the algebraic closure, and the endnotes goon to illustrate how to compute with the Frobeniuselement. Chapter 15 is about his and Drinfeld’sefforts to merge Frenkel’s earlier work on Kac-Moody algebras, which is not actually detailed inthe book, with the theory of sheaves. The readermay suppose that this merger can take place andthat the result completes the enlargement of Weil’sRosetta stone.

More than half of Chapter 15 is occupied withthe screenplay of a conversation between Drinfeldand Frenkel. Part of the screenplay reads as follows:

DRINFELD writes the symbol LG on theblackboard.

EDWARD

Is the L for Langlands?

DRINFELD

(hint of a smile)

Well, Langlands’ original motivation was tounderstand something called L-functions,so he called this group an L-group …

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The snide inclusion of this exchange is completelyuncalled for. We have seen that the term L-functiongoes back to Dirichlet or earlier and that the nameL-group and the notation LG were introduced byJacquet, not Langlands. Langlands himself hadinitially used the notation G to indicate what isnow known as the L-group.

Chapters 16 and 17 are about quantum duality,string theory, and superstring theory—all in aneffort to put them into the context of the Langlandsprogram. In a sense this is ongoing research. Toan extent it is also theoretical physics, which hasto mesh eventually with the real universe in orderto be acceptable. Langlands put comments aboutthis research on his website at [5] on April 6, 2014.He said of Frenkel’s articles in this direction,

These articles are impressive achievementsbut often freewheeling, so that, although Ihave studied them with considerable careand learned a great deal from them thatI might never have learned from othersources, I find them in a number of respectsincomplete or unsatisfactory. …

It [a certain duality that is involved] hasto be judged by different criteria. One iswhether it is physically relevant. Thereis, I believe a good deal of scepticism,which, if I am to believe my informants,is experimentally well founded. Althoughthe notions of functoriality and reciprocityhave, on the whole, been well received bymathematicians, they have had to surmountsome entrenched resistance, perhaps stilllatent. So I, at least, am uneasy aboutassociating them with vulnerable physicalnotions. …

Chapter 18Chapter 18, the last chapter, is completely differentfrom the others. It is unrelated to the theme of thebook and simply does not fit. The most charitableexplanation that comes to mind for what happenedis that inclusion of the chapter was a marketingdecision. In any case, including it was a mistake,and I choose to disregard this chapter.

SummaryThus much of the book is a personal historyabout the author. This portion is well writtenand entertaining. Chapters 7–9 present somemathematics that is at once deep and beautiful,and they do so in a way that largely can beappreciated by many readers. The later chaptershave nuggets of mathematics that are well done, butnot enough to keep the attention of most readers.Most of those later chapters come dangerouslyclose to the content-empty popular physics books

that I find merely irritating. Nevertheless, Frenkel’sbook is a valiant effort at promoting widespreadlove of mathematics in a wide audience. It couldwell be that it is actually impossible to write a bookof the scope envisioned by Frenkel that achievesthis goal fully. If he had not been so ambitious,the result might have been better. For example,stopping after Chapter 9 and including a littlemore detail might have enabled him to come closerto achieving what he wanted.

Three stars out of five.

AcknowledgmentI am grateful for helpful conversations with RobertLanglands and Martin Krieger.

References[1] A. Borel, Formes automorphes et séries de

Dirichlet (d’apres Langlands), Séminaire Bour-baki, Exp. 466 (174/75), Lect. Notes in Math. 514(1976), 183–222; Oeuvres, III, 374–398.

[2] J. W. Cremona, Algorithms for Modular Elliptic Curves,Cambridge University Press, 1992.

[3] A. W. Knapp, Elliptic Curves, Princeton University Press,1992.

[4] R. P. Langlands, “Functoriality,” http://publications.ias.edu/rpl/section/21. The “edi-torial comments” and “author’s comments” describeearly history of the Langlands program and the reasonsfor its introduction. Various papers of Langlands areaccessible from this Web location.

[5] , “Beyond endoscopy,” http://publications.ias.edu/rpl/section/25. Within the essay, see thesection called “Message to Peter Sarnak” and especiallythe subsection “Author’s comments (Apr. 6, 2014).”

[6] A. Weil, “Une lettre et un extrait de lettre à SimoneWeil,” March 26, 1940, Oeuvres Scientifiques, I, 244–255;English translation, M. H. Krieger, Notices Amer. Math.Soc. 52 (2005), 334–341.

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