israel moiseevich gelfand, part i

Israel MoiseevichGelfand, Part IVladimir Retakh, Coordinating Editor

I. M. Gelfand

Israel Moiseevich Gelfand, amathematician compared byHenri Cartan to Poincaré andHilbert, was born on Septem-ber 2, 1913, in the small townof Okny (later Red Okny) nearOdessa in the Ukraine and diedin New Brunswick, New Jersey,USA, on October 5, 2009.

Nobody guided Gelfand inhis studies. He attended theonly school in town, and hismathematics teacher couldoffer him nothing exceptencouragement—and this wasvery important. In Gelfand’s

own words: “Offering encouragement is a teacher’smost important job.” In 1923 the family movedto another place and Gelfand entered a vocationalschool for chemistry lab technicians. However,he was expelled in the ninth grade as a son ofa “bourgeois element” (“netrudovoi element” inSoviet parlance)—his father was a mill manager.After that Gelfand (he was sixteen and a half atthat time) decided to go to Moscow, where he hadsome distant relatives.

Until his move to Moscow in 1930, Gelfand livedin total mathematical isolation. The only booksavailable to him were secondary school texts andseveral community college textbooks. The most ad-vanced of these books claimed that there are threekinds of functions: analytical, defined by formulas;empirical, defined by tables; and correlational. LikeRamanujan, he was experimenting a lot. Around

Vladimir Retakh is professor of mathematics at Rutgers Uni-versity. His email address is [email protected].

He thanks Mark Saul for his help in the preparation of thiscollection.

The AMS wants to thank M. E. Bronstein, Elena Ermakova,Tatiana I. Gelfand, Tatiana V. Gelfand, and Carol Tate fortheir help with photographs for this article.

DOI: http://dx.doi.org/10.1090/noti937

age twelve Gelfand understood that some problemsin geometry cannot be solved algebraically anddrew a table of ratios of the length of the chord tothe length of the arc. Much later it became clearfor him that in fact he was drawing trigonometrictables.

From this period came his Mozartean styleand his belief in the unity and harmony ofmathematics (including applied mathematics)—the unity determined not by rigid and loudlyproclaimed programs, but rather by invisibleand sometimes hidden ties connecting seeminglydifferent areas. Gelfand described his school yearsand mathematical studies in an interview publishedin Quantum, a science magazine for high schoolstudents [1]. In Gelfand’s own words: “It is my deepconviction that mathematical ability in most futureprofessional mathematicians appears…when theyare 13 to 16 years old…. This period formed mystyle of doing mathematics. I studied differentsubjects, but the artistic form of mathematics thattook root at this time became the basis of my tastein choosing problems that continue to attract meto this day. Without understanding this motivation,I think it is impossible to make heads or tails ofthe seeming illogicality of my ways in workingand the choice of themes in my work. Because ofthis motivating force, however, they actually cometogether sequentially and logically.”

The interview also shows how a small provincialboy was jumping over centuries in his mathematicaldiscoveries. At the age of fifteen Gelfand learnedof a series for calculating the sine. He describedthis moment in the “Quantum” interview: “Beforethis I thought there were two types of mathematics,algebraic and geometric…When I discovered thatthe sine can be expressed algebraically as a series,the barriers came tumbling down, and mathematicsbecame one. To this day I see various branches ofmathematics, together with mathematical physics,as parts of a united whole.”

After arriving in Moscow, Gelfand did nothave steady work and lived on earnings from

24 Notices of the AMS Volume 60, Number 1

occasional odd jobs. At some point he had thegood fortune to work at the checkout counter atthe State (Lenin’s) Library. This gave Gelfand a rareopportunity to talk with mathematics studentsfrom Moscow University. He also started to attenduniversity seminars, where he found himself underintense psychological stress: new breezes wereblowing in mathematics with the new demandsfor rigorous proofs. It was so different from his“homemade” experiments and his romantic viewsof mathematics. He also learned that none ofhis discoveries were new. Neither this nor othercircumstances of his life deterred him, and hisinterest in mathematics continued to grow.

Just as his abrupt expulsion from school, thenext twist in Gelfand’s fate was also one of manyparadoxes of life in the Soviet Union. On one hand,as a son of a “bourgeois element” he could not be auniversity student. On the other hand, at eighteenhe was able to obtain a teaching position at oneof many newly created technical colleges and atnineteen to enter the Ph.D. program at MoscowUniversity. The reasons were simple: the Sovietstate needed knowledgeable instructors to educateits future engineers and scientists of the proper“proletarian origin”. But at that time the systemwas not rigid enough to purge or even strictlyregulate graduate schools. As a result, a talentedboy was able to enter a Ph.D. program without acollege or even a high school diploma.

At the beginning of his career Gelfand wasinfluenced by several Moscow mathematicians,especially his thesis adviser, A. N. Kolmogorov.In the Quantum interview Gelfand said that fromKolmogorov he learned “that a true mathematicianmust be a philosopher of nature.” Another influencewas the brilliant L. G. Shnirelman. In 1935 Gelfanddefended his “candidate” (Ph.D.) thesis and in 1940obtained the higher degree of Doctor of Science.In 1933 he began teaching at Moscow University,where he became a full professor in 1943 andstarted his influential seminar. He lost this positiontemporarily in 1952 during the infamous “anti-cosmopolitans” (in fact, anti-Semitic) campaign butwas allowed to continue the seminar. Gelfand alsoworked at the Steklov Institute and for many yearsat the Institute for Applied Mathematics. Therehe took part in the secret program related to theSoviet version of the Manhattan Project and itsextensions. Andrei Sakharov mentioned his workwith Gelfand in [1].

In 1953 Gelfand was elected a CorrespondingMember of the Academy of Science (an importanttitle in the Soviet hierarchy). This happened rightafter Stalin’s death and at the end of the anti-cosmopolitans campaign. According to Gelfand,the political uncertainty of the times made hiselection possible. Later the situation in the USSR

Gelfand’s parents.

stabilized, anti-Semitism be-came part of the Sovietsystem, and Gelfand be-came a full member ofthe Soviet Academy only in1984 after being elected toleading foreign academies.

In 1989 Gelfand movedto the United States. Af-ter spending some timeat Harvard and MIT, hebecame a professor at Rut-gers University, where heworked until his death.

The articles in the Notices

Gelfand at age 3, 1916.

present various descriptionsof Gelfand’s multifaceted re-search, his way of doingmathematics, and interactionwith people. A leading Amer-ican expert once told me thatafter reading all definitionsin Gelfand’s papers, he couldeasily prove all his theorems.Well, this easiness was basedon long computations and athorough consideration of avariety of carefully selectedexamples. Gelfand himselfliked to repeat a statementby a Moscow mathematician:“Gelfand cannot prove hardtheorems. He just turns anytheorem into an easy one.”

Gelfand always was surrounded by numerouscollaborators attracted by his legendary intuitionand the permanent flow of new ideas: in everydecade he was establishing a new area of research.He had completely different approaches to differentpeople, as described by A. Vershik, A. Zelevinsky,and me in our recollections. His students andcollaborators represent an unusual variety of stylesand interests. The only similarity for members ofthe Gelfand school was their inherited passion formathematics.

One cannot write about Gelfand without men-tioning his legendary seminar, which started in1943 and continued until his death. Gelfand con-sidered the seminar one of his most importantcreations. It is hard to describe the seminar in a fewwords: it was a “mathematical stock exchange”, abreeding ground for young scientists, a demonstra-tion of how to think about mathematics, a one-manshow, and much more. It was not about functionalanalysis or geometry, it was about mathematics.Some would come to the seminar just to hearGelfand’s jokes and paradoxes. For example, afterhis visit to the U.S. he stated, “[The] Mathematical

January 2013 Notices of the AMS 25

Gelfand in 1934.

world is not a metric space:The distance from Harvard toMIT is greater than the sumof distances from Harvard toMoscow and from Moscow to MIT.”

One should add that sometimesGelfand’s jokes were rather sharp,but for a young person to becomea subject of Gelfand’s joke meantto be noticed, to be knighted. Theseminar was also the right place tofind out about fresh preprints com-ing from the inaccessible West. Butmost participants were attractedby the power and originality of

Gelfand, circa 1950.

Gelfand’s approach to mathematics.He used the simplest grass-rootsexamples, but he could turn themaround in a totally unexpected way.

Gelfand always paid special at-tention to students, who formed themajority of the seminar audience.From time to time he would re-peat: “My seminar is for high schoolstudents, decent undergraduates,bright graduates, and outstandingprofessors.” One of the best descrip-tions of the seminar was given by S.Gindikin [3]. You may also see thenotes by Zelevinsky and me in thiscollection (my notes will appear in

the next issue). Dusa McDuff described the seminarimpressions of a young foreigner.

The seminar also served as a constant supply ofGelfand’s collaborators, who were already familiarwith Gelfand’s style and his way of thinking. Theirroles were quite different. Sometimes they woulddiscuss specific examples, sometimes very vagueideas; sometimes they would bring their ownsuggestions that would be ridiculed, torn apart,turned upside down, and then transformed intosomething exquisite. Only a few could bear thetask, but the pool of mathematicians in Moscowwas enormous.

The seminar was a reflection of Gelfand’s passionto teach, as he tried to teach everyone and every-where. Among his former students are F. Berezin,J. Bernstein, E. Dynkin, A. Goncharov, D. Kazh-dan, A. Kirillov, M. Kontsevich, and A. Zelevinsky.The number of his informal students is hard toestimate.

From the early period of his life also comesGelfand’s interest in education, especially in edu-cation for school students living far from researchcenters. He was among the founders of the MoscowMathematical Olympiads and later organized hisfamous Mathematics School by Correspondencefor middle and high school students (see the

recollections by Sergei Tabachnikov in the secondpart of this article). Gelfand founded, ran, andwrote several textbooks for the school. His uni-versity textbooks Linear Algebra and Calculus ofVariations (written with S. Fomin) also bear theimprint of his style and personality.

It is hard to describe all of Gelfand’s achieve-ments in mathematics. He left his unique andpowerful imprint everywhere (excluding, probably,mathematical logic). Some (but far from all) ofhis breakthroughs are described here by SimonGindikin, David Kazhdan, Bertram Kostant, PeterLax, Isadore Singer, and Anatoly Vershik. ButGelfand’s interests spread far beyond pure andapplied mathematics. He left a number of papersin biology, physiology, medicine, and other fields.

Gelfand was the first to obtain the Wolf Prize,in 1978 (together with C. L. Siegel), and had manyother awards, including the Kyoto Prize (1989) andthe MacArthur Fellowship (1994). He was electedto all leading academies and had honorary degreesfrom many universities.

This collection of articles about Gelfand alsocontains an impression of the Gelfand seminarby a foreigner (Dusa McDuff), a look at theGelfand school from inside (Andrei Zelevinsky andVladimir Retakh), a description of Gelfand’s Schoolby Correspondence (Sergei Tabachnikov), and anessay, “Gelfand at 92” by Mark Saul.

References[1] A talk with professor I. M. Gelfand, recorded by V. Re-

takh and A. Sosinsky, Kvant (1989), no. 1, 3–12 (inRussian). English translation in: Quantum (1991), no. 1,20–26.

[2] A. Sakharov, Memoirs, Alfred Knopf, New York, 1990.[3] S. Gindikin, Foreword to I. M. Gelfand Seminar, Ad-

vances in Soviet Mathematics, vol. 16, Part 1, Amer.Math. Soc., Providence, RI, 1993.

I. M. Singer

I. M. Gelfand

Israel Gelfand was one of the most influentialmathematicians of the twentieth century—I daresay, the most outstanding in the last sixty years.

Unfortunately, our society neither understandsnor appreciates mathematics. Despite its manyapplications, despite its intellectual power whichhas changed the way we do science, mathematiciansare undervalued and ignored. Its practitioners,its leaders go unrecognized. They have neitherpower nor influence. Watching the negative effects

I. M. Singer is Emeritus Institute Professor at the Mas-sachusetts Institute of Technology. His email address [email protected].

Adapted from: I. M. Singer, “Tribute to I. M. Gelfand”,Progress in Mathematics, vol. 132, Birkhäuser Boston, 1995.


popularity causes in other fields and looking atthe few superficial articles about mathematics, Ithink it is just as well.

Faced constantly with problems we can’t solve,most mathematicians tend to be modest aboutthemselves and their accomplishments. Perhapsthat is why we have failed to recognize a giantin our midst. I won’t compare Gelfand with otheroutstanding mathematicians or scientists of thetwentieth century; if I did, you would start checkingfor yourselves whether you agree with me. But focuson my point—we had a giant in our midst. I turnto other fields to find comparable achievements:Balanchine in dance or Thomas Mann in literatureor Stravinsky, better still, Mozart in music, butfor me, a better comparison is with artists likeCezanne and Matisse. I commend to you the greatpoet Paul Rilke’s letter on Cezanne. He said, “PaulCezanne has been my supreme example, becausehe has remained in the innermost center of hiswork for forty years…which explains somethingbeyond the freshness and purity of his paintings”(of course, much longer for Gelfand).

Evoking Matisse is perhaps more apt. A Matisseis breathtaking. No matter what his personalcircumstance, he turns to new frontiers with joyand energy. Particularly outstanding is his laterwork: Jazz and the remarkable “papier-decouples”,efforts done in the early 1880s.

Gelfand always dazzled us with new and pro-found ideas. One of his latest works, the book withKapranov and Zelevinsky, is a major effort thatmaps out new directions for decades to come.

In preparing this article, I asked many people fortopics I should emphasize. You will be interestedin what happened. First, there was little inter-section in the subjects my correspondents chose.Second, everyone gave me a five- to twenty-minuteenthusiastic lecture on the essence of Gelfand’scontribution—simple and profound.

Reviewing Gelfand’s contributions to mathemat-ics is an education. Let me remind you of some ofhis main work.

1. Normed rings2. C*-Algebras (with Raikov)—the GNS Con-

struction3. Representations of complex and real semisim-

ple groups (with Naimark and Graev)4. Integral Geometry—Generalizations of the

Radon Transform5. Inverse scattering of Sturm-Liouville systems

(with Levitan)6. Gelfand-Dickey on Lax operators and KdV7. The treatises on generalized functions8. Elliptic equations9. The cohomology of infinite dimensional Lie

algebras (with Fuchs)

After receiving honorary degree at Oxford in1973. Left with Gian-Carlo Rota.

10. Combinatorial characteristic classes (begin-ning with MacPherson)

11. Dilogarithms, discriminants, hypergeometricfunctions

12. The Gelfand seminarIt is impossible to review his enormous contri-

butions in a short note. I will just comment on afew results that affected me.

As a graduate student, one of the first stronginfluences on me was Gelfand’s normed ring paper.Marshall Stone had already taught us that pointscould be recaptured in Boolean algebras as maximalideals. But Gelfand combined analysis with algebrain a simple and beautiful way. Using the maximalideal in a complex commutative Banach algebra, herepresented such algebras as algebras of functions.Thus began the theory of commutative Banachalgebras. The spectral theorem and the Wiener’sTauberian theorem were elementary consequences.I was greatly influenced by the revolutionary viewbegun there.

A natural next step for Gelfand was the study ofnoncommutative C*-algebras. He represented suchalgebras as operator algebras using the famous GNSconstruction. It seemed inevitable to find unitaryrepresentations of locally compact groups usingtheir convolution algebras. The representationtheory of complex and real semisimple Lie groupsfollowed quickly. What struck me most was thegeometric approach Gelfand and his coworkerstook. Only recently it appears this subject hasbecome geometric again.

In 1963 twenty American experts in PDE wereon their way to Novosibirsk for the first visit offoreign scientists to the academic city there. Itwas in the midst of Khrushchev’s thaw. When Ilearned about it, I asked whether I could be addedto the list of visitors, citing the index theoremAtiyah and I had just proved. After reading hisearly papers, I wanted to meet Gelfand. Each day ofmy two-week stay in Novosibirsk I asked Gelfand’s


With M. Atiyah, 1973.

students whenhe was com-ing. The responsewas always “to-morrow”. Gelfandnever came. Isadly returned toMoscow. When Igot to my roomat the famous Ho-tel Ukraine, thetelephone rangand someone saidGelfand wanted to

With I. M. Singer, 1973.

meet me, couldI come down-stairs. There wasGelfand. He in-vited Peter Lax andme for a walk.During this walk,Peter tried to tellGelfand about hiswork on SL(2, R)with Ralph Phillips.Gelfand tried to ex-plain his own viewof SL(2, R) to Pe-ter, but his English

was inadequate. (He was rusty; within two days hisEnglish was fluent.) I interrupted and explainedGelfand’s program to Peter. At the corner Gelfandstopped, turned to me and said, “But you are mystudent.” I replied, “Indeed, I am your student.”(By the way, Gelfand told me he didn’t come toNovosibirsk, because he hated long conferences.)

Although it was an honor to be Gelfand’s student,it was also a burden. We tried to imitate the depthand unity Gelfand brought to mathematics. Hemade us think harder than we believed possible.Gelfand and I became close friends in a matter ofminutes, and we remained so ever since. I was illin Moscow, and Gelfand took care of me.

I didn’t see him again for ten years. He wasscheduled to receive an honorary degree at Oxford,where I was visiting. It was unclear that he would beallowed to leave the Soviet Union to visit the West.I decided not to wait and returned home. A weeklater, I received a telegram from Atiyah: Gelfandwas coming—the queen had asked the Russianambassador to intercede. I flew back to England andaccompanied Gelfand during his visit—a glorioustime. Many things stand out, but I’ll mention onlyone: our visit to a Parker fountain pen store. Thoseof you who have ever shopped with Gelfand willsmile; it was always an unforgettable experience.Within fifteen minutes, he had every salespersonscrambling for different pens. Within an hour I

knew more about the construction of fountainpens than I ever cared to know and had everbelieved possible! Gelfand’s infinite curiosity andfocused energy on details were unbelievable; those,coupled with his profound intuition of essentialfeatures, are rare among human beings. He wasbeyond category.

Talking about Oxford, let me emphasizeGelfand’s paper on elliptic equations. In 1962Atiyah and I had found the Dirac operator onspin manifolds and already had the index formulafor geometric operators coupled to any vectorbundle, although it took another nine months toprove our theorem. Gelfand’s paper was broughtto our attention by Smale. It enlarged our viewconsiderably, as Gelfand always did, and we quicklyrealized, using essentially the Bott periodicitytheorem, that we could prove the index theoremfor any elliptic operator.

I should also mention the application ofGelfand’s work to physics: Gelfand-Fuchs, forexample, on vector fields on the circle, the so-called Virasoro algebra, which Virasoro did not infact define. Although I mentioned Gelfand-Dickey,I have to stress its influence on matrix modeltheory. And I have to describe how encouraginghe was and how far ahead of his time he was inunderstanding the implication of a paper whichseemed obscure at the time.

Claude Itzykson told me that his famous pa-per with Brezin, Parisi, and Zuber that led totriangulating moduli space at the beginning wentunnoticed by scientists. The authors then receivedone request for a reprint—from Gelfand.

Ray and I were very excited about our definitionof determinants for Laplacian-like operators andits use in obtaining manifold invariants-analytictorsion. The early response in the United States wassilence; Gelfand sent us a congratulatory telegram.

In conclusion, I want to mention one specialquality of Gelfand. He was a magician. It is not verydifficult—not very difficult at all—for any of usmere mortals to keep the difference in our ages aconstant function of time. But with Gelfand, when Imet him at his fiftieth, and in his sixtieth, I thoughthe was older than I. Ten years later, I felt that wewere the same age. Later it became clear to methat Gelfand was, in fact, much younger than mostmathematicians.


David Kazhdan

Works of I. Gelfand on the Theory ofRepresentations

The theory of group representations was the centerof interest of I. Gelfand. I think this is relatedto the nature of this domain, which combinesanalysis, algebra, and topology in a very intricatefashion. But this richness of the representationtheory should not be taken as self-evident. To agreat extent we owe this understanding to works ofI. Gelfand, to his unique way of seeing mathematicsas a unity of different points of view.

In the late thirties, when Gelfand started hismathematical career, the theory of representationsof compact groups and the general principles ofharmonic analysis on compact groups were wellunderstood due to works of Hermann Weyl. Har-monic analysis on locally compact abelian groupswas developed by Lev Pontryagin. The generalstructure of operator algebras was clarified byMurray and von Neumann. But the representationtheory of noncompact noncommutative groupswas almost nonexistent. The only result I know ofis the work of Eugene Wigner on representationsof the inhomogeneous Lorentz group. Wigner hasshown that the study of physically interestingirreducible representations of this group can bereduced to the study of irreducible representationsof its compact subgroups.

It was not at all clear whether the theory ofrepresentations of real semisimple noncompactgroups is “good”, i.e., whether the set of irreduciblerepresentations could be parameterized by pointsof a “reasonable” set, and whether the unitaryrepresentations can be uniquely decomposed intoirreducible ones. The conventional wisdom wasto expect that the beautiful theory of Murray-vonNeumann factors is necessary for the descriptionof representations of real semisimple noncompactgroups. On the other hand, Gelfand, for whomGauss and Riemann were the heroes, expectedthat the theory of representations of such groupsshould possess classical beauty.

Gelfand’s first result in 1942 with D. Raikovin the theory of representations of groups isa proof of the existence of “sufficiently many”unitary representations for any locally compactgroup G. In other words, any unitary irreduciblerepresentation ofG is a direct integral of irreducibleones. The proof of this result is based on the veryimportant observation that the representationtheory of the group G is identical to the theoryof representations of the convolution algebra of

David Kazhdan is professor of mathematics at the Ein-stein Institute of Mathematics. His email address [email protected].

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On Raoul Bott’s porch, 1976. Left toright, I. Gelfand, R. MacPherson,R. Bott, D. Kazhdan.

measures onGwithcompact supportand on an appli-cation of Gelfand’stheory of normedrings.

Next, in thelate forties, therewas a stream ofpapers (most ofthem jointly withM. Naimark) thatdeveloped the mainconcepts of the rep-resentation theoryof noncompact classical groups G. It would be amuch simpler task to describe the concepts thatappeared later than to describe the richness of thiswork.

Gelfand believed that the space G of irreduciblerepresentations of G is a reasonable “classical”space. If I understand correctly, the first indicationof the correctness of this intuition came from thetheory of spherical functions, developed in theearly forties but published only in 1950. Let K ⊂ Gbe the maximal compact subgroup and G0 ⊂ G bethe subset of irreducible representations of class 1(that is, the representations (π,V) of G such thatVK ≠ 0). Gelfand observed that the subset G0 isequal to the set of irreducible representations ofthe subalgebra of two-sided K-invariant functionson G, proved the commutativity of this algebra,and identified the space of its maximal ideals withthe quotient LT/W where LT is the torus dualto the maximal split torus T ⊂ G and W is theWeyl group. The generalization of this approachdeveloped in the fifties by Harish-Chandra andGodement led to the proof of the uniqueness ofthe decomposition of any representation of thegroup G into irreducibles.

Given such a nice classification of irreduciblerepresentations of class 1, it was natural toguess that the total space G is also an algebraicvariety. But for this purpose one had to find away to construct irreducible representations ofG. Gelfand introduced the notion of parabolicinduction (for classical groups) and, in particular,studied representations πξ of G induced froma character ξ of a Borel subgroup B ⊂ G. Heshowed that, for generic character ξ of T = B/U ,the representation πξ is irreducible and that therepresentations πξ , πξ′ are equivalent if and onlyif the characters ξ, ξ′ of T are conjugate underthe action of the Weyl group W . The proof isbased on the decomposition G =

⋃w∈W BwB for

classical groups. This decomposition was extendedby Harish-Chandra to the case of an arbitrary


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60th birthday.

semisimple group and isknown now as the Bruhatdecomposition.

This construction givesmany irreducible repre-sentations. But how toshow that not much ismissing? In the case ofa compact group G itis well known that allthe representations ofG are constituents ofthe regular representa-tion. Therefore, to seethat a list of represen-tations πa, a ∈ A, of G iscomplete, it is sufficientto show that one can writethe delta function δe on

G as a linear combination of the characters tr(πa).But for representations (π,V) of a noncompactgroup G, which are typically infinite-dimensional,the trace of the operator π(g), g ∈ G, is notdefined. The ingenious idea of Gelfand was todefine characters tr(π) as distributions. That is,he showed that for any smooth function f (g) withcompact support, the operator

πξ(f ) :=∫f (g)πξ(g)dg

is of trace class and defined the distribution tr(π)by tr(π)(f ) := tr(π(f )). Now one could lookfor a W -invariant measure (called the Plancherelmeasure) µX on the space X of unitary charactersof T such that

(1) δe =∫ξ ∈ Xtr(πξ)µX .

It is not difficult to see that such a measure µXis unique (if it exists), and the knowledge of µXis equivalent to the explicit decomposition of theregular representation L2(G) into irreducible ones.

In a series of joint works with M. Naimark,Gelfand was able to guess a beautiful algebraicexpression for the Plancherel measure µX in thecase of classical complex groups and to proveequality 1 by very intricate explicit calculations—agreat reward for difficult work.

As a continuation of this series of works, Gelfandposed a number of questions (he was able to answerthem only in particular cases) which influenced thedevelopment of representation theory for manyyears.

1. In joint work with M. Graev, Gelfand classifiedgeneric irreducible representations of the groupSL(n,R). They found that G is a union of piecescalled series which correspond to conjugacy classesof maximal tori. Moreover, the series correspond-ing to nonsplit tori have realizations in spaces of

(partially) analytic functions. Gelfand conjecturedthat the analogous description of the space Gshould be true for all real semisimple groupsand that it should be possible to realize discreteseries in appropriate spaces of analytic functions.The first part of the conjecture was justified byHarish-Chandra, who constructed discrete seriesof representations for real semisimple groups andfound the Plancherel measure concentrated on uni-tary representations induced from discrete seriesof Levi subgroups. The second part was modifiedby Langlands, who suggested the realization ofdiscrete series (which exist if and only if there isa maximal compact torus T c ⊂ G) in the spaceof cohomologies Hi(G/T c ,F) of homogeneousholomorphic vector bundles F on G/T c (whileGelfand considered only the realization in sectionsof such bundles).

2. In joint work with Graev, Gelfand constructedthe analog of the Paley-Wiener theorem for groupsSL2(C) and SL2(R) (that is, a decomposition of therepresentation of G on the space C∞c (G) of smoothfunctions with compact support) and raised thequestion about the extension of this result toother groups. This generalization was obtained byJ. Arthur in 1983.

3. Also in joint work with Graev, Gelfand con-structed the decomposition of representations ofthe group SL2(C) on the space L2(SL2(C)/SL2(R)and asked the question about the decomposition ofrepresentations of the group G on L2(G/H) whereH ⊂ G is the set of fixed points of an involution.An extension of this result to arbitrary such pairs(G,H) was achieved only recently (see the talk byP. Delorme at the ICM Congress, 2002).

4. Gelfand showed that many special functions,such as Bessel and Whittaker functions and Ja-cobi and Legendre polynomials, appear as matrixcoefficients of irreducible representations. Thisinterpretation of special functions immediatelyexplains the functional and differential equationsfor these functions. It is clear now that (almost)all special functions studied in the nineteenthand twentieth centuries can be interpreted asmatrix coefficients or traces of representationsof groups or their quantum analogs (see, for ex-ample, papers of Tsuchiya-Kanie, Koelink, Noumi,Rosengren, Stokman, Sugitani, and others on repre-sentation theoretic interpretation of Askey-Wilson,Macdonald, and Koornwinder polynomials).

The next series of Gelfand’s work (with M. Tsetlin)was on irreducible finite-dimensional representa-tions of classical groups G. The classification ofsuch representations (πλ, Vλ) was known, butGelfand asked a new question, partially influencedby his interest in physics: how to find a “good”realization of these representations. In other words,how to find a basis in Vλ which allows one to


compute matrix coefficients of πλ(g), g ∈ G, inthese bases. Such a basis (Gelfand-Tsetlin basis)was constructed for irreducible representations ofgroups SLn and SOn and became the core of manyworks in representation theory and combinatorics.

Gelfand and Graev found the expression forthe matrix coefficients of representations πλ interms of discrete versions of Γ -functions. Thisrealization of finite-dimensional representationshas an important analog for infinite-dimensionalrepresentations of groups over local fields.

As part of the theory of finite-dimensionalrepresentations, Gelfand studied the Clebsch-Gordan coefficients, which give a decomposition oftensor products of irreducible representations intoirreducible components. He noticed (at least forG = SL2) that Clebsch-Gordan coefficients of G arediscrete analogs of Jacobi polynomials which arematrix coefficients of irreducible representationsof G. Possibly an explanation of this can be givenby using the theory of quantum groups, wheremultiplication and comultiplication are almostsymmetric to each other.

The next series of Gelfand’s work was onrepresentations of groups over finite and localfields F . The basic results here are the proof of theuniqueness of a Whittaker vector, the existenceof a Whittaker vector for cuspidal representationsof GLn(F), the construction of an analog of theGelfand-Tsetlin basis for such representations,and the description of cuspidal representationsof GLn(F) in terms of Γ -functions (joint workwith M. Graev and D. Kazhdan). But I thinkthat his most important work in this area is thecomplete description of irreducible representationsof groups SL2(F) and GL2(F) for local fields withthe residue characteristic different from 2 (jointwork with M. Graev and A. Kirillov). They showedthat irreducible representations of GL2(F) areessentially parameterized by conjugacy classesof pairs (T , ξ) where T ⊂ GL2(F) is a maximaltorus and ξ : T → C is a character. Moreover,they found a formula for the characters trTξ (g) ofthese representations and an explicit expressionfor the Plancherel measure. A striking and untilnow unexplained feature of these formulas is thatthey are essentially algebraic. For example, theMellin transform L(g, t) in ξ of trTξ (g), which is afunction of GL2(F)× T , is given by

L(g, t) = δ(det(g),Nm(t))εT× (tr(g)− tr(t))/|tr(g)− tr(t)|.

Here T = E∗ where E is a quadratic extensionof F , εT : F∗ → ± is the quadratic charactercorresponding to E, tr is the matrix trace, and trand Nm are the trace and norm maps from E to F .

The understanding of the existence of anintrinsic connection between the structure of

Talking at 90th birthday conference,2003.

irreducible represen-tations of groupsover local fieldsand number theorywas greatly clarifiedby Langlands. Onthe other hand, ageneralization of al-gebraic formulas forthe Mellin transformof characters andfor the Plancherelmeasure was neverfound.

In other work,Gelfand and Graevfound a descriptionof the irreducible representations of the multiplica-tive group D∗ of quaternions over F as inducedfrom 1-dimensional representations of appropriatesubgroups. This was of constructions of irreduciblerepresentations of D∗ that were later generalizedby R. Howe to other p-adic groups.

The description of representations of the groupsSL2(F) and GL2(F) for local fields is presented inthe book Generalized Functions, Volume 6 writtenwith Graev and I. Piatetski-Shapiro. In the samebook, Gelfand developed the theory of representa-tions of semisimple adelic groups G(AK) for globalfields K. He defined the cuspidal part

L20(G(AK)/G(K)) ⊂ L2(G(AK)/G(K))

of the space of automorphic forms, proved thatthe representation of G(AK) on L2

0(G(AK)/G(K))is a direct sum of irreducible representations, anddeveloped a representation-theoretical interpreta-tion of the theory of modular forms. The work ofLanglands is very much influenced by these resultsof Gelfand.

It became clear that generic representations ofany semisimple Lie group or Lie algebra almostdo not depend on a choice of a particular group,so Gelfand tried to find a way to express thissimilarity in an intrinsic way. In a series of paperswith A. Kirillov he studied the structure of theskew-field F(g) of fractions for the universalenveloping algebra of a Lie algebra g. He foundthat the skew-fields F(G) are almost defined bythe transcendence degree of the center Z(g) (equalto the rank r(g) of g) and by their Gelfand-Kirillovdimension (equal to (dim(g) − r(g))/2). Theseresults are the foundation of works of A. Josephon the structure of the category of Harish-Chandramodules.

The last series of work of Gelfand on represen-tation theory was on category O of representationsof a semisimple Lie algebra. This category ofrepresentations was defined by Verma, but the


basic results are due to J. Bernstein, I. Gelfand,and S. Gelfand. They constructed a resolutionof finite-dimensional representations by Vermamodules Vw , w ∈ W (known as the BGG-resolution),discovered the duality between irreducible and pro-jective modules in the category O, and found therelation between the category O and the categoryof Harish-Chandra modules. These results formthe cornerstone of the theory of representations ofsemisimple Lie algebras and their affine analogs.

However, their main discovery was the existenceof a strong connection between algebraic geometryof the flag space B of a semisimple group andthe structure of the category O. For example,they showed that there is an embedding of Vwinto Vw ′ if and only if the Bruhat cell BwB ⊂ Bbelongs to the closure of Bw ′B. This connectionbetween algebraic geometry and the category ofrepresentations is the basis for the recent geometrictheory of representations.

There are other important works of Gelfandon representation theory (such as indecompos-able representations of semisimple Lie groups,models of representations, and representationsof infinite-dimensional groups), but I want tomention two series of works that originated inrepresentation theory but have an independentlife. The appearance of such works is very natural,since for Gelfand representation theory was a partof a much broader structure of analysis.

Integral geometry is an offshoot of representa-tion theory. The proof of the Plancherel theoremfor complex groups is equivalent to the construc-tion of the inversion formula, which gives thevalue of a function in terms of its integrals overhorocycles. Gelfand (in a series of joint workswith Graev, Z. Shapiro, S. Gindikin, and others)found inversion formulas for reconstruction ofthe value of a function on a manifold in termsof its integrals over an appropriate family ofsubmanifolds. The existence of such inversionformulas found applications in such areas assymplectic geometry, multi-dimensional complexanalysis, algebraic analysis, nonlinear differentialequations, and Riemannian geometry, as well asin applied mathematics (tomography). (A moredetailed description of Gelfand’s work on integralgeometry is given by S. Gindikin in the currentissue of the Notices.)

Analogously, the work of Gelfand with V. Pono-marev and, later, with J. Bernstein on quiverswas motivated by the problems in representationtheory—the description of indecomposable repre-sentations for the Lorentz group. But the innerdevelopment of this subject led to a beautiful anddeep theory which later made a full circle in worksof Ringel, Lusztig, and Nakajima and became the

foundation for geometric representation theory ofLie algebras and quantum groups.

Anatoly Vershik

Gelfand, My Inspiration

The achievement of Israel Moiseevich Gelfandwas an unusual phenomenon of twentieth-centurymathematics. His name must be included on anyshort list of those who formed the mathematics ofthat century. He pioneered many new ideas andcreated whole new provinces of knowledge. An ex-ceptional intuition, a depth and breadth of thought,a liveliness in comprehending mathematics—thesewere his chief qualities.

But the most important quality of this wonderfulmathematician was his ability to inspire others.Many other mathematicians, both in Russia andabroad, felt an amazing sense of inspiration intalking to or corresponding with Gelfand. His ideasand his advice stimulated many discoveries andmuch research done by others.

Gelfand and Leningrad MathematicsI was a student in Leningrad in the 1950s, whenGelfand’s name was well known to all Leningradmathematicians, including students interested inthe subject. This is not surprising. The interestsof most Leningrad mathematicians (includingL. V. Kantorovich, V. I. Smirnov, G. M. Fikhtengoltz,and others) centered on functional analysis andits applications, and this “Leningrad approach”was one of the branches of functional analysisthat developed in the Soviet Union in the years1930–1950. I. M. Gelfand was an iconic figure inthis development.

The Leningrad approach differed from thattaken in Moscow (by A. N. Kolmogorov, I. M.Gelfand, L. A. Lyusternik and A. I. Plessner) andin the Ukraine (by M. G. Krein and N. I. Akhiezer).It was centered around the theory of functions,operator theory, and its applications to differentialequations and methods of computation. The maindifference between the Moscow approach and theother two was the study advocated by Gelfand ofinfinite-dimensional representations of groups andalgebras and an interest not so much in Banachspaces as in general problems of noncommutativeFourier analysis and, more broadly, in a synthesisof algebra, classical analysis, and the theoryof functions. These ideas were not present inLeningrad for many years.

Anatoly Vershik is professor of mathematics at the St.Petersburg Department of the Steklov Mathematical InstituteRAS. His email address is [email protected].

This segment of the article was translated from the Russianby Vladimir Retakh and Mark Saul.


Gelfand’s fame and authority went unchallengedat Leningrad seminars. While still young, Gelfandhad broken into the elite club of Moscow mathe-maticians of the highest level, amazing everyonewith his theory of maximal ideals of commutativeBanach algebras (the so-called “Gelfand trans-form”), and he remained a leader of this club untilhis death.

Among Leningrad mathematicians Gelfand es-pecially valued L. V. Kantorovich, D. K. Faddeev,and V. A. Rokhlin, whom he remembered from hisstudent days. Like Kantorovich, Gelfand played anactive role in the atomic research project and gen-erally knew the work of Kantorovich on functionalanalysis. Both these outstanding mathematiciansdid much work on applied problems.

In 1986, at the funeral of Kantorovich, Gelfandtold me that of all of Kantorovich’s results, the onethat impressed him the most was his pioneeringwork on linear programming. That was my view aswell. On the other hand, when I asked Kantorovichto submit my paper to Doklady in 1971, he notedthat the paper really represented the work ofGelfand and that it was disgraceful that Gelfandwas not still a full member of the Soviet Academyof Science. Gelfand had direct contact with only afew of the younger Leningrad mathematicians inaddition to me—in particular, with L. D. Faddeevand M. S. Birman.

My First Acquaintance with Gelfand’s WorkThe first significant mathematical papers that Istudied were the papers of I. M. Gelfand, D. A.Raikov, and G. E. Shilov (Gimdargesh, as I calledthem) on commutative normed rings and their pa-pers that followed on generalized Fourier analysis.This theory was a mathematical awakening for me.I was enchanted by its beauty and simplicity, itsgenerality and depth.

Before this I had been hesitant. I could jointhe Department of Algebra, where I attendedlectures by D. K. Faddeev. Or I could work in theDepartment of Analysis, where my first mentor wasG. P. Akilov and where I might choose to specializein complex analysis (V. I. Smirnov, N. A. Lebedev)or real and functional analysis (G. M. Fikhtengoltz,L. V. Kantorovich, G. P. Akilov). Now functionalanalysis was my only choice. And my interest wasmostly in the analysis of the Moscow school—Gelfand’s functional analysis, as I described earlier.Since that time the work of Gelfand and his schoolin various areas have become my mathematicalguide.

One could say a lot about the great varietyof Gelfand’s interests and works. In fact, hewas interested in everything: biology, music, and

Lab of Function Theory at Moscow StateUniversity, circa 1958. Left to right sitting,I. Gelfand, Polyakov, D. E. Menshov,N. K. Bari, G. P. Tolstov; standing,P. L. Ulyanov, A. G. Kostychenko,F. A. Berezin, G. E. Shilov, R. A. Minlos.

politics, not justmathematics. Iwould like towrite here aboutthe area ofhis mathemati-cal activity thatwas closest tomy interests andabout the prob-lems on which Iwas fortunate towork with him.

I would go sofar as to say thatthe main legacyof the work ofGelfand’s firstperiod was his foundation (along with the group ofstudents he led) of the theory of normed commu-tative rings (commutative Banach algebras, as theyare now called) and, most importantly, the theoryof unitary infinite-dimensional representations oflocally compact groups. These papers becameclassics of mathematics despite the youth of theauthors.

As a byproduct of this theory, Gelfand andNaimark gave a definition of general C∗-algebrasthat is now in common usage. Once, as earlyas the 1970s, Gelfand said to me, “If I couldonly talk to von Neumann, I would explain tohim why our C∗-algebras are more importantthan W∗-algebras (von Neumann algebras).” I canadd that Gelfand was always more interestedin topological and smooth problems (which areconnected with C∗-algebras) than in theoreticalmetric problems (which are formulated in termsof W∗-algebras). The representation theory thatwill forever be connected with his name alwayswas a favorite subject of Gelfand’s, but his ideasabout relationships and details within this theorykept changing. Among his wonderful traits wasa fearless irony towards his own earlier results.When I told him that as a student I was impressedby his work on Banach algebras, Gelfand replied,“But why did we consider only maximal and notprime ideals?” This remark is similar to one hemade in a talk in the 1970s, when I mentioned tohim my interest in his early papers. “I am alwayspraised for my work that was done five or moreyears ago, and the same people are not satisfiedwith what I am doing right now,” he explainedabout his not uncommon critics. In fact, he wasalways ahead of his time, he was always creating anew fashion.

I remember being strongly influenced byGelfand’s talk in 1956 about problems in func-tional analysis (he was probably following the


example of a similar talk by von Neumann at theInternational Congress of Mathematicians in 1954).In this address he posed several general problemsand talked about their relationship to variousmathematical phenomena. One of these problemswas a conjecture about a possible connectionbetween Wiener measures and von Neumannfactors and of its possible utility in future work.“Such beauty must not perish unused,” notedGelfand. I was a student at that time and I wasdeeply impressed by this strong assertion of theprimacy of aesthetic estimation of mathematicaltheories above other criteria.

A Few Comments on Gelfand’s Work of the1950s and 1960sPerhaps Gelfand’s highest achievement was thetheory of infinite-dimensional unitary representa-tions of complex semisimple Lie groups, createdtogether with M. A. Naimark, M. I. Graev, andothers.

It is hardly possible to comment on this devel-opment completely here, but it is worth notingthat Gelfand faced many difficulties common to allpioneers: many details had to be clarified later orproofs rewritten, and so it would happen that attri-butions of priority or authorship were given to otherresearchers. But one must remember that Gelfandwas the first to initiate and develop this hugearea of mathematics, which is so useful in physics.There was no theory of infinite-dimensional rep-resentations before Gelfand’s. From reading therecently published diaries of A. N. Kolmogorov,one can understand how impressed the world ofmathematics was by these works in the 1940s. It isremarkable how confident Gelfand was in startingup completely new areas.

Gelfand’s next great passion in the late 1950sand 1960s—an interest common to many centersof mathematics—was the theory of distributionsof L. Schwartz (called in Russia “generalizedfunctions”). Several Russian mathematicians hada keen interest in the subject but with a touch ofbitterness and perhaps chagrin. The reason wassimple: Gelfand, and before him N. M. Gyunter(a famous Petersburg mathematician) and L. V.Kantorovich and certainly S. L. Sobolev in factnot just made use of but created the theory ofgeneralized functions. In one of his lectures inLeningrad in the early1960s, Gelfand stated directlythat he and Naimark used distributions in theirwork on representation theory. However, the meritof L. Schwartz’s work is that he understood theimportance of the general theory and substantiatedit with many examples. It is important to add that,as often happens in the theory of functions, thisreformulation actually gave a new language andnew understanding to many concepts.

However, the hope that arose at that time thatthis reformulation would give essentially newresults did not exactly come true, and we cannottalk about a serious change from the theory ofBanach spaces to the theory of locally convexspaces, despite the expectations of that time.

The celebrated six-volume series of books Gen-eralized Functions by Gelfand and his coauthorsdemonstrated their unparalleled depth of under-standing of the subject. The breadth of the areacovered, from partial differential equations to rep-resentations and number theory, is characteristicof Gelfand’s style.

In particular, Gelfand was among the first to un-derstand the importance of turning from Banach’smethods in functional analysis to more generallinear topological (nuclear) spaces and their valuefor the spectral theory of operators (the Gelfand-Kostyuchenko theorem) and the theory of measurein linear spaces (the Minlos theorem). Gelfandalso immediately understood the usefulness ofgeneralized stochastic processes (Gelfand-Ito pro-cesses). This stimulated the construction of thetheory of measures in linear spaces started byA. N. Kolmogorov in the 1930s. The famous “holytrinity” (triples of Hilbert spaces), generalizedspectral decompositions, an original approach toLevi processes and to Gelfand-Segal constructions,quasi-invariant measures, and so on were studiedand developed by hundreds of mathematicians.

My First Meeting with GelfandNot counting a few brief discussions with Gelfandduring the Mathematical Congress in 1966 or in thelate 1960s or on my rare visits to his seminar whenI happened to be in Moscow, our first close meetingtook place in the spring of 1972 when I came toMoscow for two months. After the seminar I walkedto his home with several participants. I startedtalking about the work I had recently startedon asymptotic statistics of the lengths of cyclesof random permutations. This paper initiateda long series written by me and my studentson what I later called the asymptotic theory ofgroup representations. Gelfand was interested andinvited me to his home the next day to continue theconversation. I remember that Dima Kazhdan, whowas also there, understood me sooner than Gelfand,who often asked me to repeat what I had said.His active and sometimes aggressive questioningwas very helpful to the development of the themeand the improvement of the talk, if you couldignore the form of the criticism. But I had longago heard about Gelfand’s manner in such things.In talking about mathematics he would openlyexpress his displeasure to his listener. One of hisexpressions, which is useful in the upbringing ofyoung mathematicians, was “Keep your work and


your self-esteem separate.” In other words, do nottake even harsh intellectual criticism personally.

I told Gelfand about my plans to study thesymmetric group and its representations. Mymotivation for this research lay not only in the studyof the subject by itself but also in its applicationsto optimization theory, linear programming, andcombinatorics. (At that time I worked in theDepartment of Computational Mathematics andOperational Research.) There was also a wayof connecting asymptotic theory with ergodictheory. In our discussions, Gelfand remarkedthat everything seemed to be clear with regardto representations of finite groups, then startedtalking enthusiastically about symmetric functions.He recommended that I look at a mysterious paperof E. Thoma about characteristics of the infinitesymmetric group, which were of special interest tome.

Many things about the symmetric group wereunclear to me, and I was not satisfied with theclassical exposition of the theory of its representa-tions despite the fact that it had been developedby Frobenius, Schur, and Young, and also by suchgiants as von Neumann and H. Weyl. It seemed tome then that the foundations of this theory andits relations with combinatorics were not yet clear.Gelfand gave his final conclusion many years later,after he had gotten to know and appreciate mypapers with S. Kerov and later with A. Okounkov.He said something like, “It is all clear now.” Butthat was all years later. In fact, the main idea of myapproach was, roughly speaking, an application ofthe Gelfand-Tsetlin method, created in the early1950s for representation of compact Lie groups, tothe theory of symmetric groups. It is worth notingthat a dissatisfaction with established theory oftenbegins with those studying the theory in theirmature years. That is what happened in my case. Istarted to study representations theory later thanusual and got a certain advantage from this.

In Moscow, during our first conversation athis home, Gelfand once again surprised me. Herepeated part of our discussion to Dima Kazhdan,who had come in later, and said, as if it had alreadybeen done, that I was studying the asymptotics ofYoung diagrams. However, I had not mentionedthis and was just planning to do it. When Icorrected him, Gelfand said, “Yeah, yeah, but youwill do it later.” This unique ability of Gelfandto see “three meters under the ground wherehis interlocutor was standing” was one of hismost remarkable qualities. It instilled trust in hisjudgment by many who got to know him, butalso drove off many who feared his ability tosee through them. He immediately and (usually)correctly guessed what his companions were aboutto say or what they had in mind but did not express,

With M. I. Graev, 1999.

and generallyknew how thediscussion wouldturn out. Thereare many ex-amples of this,the most strikingwas his mathe-matical intuition,which allowedhim to guess theresult (often with-out calculations)and to foreseewhat to expectfrom a given method or even from the fur-ther work of a mathematician or a whole team.During that discussion Gelfand approved of myideas (later called asymptotic representations the-ory). Many of these plans were then acted on by S.Kerov, G. Olshanski, A. Okounkov, A. Borodin, anda number of Western mathematicians.

At Gelfand’s seminar in 1977 I talked withS. Kerov about my results on the limiting formof Young diagrams and about our approximationapproach to representations of the infinite symmet-ric group. At that time Gelfand, Graev, and I hadalready started our work on the representations ofcurrent groups. I. M. always commented on everytalk at his seminar. During my talk he said (andrepeated this many times) that combinatorics isbecoming a central part of mathematics of thefuture. As a permanent seminar participant saidto me, “He was excited.”

Our CollaborationLet me return to 1972 and to the history of ourcollaboration. After our conversation I said good-bye and intended to leave for Leningrad. In MoscowI always stayed with my old friend, the virologistN. V. Kaverin. He once attended a Gelfand seminarin biology. Gelfand remembered him, but they hadno other contact. I mentioned to Gelfand that I wasstaying with Kaverin. Suddenly, on the day of mydeparture, Gelfand called me at Kaverin’s (havingfound the phone number with some difficulty1)and asked me to come over immediately. He alsoinvited M. I. Graev, and during our long walktogether started to talk about a construction ofnoncommutative integrals of representations ofsemisimple groups, primarily for SL(2, R). He saidthat he had been incubating this problem for a longtime and had suggested it to his other students,but he was confident that the problem was perfectfor me.

1There was no “official” way to get someone’s telephonenumber in Moscow at the time.


At the time of the ICM, 1966. Secondfrom left, I. Gelfand; third from left, A.

Kolmogorov; fifth from left, S. V. Fomin;far right, O. A. Oleinik.

I was a bitsurprised thatGelfand wasaware of myknowledge ofthe represen-tation theoryof Lie groups,in particularof SL(2, R). Wehad not talkedabout this. ButGelfand wasright: the prob-lem was posedat just the right

time for me. In the early 1970s, independent of myother work, I was lecturing on representations ofgroups, C∗-algebras, and factors. Perhaps Gelfandhad heard about this, but not from me. Morelikely, this was the result of his ability to foreseethings that I mentioned earlier. Gelfand said thatin order to construct the multiplicative integralof representations we have to study “infinitesi-mal” representations, i.e., a neighborhood of theidentity representation. In the summer of 1972Gelfand came to Leningrad for the thesis defenseof M. Gromov, and I told him about my preliminaryexperiments with the Heisenberg group, where aconstruction of the integral by other methods wasalready known. In December of 1972 we founda solution, a spherical function of the requiredrepresentation of SL(2, R), or the “canonical state”,as Gelfand suggested we call it. I came to Moscowand we drafted the text. The problem was solvedin a few months. The next spring Gelfand came toLeningrad with M. I. Graev and stayed with us. Mydomestic helpers prepared food as instructed byZorya Yakovlevna.2 While we were working on thepaper with Graev, I. M. talked with my wife, Rita,about his favorite music and paintings. Of course,we went to the Hermitage Museum, where Gelfandtalked a lot about a painting of El Greco (SaintsPeter and Paul) and where we also discussed howto continue our work.

I recall next Gelfand’s last visit to Leningrad in1984. I invited him to talk about his own life at ameeting of the Leningrad Mathematical Society (hehad turned seventy-six months earlier). It was aninteresting talk with many details about his firststeps in mathematics, including his discovery at ayoung age of the Euler-Maclaurin formula (recallthat he taught himself almost entirely and neverattended a high school).

2Zorya Yakovlevna Shapiro was then Gelfand’s wife, withwhom he wrote several important papers.

Our first paper was published in Uspekhi in1973 in an issue honoring the seventieth birthdayof Kolmogorov, and this was the beginning ofmy collaboration with Gelfand and Graev, whichcontinued with some breaks over ten years. (Atsome point I will write more about this.) The firstpaper in this series (in Gelfand’s opinion and mine,the best one) touched on many subjects that wereimportant at that time. In particular, we discussedcohomology with coefficients in irreducible rep-resentations. In the next paper we described thecohomology of all semisimple groups without theKazhdan property and gave explicit formulas forsemisimple groups of rank 1. We also constructedirreducible nonlocal representations of the cor-responding current groups. The next paper wasdevoted to representations of groups of diffeo-morphisms and the start of the development ofthe geometry of configurations and its applicationto representation theory. We had no doubts (andit was later confirmed and reconfirmed) that thisseries of papers would have various applicationsand that the work would continue. In recent yearsM. I. Graev and I have found new constructionsand new representations of functional groups, andthis idea certainly has a future.

I would like to recall here one particular story.Our first paper had a natural continuation relatedto representations of groups of smooth functionson a manifold with values in a simple compact Liegroup. The properties of these so-called “energy”representations depend on the dimension of themanifold.

This is not the place for details, but we wereable to prove that the representations are reduciblewhen the dimension is three or higher and whenthe dimension is two under some conditions onthe length of a simple root (see our papers and alsopapers by R. Ismagilov, R. Hoeg-Krohn, S. Albaverio,and N. Wallach). Dimension one is special, andeven at the beginning of our work, Gelfand saidthat we would probably have to deal with a vonNeumann factor-representation. There were noobvious reasons or even preliminary estimatesfor this at that time; we had just started towork on the subject. After some years Gelfand’sprediction was confirmed: in dimension one theenergy representation is a factor-representation oftype III1, and it turned out to be connected witha Wiener measure (not with the scalar one, butwith a measure generated by Brownian motion ona compact group). Do you remember the mysticprediction by Gelfand fifty years ago that factorsshould connect with Wiener measures? Today thesubject is actively being developed.


The SeminarGelfand left us an enormous number of statementson different mathematical subjects. Someone withpatience should collect them. Similarly, someoneshould have kept a diary of the Gelfand Seminar.Unfortunately, no one took the responsibility. Nowwe can pick up only bits and pieces. The seminarswere like one-man shows, sometimes successful,sometimes rough, with salty humor, often relevantand instructive. I gave several talks at GelfandSeminars in Moscow and later at Rutgers. I havealready described my first very successful talk.The second was shorter and less successful butalso distinctive. When I mentioned a not veryinteresting but very new fact, Gelfand called agraduate student to the blackboard and asked himto give a proof right away. Should I take this as anoffense? Certainly not.

I attended the seminar for the first time when Iwas a graduate student. A French mathematicianwas talking about a paper by von Neumann. The talkwas not very clear. Then Gelfand said, “Next time so-and-so will give a talk on this paper” and mentionedseveral names of professors and graduate studentsand immediately started to evaluate the futuretalks. He said that one professor’s talk would givean illusion of understanding, but in reality neitherthe speaker nor the listeners would have any ideaof what was going on. Another one would speakabout something else, including his own results,and so on. All this added some drama to theseminar. One could take it seriously or with a smile.Much later Gelfand told me that L. D. Landau ran aseminar in the style of Pauli (which bore a surfaceresemblance to Gelfand’s but was somewhat morerude). Gelfand himself attended Landau’s seminarand once repeated to me a joke made by M. Migdal:“Gelfand goes to the physicists as an intellectual tothe peasantry.”3

At the same time, we must note that the Gelfandstyle was not for everyone. It drove some awayand even complicated their lives. For example,one of his first, and most beloved, students wasF. A. Berezin. At one point they parted wayscompletely. In the 1970s I tried unsuccessfully toget them to meet and talk. Nowadays, Berezin’swork, especially in supermathematics, has gottenworldwide recognition, but he did not live to seethis. Because I was not nearby and did not talkto Gelfand very often, I was able to avoid thefrustration of some of his coauthors who workedwith him more closely. On the other hand, withme he was always extremely polite and friendly.

3A reference to the political movement in Russia in the1870s known as “Going to the People” (Ho�denie k nar-odu), during which intellectuals went to the countryside inexpectations of “enlightening” the peasants.

His telegrams on my birthday were always verycomplimentary.

Life in the USSRIt is more and more difficult nowadays to explainto my Western colleagues, as well as young peoplein Russia, what academic life was like in the USSR,or even what other parts of life were like. Forexample, why did Gelfand, with all his achievementsin science and contributions to classified stateprojects (which were considered exceptionallyimportant), become a full member of the SovietAcademy of Science only after a shameful delay? Ofcourse, one of the reasons is an official academicand even governmental anti-Semitism. But, forexample, this feeling was not so strong amongphysicists. There were other reasons as well.

Once Gelfand told me, “The situation was quitesimple for me in the beginning of the 1950s [thetime of the infamous “fight with cosmopolitism”,4

when Gelfand lost his position in Moscow StateUniversity]: only those who were really interestedin mathematics became my students.” It wastrue at that time (and also later) that it wasbetter to have another advisor for one’s career.But he still had quite a number of students.And here we face one of the most importantand, in some ways, transcendental qualities ofGelfand: he could attract very different kinds ofpeople. He surrounded himself with a whirlpoolof established mathematicians, newcomers tothe field, biologists, and others. His seminarwas a place for meetings, corridor discussions,exchanges of news, opinions, and so on. It wasnot by chance that the famous “letter from 99mathematicians” defending A. Esenin-Volpin5 wasopenly distributed for signatures at his seminar. Itwas met by authorities with resentment and fearat the highest levels. In a totalitarian state such asthe USSR, only those who were trusted and testedby the state could be a magnet for people. For thisreason, officials, including official mathematicians,did not like Gelfand and his circle, not just outof consideration of their origins but also by theprinciple of “either with us or against us”.

“Of course, we live in a prison,” Gelfand oncetold me during a conference outside Moscow inthe late 1970s during lively discussions betweenour scientists and others visiting from the West.Still, he always refused to take any Samizdat booksfrom me. People should write more about this.

Before any of his few visits abroad, Gelfandhad to pass a number of unpleasant procedures,as did almost all Soviet people. My friends and I,

4A euphemism for an anti-Semitic government program.5In 1969 Esenin-Volpin was thrown into a psychiatric“hospital” for political reasons.


who were openly denied the right to travel abroadfor any purpose, were for that reason sparedthe examinations by party commissions of ourbehavior and of our knowledge of party politics.These seemed to us unfortunate, but at the sametime as rather a prestigious distinction.

At the end of the 1980s and the beginning ofthe 1990s the desire of many scientists to leaveRussia (quite understandably, especially at thattime) touched Gelfand as well. His Moscow seminardied soon after that. It was impossible to replace itsleader. A relatively modest version of the seminarwas reestablished at Rutgers. One can only guesswhat might have happened had Gelfand stayed inMoscow.

I met Gelfand in the U.S. several times. Whilemathematical life was different in the U.S., Gelfandfound his place. The celebration of his ninetiethbirthday at Harvard was perfectly organized andbecame a real scientific event with his brilliant talk.

I have never had any doubts that the name of I. M.Gelfand will be one of the icons of mathematicsin the twentieth century, because the century wasnot just a century of outstanding achievementsbut also of new conceptions. I. M. Gelfand valuedand created exactly this sort of mathematics.

Bertram Kostant

I. M. Gelfand

I first heard of I. M. Gelfand when I was a graduatestudent in Chicago in the early 1950s. At that time,Gelfand’s paper “Normed rings” played a majorrole in the area of modern harmonic analysis, whichwas then very popular with students. My thesisadvisor, Irving Segal, had arranged for me to spendthe years 1953–1955 at the Institute for AdvancedStudy in Princeton. I already knew Chern andWeil from my years in Chicago, and at PrincetonI became friendly with Lefschetz, Hermann Weyl,von Neumann, and Einstein. Certainly I consideredGelfand to be in the same class as these twentieth-century math and physics luminaries, and I lookedforward to meeting him as well.

My chance for this meeting came about when Iwas invited to a 1971 summer school in Budapest.The activities of the school were organized byGelfand himself, and I believe this was the firsttime he had been given permission to attenda conference outside the Soviet Union. A largenumber of his students and colleagues came withhim. It was an unforgettable experience for me tobe a participant in one of his famous multihourseminars. Gelfand is a coauthor in a vast number

Bertram Kostant is professor emeritus at the Mas-sachusetts Institute of Technology. His email address [email protected].

of papers. No doubt many of these papers wereoutgrowths of these seminars. He seemed to havea very distinctive style of inspiring research byposing probing questions to potential collaboratorsand insisting on not letting go until there wassome sort of resolution. I went to Budapestcarrying a recently written paper on the sphericalprincipal series. Gelfand requested that I submitthe paper to the proceedings (Lie Groups andTheir Representations) of the Budapest conference.Even though this paper was later (1990) to winthe Steele Prize, it was probably a mistake for meto publish the paper in the proceedings of theBudapest conference. The initial publisher, HalstedPress, went out of business, and the book did notappear until 1974. When it finally did appear, manypeople reported that it was very hard to find.

In June 1972, responding to an invitation, I wentto Moscow, accompanied by my wife, Ann, for athree-week stay. I have to say that I was deeplytouched and happy to be the recipient of Gelfand’swarmth, friendship, and respect. In his 1970 ICMreport he had included me in a very small (5) groupof people whom he said had made outstandingcontributions to representation theory. In Moscow,Gelfand made arrangements for me to addressa meeting of the Moscow Mathematics Society,chaired by Shafarevich, with whom I had a pleasantlunch. Gelfand also made arrangements for meto meet on a regular basis with a number of hisstudents, including Kazhdan, Bernstein, Kirillov,Gindikin, and his son, Sergei. I was also introducedto his colleagues Manin, Novikov, and Graev, andI spent an afternoon with Berezin in Gorky Parktalking about (from my perspective, geometric)quantization.

One of the mathematical topics that came upduring my conversations with Gelfand was theresults in the first of the BGG, [BGG-71] papers.Somewhat earlier I had solved a problem of Bottwhich asked for a determination of the polynomialfunctions on the dual of a Cartan subalgebrawhich mapped, via Borel transgression, onto thedual basis of the Schubert classes of a general flagmanifold. Acknowledging my precedence in solvingthis problem, [BGG-71] made a penetrating furtherdevelopment in the solution of this problem byintroducing the very important BGG operators.

What was apparent to me during my staywas that Gelfand seemed to have created anenvironment where he was involved with both thepersonal as well as the professional lives of themany people around him. Gelfand’s apartment waslike Grand Central Station, with any number ofpeople going in and out. He seemed to be carryingon n conversations. I can’t speak for the peopleorbiting around him, but as an outsider lookingin I was charmed. Doing mathematics for me has


had many lonely moments, but in this excitingenvironment I sensed that mathematics (like muchof physics) could be done in a communal, sociallysatisfying way.

There are any number of unforgettable incidentswhich occurred during my Moscow visit. Thefollowing was very embarrassing for me (but notwithout its comic aspects). One day Gelfand invitedAnn and me to have some borscht with him at therestaurant of the Soviet Academy. All the tables inthe room were filled with people, except for thetable opposite us. That table was occupied by onlyone man. Suddenly Gelfand started whispering tome, but I couldn’t make out what he was saying.Then he repeated it somewhat louder. Finally, whenI heard what he was saying, I foolishly blurtedout loud the name Lysenko. The room becamestill, and then I realized what I had done. “Oh myGod,” I thought. “I have gotten Gelfand into troublewith the authorities.” For a moment visions of himbeing dragged off to the Gulag went through myhead. But all turned out well. By the early seventiesLysenko had apparently been defanged.

Much later Ann and I saw a great deal of Gelfandwhen he was invited to Cambridge, Massachusetts,in the United States to receive an honorary degreefrom Harvard. A number of scenes during thatvisit still stick in my head. One was at my houseduring a party as I watched him on the floor with abunch of young kids, somehow managing to keeptheir interest by telling them some mathematicalgems. Another was the scene at the breakfast tableafter Gelfand spent the night in our house. Gelfandwas complaining about the absence of good breadin the U.S. and the difficulty of finding healthyfood. I countered by pointing to the Swiss muesliin my breakfast bowl. Exhibiting a sample of hisdelicious (no pun intended) sense of humor, hethen proceeded to eat the fruits and raisins in mybowl, all the while cautioning me not to have thatfor breakfast because it wasn’t good for me.

Sometime during his visit I met him in NewYork City and introduced him to my son, Steven,then engaged in filmmaking. Gelfand gave a longdiscourse to Steven and his friends on Stanislavski.Apparently method acting was one of Gelfand’smany artistic interests. I also took him to see JeanRenoir’s famous film Le Grande Illusion, whosemain theme was the disintegration of Europeanaristocracy in the wake of World War I. I wastotally surprised by his reaction to the film. Sinceit was made in the late thirties, he thought itwas unconscionable that a prominent film wasproduced on such a topic at a time when Hitlerwas planning his march across Europe.

Another scene that sticks in my head waswhen, while driving Gelfand to the airport

With Serre and MacPherson atMonday night seminar, MoscowState University, 1984.

for his flight back to Rus-sia, he started to recitepoems of Ossip Mandel-stam, Anna Akhmatova,and others. I rememberrecording him on tape,but unfortunately I seemto have misplaced the

Lecture at MIT, 1990.

tape.Several years later my

mathematical interestsintersected with thoseof Gelfand in the areaof completely integrablesystems. In 1979 I hadwritten a paper show-ing that the completeintegrability of the openToda lattice arises froma consideration of acertain coadjoint orbitof the Borel subgroup.Gelfand brilliantly putthis result in an infinite-dimensional context andapplied it to the theoryof pseudodifferential op-erators. I believe asimilar observation was made by Mark Adler.

I would like to end here by citing a mathemat-ically philosophical statement of Gelfand whichI think deserves considerable attention. It alsoopens a little window, presenting us with a view ofthe way Gelfand’s mind sometimes worked. One ofmy first papers gave a formula for the multiplicityof a weight in finite-dimensional (Cartan-Weyl) rep-resentation theory. A key ingredient of the formulawas the introduction of a partition function onthe positive part of the root lattice. The partitionfunction was very easy to define combinatorially,but giving an expression for its value at a particularlattice point was altogether a different matter.Gelfand was very interested in this partition func-tion and mentioned it on many occasions. Hefinally convinced himself that no algebraic formulaexisted which would give its values everywhere.He dealt with this realization as follows. Oneday he said to me that in any good mathematicaltheory there should be at least one “transcendental”element and this transcendental element shouldaccount for many of the subtleties of the theory. Inthe Cartan-Weyl theory, he said that my partitionfunction was the transcendental element.


Simon Gindikin

50 Years of Gelfand’s Integral Geometry

As I begin this paper, the entire fifty-year history ofintegral geometry in Gelfand’s life unfolds beforemy eyes. I was fortunate enough to collaboratewith him during some key moments. I would liketo discuss here those points that appear to me tobe the most important in this wonderful endeavor.

First Presentation

I remember well a meeting of Dynkin’s seminar onLie groups, most likely in the spring of 1959. Theseminar, which had been only for undergraduatestudents at first, at this point combined bothundergraduate students (Kirillov, Vinberg, me) andwell-known mathematicians (Karpelevich, Berezin,Piatetski-Shapiro). Suddenly, Gelfand appeared(late as usual and accompanied by Graev), and withenormous enthusiasm he began talking about theirnew work [1]. This was the first time their work onintegral geometry was presented.

It may appear strange that Gelfand did notpresent this in his own seminar (in fact, he almostnever presented his new results there) and thathe selected what was mostly a student seminar.Without a doubt, this was no accident, since Gelfandwas always very precise in selecting venues for histalks, and this fit well into traditions of Moscow’smathematical life. My mind’s eye does not seeany specifics of the presentation but rather recallsGelfand’s excitement, obvious to the listeners, andhis certainty that something significant had openedbefore him—a new direction of geometric analysis,which he proposed to call “integral geometry”, asit was equaled in importance only by differentialgeometry. He commented that Blaschke used thisterm for a certain class of problems connectedwith calculations of geometric measures but thatthis narrow area did not deserve such an ambitioustitle, and so he felt justified in appropriating it forthis new field. The contentious nature of such aposition is obvious. Such things worried Gelfandonly a little, and I will not venture an opinion on thesubject. I think this is important for understandingGelfand’s emotional state. He reminded us of hisfrequent saying that “representation theory is all ofmathematics.” (I heard this many times, and Maninonce recalled that he also heard from Gelfand that“all mathematics is representation theory,” notingthe delicate difference between these aphorisms.)From now on, Gelfand said, he considered that“integral geometry is all of mathematics.”

Simon Gindikin is Board of Governors Professor at RutgersUniversity. His email address is [email protected].

Now a word about Gelfand’s mathematical mo-tivation. He considered that the main problem ofthe theory of representations was the decompo-sition of certain reducible representations—mostimportantly, regular representations on homoge-neous spaces—into a sum (maybe continuous) ofirreducible representations (Plancherel formulas).The class of groups and homogeneous spaces wasnot specified in the talk. It was only clear thatcomplex semisimple Lie groups, together with theirhomogeneous spaces with maximally compact andCartan isotropy subgroups, were in this class. Thefirst two spaces are symmetric, but the third isnot. This is of principal importance—spaces thatare not symmetric are considered. From the verybeginning the decomposition into irreducible rep-resentations is interpreted as a noncommutativeanalog of the Fourier integral: under the conditionsof a continuous simple spectrum, we can talkabout the “projections” of a function on a homo-geneous space onto irreducible representations.The usual Fourier transform on Rn has a geomet-ric twin in the Radon transform: integration onhyperplanes. They are connected with each othervia the one-dimensional Fourier transform. Themarvelous discovery of Gelfand-Graev was that,in the case of semisimple Lie groups, there is ananalogous geometric twin: on homogeneous spaceswe consider horospheres—the set of orbits of allmaximally unipotent subgroups—and the horo-spherical transform (the operator of integration onhorospheres). The generalized Fourier transformand the horospherical transform are connected bythe (commutative) Mellin transform, equivalently,with the commutative Fourier transform. For thisreason, problems about the Plancherel formulaand inversion of the horospherical transform on ahomogeneous space are trivially reduced to eachother. Horospheres in hyperbolic geometry werewell known (and highly valued) already by theircreators: they are spheres of infinite radius withcenters at infinity and different from hyperbolichyperplanes. For other homogeneous spaces (in-cluding symmetric spaces), they had remainedunnoticed before the work of Gelfand-Graev.

A posteriori, the idea of the horosphericaltransform seems almost obvious. Let G be acomplex semisimple Lie group and let B = MANbe a Borel subgroup; hereN is the nilpotent radical,MA is a Levi subgroup (which is commutative inthis case) with M and A its compact and vectorsubgroups, respectively. Then the manifold ofhorospheres is Ξ = G/N. Gelfand and Graev calledit the “principal affine space”. On this space thereare two commuting actions: that of the group Gacting by left shifts and that of the group MAacting by right shifts; the latter action is welldefined because MA normalizes N.


The decomposition of the regular representationon Ξ relative to the action ofMA gives precisely theirreducible representations of group G realized inthe spaces of sections of line bundles over the flagmanifold F = G/B (the “principal projective space”in Gelfand-Graev’s terminology). This is simply arephrasing of the conventional realization of theprincipal series representations. Therefore, if thehorospherical transform (which is an intertwiningoperator) is injective, the decomposition of theregular representation is reduced to the decompo-sition relative to the action of MA on the spaceof horospheres and the above-mentioned Mellintransform along the abelian subgroup A.

Prehistory

Needless to say, the real road to the discovery justoutlined was entirely different. I was lucky to hearabout it from Gelfand, and I want to share it here.Three papers appeared in 1947—by Bargmann,Gelfand-Naimark, and Harish-Chandra—in whichunitary representations of the Lorentz group werefound. I believe that I once heard Gelfand’s ob-servation that important things in mathematicsspring forward independently from three placesat once (the history of mathematics indeed de-livers striking examples of this, beginning withnon-Euclidean geometry). The Lorentz group islocally isomorphic to group SL(2;C). Not sur-prisingly, the initiative in this problem belongedto the physicists, and the authors were target-ing physical applications. (Bargmann was solvingPauli’s problem and discussed it with Wigner andvon Neumann; the problem was suggested toHarish-Chandra by Dirac; the first publication ofGelfand-Naimark was in a physics journal.) Becauserepresentations of the rotation group play a keyrole in nonrelativistic quantum mechanics, it wasnatural to expect representations of the Lorentzgroup to play an analogous role in the relativisticcase. The difficulty lay in the fact that, while therotation group is compact and all its irreducibleunitary representations are finite dimensional, theLorentz group is not compact and its only finite-dimensional unitary representation is the trivialone-dimensional representation. (It is preciselythe unitary representations that have a physicalinterpretation.)

In the 1930s Dirac and Wigner considered certaininfinite-dimensional unitary representations of theLorentz group. From the other direction, in theearly 1940s Gelfand-Raikov constructed an analogto the theory of Peter-Weyl for general locallycompact groups using infinite-dimensional unitaryrepresentations. It seems likely that there wasa consensus that the description of all unitaryrepresentations of the Lorentz group could not bedone in an explicit form and had to include difficult

Lecture at Collège de France, 1978.

considerations ofvon Neumann fac-tors. Doubtless thebiggest surprisefor the authorswas how explicitlyand simply all ir-reducible unitaryrepresentations ofthe Lorentz groupcan be described.

Bargmann alsoconsidered representations of the “real” Lorentzgroup SL(2;R) and discovered representationsof the discrete series, realized in holomor-phic functions on the disk. The central resultis the completeness of the constructed unitaryrepresentations, although Bargmann and Harish-Chandra proved only an infinitesimal version ofthe completeness. Bargmann also proved a cer-tain statement of the completeness of matrixcoefficients.

At the same time, Gelfand and Naimark [2]went much further, establishing essentially allthe principal concepts of the theory of unitaryrepresentations. They introduced characters ofirreducible unitary representations as distributions,showed that characters define representations up toequivalence, and computed explicitly the charactervalues on regular elements, thus obtaining an exactanalog of the classical Weyl character formula.The paper contains a complicated analytical proofof the fact that their list of irreducible unitaryrepresentations is complete; however, the focus ofthe work is an analog of the Plancherel formula.This provides the (continuous) decomposition intoirreducibles of the two-sided regular representationon theL2 space of the group relative to the invariantmeasure.

It turned out that in this decomposition onlysome unitary representations appear; they werecalled representatives of the principal series (therest of the unitary representations make up thecomplementary series). It is remarkable that thePlancherel measure, which appears in the expres-sion of a norm of a function on a group through thenorms of its projections on irreducible components,was computed explicitly. However, this remarkableformula, reminiscent of the classical Weyl formulafor finite-dimensional representations, was a resultof almost ten pages of dense computations withoutany visible attempts to uncover the conceptualstructure. Later this was pointed out by Mautnerwhile refereeing a more conceptual proof found byHarish-Chandra. This fitted the style of Gelfand,who at the beginning was more concerned with thebeauty and clarity of the final result than with asimplification of the proof.


Ph

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Talking with René Thom atIHÉS, 1974.

However, it is typical ofGelfand that years after-wards he persistently tried toclarify the situation. Alreadyat the time of the publica-tion of [2] several notes hadappeared announcing gen-eralizations to the case ofthe group SL(n;C). In 1950a big book of Gelfand andNaimark appeared dedicatedto unitary representations ofthe classical groups (Gelfandliked to call the book “Blue”because of the color of thecover, and the paper aboutn = 2 “Red”). This book is a re-sult of an incredible amountof work, which to me appearsto be nothing short of heroic.The style is clear: derivationof explicit formulas through

intense analytic attacks, overcoming many obsta-cles. The proof of an analog of the Plancherelformula, the peak of the theory, especially standsout. The authors were unable to generalize analready difficult proof from the paper about theLorentz group; the proof given in the book isexceptionally difficult and contains many bril-liant inventions. Gelfand especially prized thecalculations in a certain generalization of ellipticcoordinates.

Around 1950 significant changes took placein Gelfand’s mathematical life. First of all, heconcluded his extraordinarily fruitful collabora-tion with Naimark. More than that, he decisivelychanged the organization of his work. If before,he concentrated on some one direction (Banachspaces, Banach algebras, theory of representations)at every moment, now he worked simultaneouslywith several coauthors on problems from verydifferent areas. He did not abandon the theory ofrepresentations, and for many years Graev becamehis primary collaborator. Soon Harish-Chandragave a conceptual proof of the Plancherel formulafor all complex semisimple Lie groups and alsofor SL(2;R). However, something kept Gelfandfrom being satisfied, and from time to time hecontinued returning to the Plancherel formula. In1953 he and Graev offered an exceptionally elegantway to derive it through application of the resultof M. Riesz about regularization of powers of aquadratic form.

Gelfand kept coming back to his proof of thePlancherel formula for the Lorentz group, whichhad not been generalized to other groups andin which he felt something important was leftnot understood. In 1958 suddenly he saw in

this proof something that had remained hiddenfor almost ten years, namely, that a large partof the proof was dedicated to the solution ofthe following elementary-sounding problem ofgeometrical analysis: Consider a function f (α,β, δ)of three complex variables. Integrate this function(in the real sense) on all complex lines intersectingthe hyperbola α = λ, δ = λ−1, β = 0, whereλ ∈ C. Then reconstruct f through all of theseintegrals. What connection do these lines have tothe Lorentz group SL(2;C)? Consider the set ofhorocycles in this group: the two-sided shifts ofthe unipotent subgroup N of unit upper triangular

matrices(

1 u0 1

)with u ∈ C. This set turns out

to be exactly the three-parameter family (on C)of all (complex) lines in the group, considered asa hyperboloid αδ − βγ = 1 in C4. If we projectthe hyperboloid onto the plane with coordinates(α,β, δ), then almost all the horocycles (excludingthose that project into points) transform into linesintersecting the hyperbola. The solution to theinversion problem in [1] is a very simple formula,reminiscent of the Radon inversion formula. Forthe inverse of the generalized Fourier transformon the group (Plancherel formula), it remainsonly to invert the usual Mellin transform. Afterthis interpretation, it was straightforward forGelfand and Graev to define horospheres andthe horospherical transform in the general case.The authors of [2] apparently did not know ofthis geometrical interpretation, but looking at thepaper, one can’t stop marveling at how closely theyfollowed it in their calculations (the correspondingtext about this problem of integral geometry in thefifth volume of Generalized Functions [4] differsfrom the text of the 1947 paper only in the additionof certain words without any substantial changesto any of the formulas).

However, at this point the romantic side ofthe story concludes and prosaic everyday mathe-matical problems begin. There is no doubt aboutthe beauty of the horospherical transform and itssignificance, but it is fair to raise the question: whatdoes it contribute to the theory of representations?From the very beginning, Gelfand had the ideathat there have to be direct methods to invertcertain generalized Radon transforms, includingthe horospherical transforms, and there had to bea more natural way to derive Plancherel formulas.However, in 1959 such a way was not uncovered,and in [1] the situation is essentially the opposite:the formula for the inversion of the horosphericaltransform for complex semisimple Lie groups andcertain homogeneous spaces was derived fromthe already-known Plancherel formula. A remark-able formula was produced, which resembled thethe Radon inversion formula in odd-dimensionalspaces but which also showed which modifications


are required for spaces of rank greater than 1: oneneeded to average (over the set of horospherespassing through a point) the result of applying anexplicit differential operator (of the order equalto the rank) that acts along parallel horospheres.It remained to develop direct methods of integralgeometry and to understand what advantagesthis approach had to harmonic analysis. This wasaccomplished, partially, only ten years later [5].

Even so, some important supporting obser-vations slowly accumulated. Gelfand noticedin the very beginning that the symmetric Rie-mannian space for the group SL(2;C) is thethree-dimensional hyperbolic space, with horo-spheres in the classical sense. It turned out thatthe inversion formula for this horospherical trans-form, with the appropriate definitions, exactlymatches the formula for the inversion of the Radontransform in three-dimensional Euclidean space.By contrast, the Plancherel formulas for the Fouriertransform in Euclidean and hyperbolic spacesdiffer dramatically. In a sense, the horosphericaltransform is independent of curvature! There wasno doubt of the importance of this observation, butfor a surprisingly long time there were no attemptsto incorporate this into some general result. Todayit is clear that if one considers for a Riemanniansymmetric space of negative curvature its “flat”twin (sending curvature to zero), then the inversionformulas for the horospherical transforms willbe identical for these two spaces. It seems to methat this explains the nature of the simple explicitformulas for the representations of semisimpleLie groups (which surprised their creators): fromthe point of view of the horospherical approach,the problems turn out to be equivalent to flat ones.There exists another important circumstance whichdoubtlessly was tacitly understood, even thoughI’ve never encountered a discussion of it. Namely,in the standard approach to the Plancherel formula,integrals on conjugacy classes are regularized soas to be meaningful as “integrals” on singularconjugacy classes, eventually, on the unit class.Horospheres, in a certain sense, are generators ofconjugacy classes; bringing them into the analysisilluminates the consideration of singular classes.

Integral Geometry of Lines and Curves

The biggest success in the following period wasconnected with the interpretation of horocycles onthe Lorentz group as lines intersecting a hyperbola.Everything began with a natural question: Whathappens if the hyperbola is replaced by anothercurve? Kirillov proved that under such a replace-ment the inversion formula remains unchanged.This meant something very important: with areplacement of the hyperbola by another curve,the group disappeared, and it became clear that a

significant part of the theory of representationsof the Lorentz group is connected not with thegroup structure, but rather with some sort ofmore general geometrical structure. This was thefirst supporting evidence for Gelfand’s project toembed the theory of representations into a widerarea of geometrical analysis.

The next step was again evident: understandthe nature of the condition on a family of lines “tointersect a fixed curve”. The family of all lines in C3

depends on four (complex) parameters. For integralgeometry it is natural to consider families withthree parameters (complexes of lines in classicalterminology), because then integration transformsa function of three variables into a function ofthree variables. For which complexes of lines arethere inversion formulas of Radon’s type? Gelfandand Graev [4] showed that this is possible not onlyfor complexes of lines intersecting a fixed curvebut also for complexes of lines tangent to a fixedsurface and not for any others. They called suchcomplexes admissible. It is worthwhile to say a fewwords about the proof of this astounding result.

F. John discovered that in the real case theimage of the operator of integration along linesin a three-dimensional space is described bythe ultrahyperbolic differential equation. In thecomplex case one has to consider the holomorphicand antiholomorphic versions of this differentialequation. It turns out that existence of the requiredinversion formula is equivalent to solvability of theGoursat problem and therefore coincides with thecharacteristic condition for this operator. This is anonlinear equation that can be integrated using themethod of Hamilton-Jacobi. The consideration ofbicharacteristics gives the description of admissiblecomplexes. These complexes had already appearedin classical differential geometry. They are, in anatural sense, maximally degenerate: the lines ofthe complex that intersect one fixed line of thecomplex lie (infinitesimally) in one plane.

This result is generalized to complexes of linesin general position in spaces of any dimension. Thenatural next step is to consider the possibility ofthe replacement of lines by curves. The final resultin this direction was obtained by Bernstein and me[12] (after Gelfand, Shapiro, and I previously founda universal structure of inversion formulas forcomplexes of curves [6]). Admissible complexes ofcurves turned out to be exactly infinitesimally fullfamilies of rational curves (infinitesimally, they arespaces of all sections of a certain vector bundleon the projected line). Here the most significantcondition is for the curves to be simultaneouslyrational. This is a quite effective condition, allow-ing the construction of many explicit examplesof families of curves with inversion formulas ofRadon type, for example, curves of the second


order. In these constructions it is natural to giveup the requirement that the family of curves be acomplex (the number of parameters of the familyequals the dimension of the manifold). In thiscase, the possibility exists to consider a certaingeneralization of a problem of integral geometry,but more instructive are the possibilities of applica-tions beyond the theory of representations. Thereexists, for example, a connection with the twistortheory of Penrose: the four-parameter admissi-ble family of curves corresponds to conformalright-flat four-metrics (the self-dual part of Weylcurvature equals zero). This allowed us to developa method of construction of explicit solutions ofthe Einstein self-dual equation. It seems to me that,in the case of curves, integral geometry has beensuccessfully fully developed. It appears to be somepart of nonlinear analysis, with a focus on explicitintegrable problems, and it is connected with themethod of the inverse problem for the case of onespectral parameter.

Complexes of Planes

The situation with submanifolds of dimensionhigher than one is significantly more complicated,and there may be no opportunity in this casefor work to advance as far. It appears that it issimilar to the possibility of the integrability ininverse problems with several spectral parameters.However, a few essential results have been achieved.Let us recall that beginning with the first workon integral geometry [1], the grand challengewas to find a proof of the Plancherel formulathrough inversion of the horospherical transform.Ten years passed before Gelfand, Graev, andShapiro did this [6], [5] for the group SL(n;C).We have already discussed the fact that whenn = 2, horospheres can be considered as lines. Forarbitrary n, horospheres for this group can also beconsidered as planes of dimensionk = n(n−1)/2 inCN , whereN = n2−1. The dimension of this familyof planes coincides with N, so we have a complexof k-planes. The key idea is to study the problemof reconstructing a function through its integralson planes from an arbitrary complex, not limitingourselves to a complex of horospheres. Let usnote that the problem of reconstructing a functionthrough its integrals on all planes is extremelyoverdetermined and the transition to complexes isa natural way to make the problem well defined.It turns out that for reconstruction at a point itis always possible to write a formula reminiscentof the Radon inversion formula. This formulainvolves averaging a certain explicit differentialoperator (of order 2k) along the set of planes ofthe complex which pass through a point. Thisoperator is defined on the set of all k-planes;however, we can compute it using its restriction to

the set of planes of the complex only under verystrong conditions. Let us call complexes satisfyingthese conditions admissible. There again arisesa certain condition involving characteristics. Wecan describe the situation slightly differently. Theimage of the operator of integration over k-planesis described via a certain system of differentialequations of the second order, generalizing theJohn equation, and admissible complexes must becharacteristic for this system. In some sense, theadmissible complexes are maximally degenerate.

The nonlinear problem of the description ofadmissible complexes is unlikely to be integrablefor k > 1. However, Gelfand and Graev [5] checkeddirectly that the complex of horospheres inSL(n;C)is admissible. As a result, we get an inversion ofthe horospherical transform and as a consequencethe Plancherel formula. I am certain that there issomething very significant in the degeneracy of themanifold of horospheres. I think that the geometricbackground of the theory of representations ofsemisimple Lie groups lies in this phenomenon.Clearly, this result makes substantial use of thepossibility of considering horospheres on SL(n;C)as planes. For other semisimple Lie groups thisis impossible, but I showed that the constructiondescribed above may be generalized for thesegroups by considering admissible complexes ofarbitrary submanifolds [8].

Nonlocal Problems and Integral Geometry forDiscrete Series

In the case of complex groups the structure ofinversion formulas with averaging of a differentialoperator means that, in particular, these formulasare local: for reconstruction of the function at apoint, it is sufficient to know the integrals on thehorospheres close to this point. In the case ofthe Radon transform, this is the case only in odd-dimensional spaces. In even-dimensional spacesthe inversion formula is nonlocal: in this situationa pseudodifferential operator is averaged. If wewere to go from complex groups to real ones andtheir homogeneous spaces, the first new difficultywould be connected with the inevitable appearanceof nonlocal formulas. It is natural to begin withRiemannian symmetric spaces of noncompacttype X = G/K, where G is a real semisimple Liegroup and K is a maximally compact subgroupof G. Then the horospherical transform is alwaysinjective, and the inversion formula is local if andonly if the multiplicities of all roots are even. Inthis case the inversion formula can be derivedusing the method described above for complexgroups. However, the general case must involvenonlocal inversion formulas and requires new ideas.After Karpelevich and I computed the Plancherelmeasure on these spaces through the product


formula for the Harish-Chandra c-function, wefound [13] in this case the analog of the approachof Gelfand-Graev to complex groups in their firstpaper [1]: we computed the kernel of the inversehorospherical transform as the inverse Mellintransform of the Plancherel density. We made thiscalculation for classical groups. Later, Beerendsmade it in the general case. The next step shouldbe a direct derivation of these inversion formulasusing methods of geometric analysis (as in the caseof even multiplicities), but so far this has not beendone. We tried with Gelfand [9] to move in thisdirection, developing a certain symmetric analogof differential forms on real Grassmannians, whichdoubtless has its own interest. However, we didnot achieve significant progress in this problem. Itseems to me that today many points look clearer,and this problem looks realistically solvable.

Finally, we can begin discussing the mainobstacle to the development of the theory ofrepresentations on the basis of the horosphericaltransform, which was clear from the very beginning.For real semisimple Lie groups the horosphericaltransform as a rule has a kernel, which consistsof all of its representation series except for themaximally continuous ones. This is already truefor the group SL(2;R): if we decompose its regularrepresentation into the sum of subspaces Lc , Ldcorresponding to continuous and discrete seriesrespectively, then the kernel of the horosphericaltransform coincides with Ld and the image isisomorphic to Lc . There does not appear to bea way to invert the horospherical transform orto derive in this way the Plancherel formula. Itis easy to interpret this as a dramatic limitationof the power of integral geometry in the case ofreal groups, where discrete series play a centralrole. This appears to be the reason that Gelfand’sidea of applying integral geometry to the theory ofrepresentations has not evoked much enthusiasmamongst experts on representation theory.

I can be a witness to the fact that Gelfandnever believed that the area of applications ofhorospheres is bounded by continuous series.In his opinion it was necessary to understandwhat corresponds to horospheres in the case ofdiscrete series, and this is perfectly possible. Thisresonated with his faith in the aesthetic harmony ofmathematics. In his first talk, which it befell me tohear (in 1955), he said about von Neumann factorsof the second type (which did not have any knownapplications at that time): “Such beauty must notvanish!” Only in one case—an imaginary three-dimensional hyperbolic space—did Gelfand andGraev manage [7] to find an appropriate expansion:the discrete series there was connected withcertain degenerate horospheres. However, this was

connected with certain very special circumstances,and there was no chance of direct generalization.

Budapest swimming pool,1966. First row, P. Dirac,I. Gelfand; second row,B. Bollobas, M. Arato.

The case of SL(2;R) re-mained the first call toaction. In 1977 Gelfand andI attempted to understandit [10]. The philosophy ofour approach was the factthat the Plancherel formulahas to be derived in twostages. First, one has to findprojections into subspacescorresponding to the repre-sentation series. Series ofrepresentations correspondto equivalence classes ofCartan subgroups. The de-composition of each seriesinto irreducible subspacesis reduced to a commu-tative Fourier transform(continuous or discrete)corresponding to the asso-ciated Cartan subgroup. Sothe first stage is the princi-pal one. Two of the main connected problems arethe finding of projections into the series and theinternal analytic characterization of subspaces forthe series. We solved these problems for SL(2;R).Subspaces corresponding to holomorphic and an-tiholomorphic discrete series can be characterizedas boundary values on SL(2;R) of holomorphicfunctions in certain tube domains in the complexgroup SL(2;C). Projections may be interpreted ascertain analogs of the Cauchy integral formula.I remember how happy Gelfand was with thisresult. He often said that new, significant thingsin representations have to be already nontrivialfor SL(2). We expected that in the general case aswell the series would be connected with certaintube domains, which may not be Stein manifolds,and then it would be required to consider ∂cohomology on these domains. Later this wascalled the Gelfand-Gindikin program. Significantprogress was achieved in this program, but onlyfor holomorphic discrete series.

It could have been expected that the projectionsinto the series had to be somehow connectedwith integral geometry, but at the time it provedimpossible to find an appropriate generalization ofthe horospheres. Much later I discovered a certainnatural construction [14]. On G = SL(2;C) thereare two classes of horospheres, each of which al-lows the possibility of constructing a horosphericaltransform. We have discussed the one-dimensionalhorocycles. However, it is also possible to considertwo-dimensional horospheres—orbits of maximalunipotent groups in SL(2;C)× SL(2;C) under the


two-sided action. Their geometrical characteriza-tion consists of sections of the group, considered asa hyperboloid, by isotropic planes. Horocycles arelinear generators of two-dimensional horospheres;for this reason, both versions of the horosphericaltransform are equivalent. For SL(2;R) both trans-forms have kernels. The idea is that since there arenot enough real horospheres, we must considercertain complex ones.

Let us consider those complex horosphereswhich do not intersect the real subgroup SL(2;R).There are three types of such horospheres. Insteadof integration on real horospheres, we consider theconvolution (on the real group) of Cauchy kernelswith singularities on the complex horosphereswithout real points. As a result, the horosphericaltransform of their three components is defined.It already does not have a kernel, and images ofthe various components can be decomposed usingrepresentations of various series. The inversion ofthe horospherical representation gives projectorsonto series and then also the Plancherel formula.At the same time, the continuation of functionsfrom the continuous series is obtained as aone-dimensional ∂-cohomology in a certain tubedomain, in agreement with our hypothesis. Itis my hope that the correct development ofthe understanding of complex horospheres forreal semisimple Lie groups is sufficient for theconstruction of the integral-geometrical equivalentof the theory of representations. Interestingly, thiscan be done even for compact Lie groups, for whichthere are no real horospheres at all.

Gelfand (in collaboration with Graev and me)worked on many other aspects of integral geometry.I have not discussed these results because Iwanted to concentrate here on integral geometryconnecting with the theory of representations,which seems to me to be the most important partof this project. It is intriguing that Gelfand wasinterested in the Radon transform long before itsconnection with the theory of representations wasdiscovered. He liked to offer problems concerningthe Radon transform to students, though he neverworked in this direction himself. This was almosta premonition of future connections, an abilityoften shown by exceptional mathematicians andtypical for him. Integral geometry was not oneof his most successful projects. It did not gainhim broad recognition or a multitude of followers.Yet we find in it, in a very pronounced way, hisinimitable approach to mathematics. Like everyother mathematician lucky enough to have workedwith Gelfand, I have gotten used to trusting hisextraordinary intuition, and I am certain that thestory is not over. Perhaps I will see the realizationof his fantastic project yet.

References[1] I. M. Gelfand, M. I. Graev, Geometry of homogeneous

spaces, representations of groups in homogeneousspaces and related questions of integral geometry,Amer. Math. Soc. Transl. (2), 37 (1964), pp. 351–429.

[2] I. M. Gelfand, M. A. Naimark, Unitary representationsof the Lorentz group, Izv. Acad. Nauk SSSR, Ser. Mat.,11 (1947), pp. 411–504.

[3] I. M. Gelfand, Integral geometry and its relation tothe theory of group representations, Russ. Math. Surv.,15 (1960), no. 2, pp. 143–151.

[4] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Gen-eralized Functions, Vol. 5. Integral Geometry andRepresentation Theory, Academic Press, 1966.

[5] I. M. Gelfand, M. I. Graev, Complexes of k-dimensional planes in the space Cn and Plancherel’sformula for the group GL(n,C), Soviet Math. Dokl., 9(1968), pp. 394–398.

[6] I. M. Gelfand, M. I. Graev, Z. Ya. Shapiro, Integralgeometry on k-dimensional planes, Funct. Anal. Appl.,1 (1967), no. 1, pp. 14–27.

[7] I. M. Gelfand, M. I. Graev, An application of the horo-spheres method to the spectral analysis of functionsin real and imaginary Lobachevsky space, Trudi Mosk.Math. Soc., 11 (1962), pp. 243–308.

[8] S. G. Gindikin, Integral geometry on symmetric man-ifolds, Amer. Math. Soc. Transl. (2), 148 (1991), pp.29–37.

[9] I. M. Gelfand, S. G. Gindikin, Nonlocal inversion for-mulas in real integral geometry, Funct. Anal. Appl., 11(1977), pp. 173–179.

[10] , Complex manifolds whose skeletons aresemisimple real Lie groups and holomorphic discreteseries of representations, Funct. Anal. Appl., 11 (1977),pp. 258–265.

[11] I. M. Gelfand, S. G. Gindikin, Z. Ya. Shapiro, Thelocal problem of integral geometry in the space ofcurves, Funct. Anal. Appl., 13 (1979), pp. 87–102.

[12] J. Bernstein, S. Gindikin, Notes on integral geometryfor manifolds of curves, Amer. Math. Soc. Transl. (2),210 (2003), pp. 57–80.

[13] S. G. Gindikin, F. I. Karpelevich, A problem ofintegral geometry, Selecta Math. Sov., 1 (1981), pp.160–184.

[14] S. Gindikin, Integral geometry on SL(2;R), Math. Res.Lett., 7 (2000), pp. 417–432.

Peter Lax

I. M. Gelfand

Like everyone else, I first heard Gelfand’s nameas the author of the famous theorem on maximalideals in commutative Banach algebras and itsapplication to prove Wiener’s theorem aboutfunctions with absolutely convergent Fourier series.The following story describes the reception of thisresult in the U.S.

Shortly after its publication, Ralph Phillipspresented this result at Harvard. It created such a

Peter Lax is professor emeritus at the Courant Insti-tute of Mathematical Sciences. His email address [email protected].


stir that he was asked to repeat the lecture, thistime to the whole faculty. And then he was askedto present it a third time to G. D. Birkhoff alone.

Other early basic results were the Gelfand for-mula for the spectral radius and the Gelfand-Levitaninverse spectral theory for ordinary differentialoperators. If you have a completely integrablesystem, you are supposed to be able to integrate itcompletely. For the KdV equation, the landmarkresult of Gelfand and Levitan on the inverse spec-tral problem for second-order ODEs turned out tobe the key to perform the integration.

It is a measure of Gelfand’s lifetime achieve-ments that these spectacular early results areviewed today as merely a small part of his totalwork.

Here is a story that sheds light on Gelfand’smodesty. I was one of three members of a committeeto award the first Wolf Prize. We all agreed thatthe prize should be given to the greatest livingmathematician. That person, in my opinion, wasCarl Ludwig Siegel, but another member of thecommittee insisted that it was Israel MoiseevichGelfand. So we argued back and forth fruitlesslyuntil we decided to divide the prize. Some timelater Gelfand remarked in a chance conversation,“It was a great honor for me to share a prize withSiegel.”

The world shall not see the like of Israel M.Gelfand for a long, long time.

Andrei Zelevinsky

Remembering I. M. Gelfand

On December 6, 2009, two months after I. M.Gelfand passed away, Rutgers University, his lastplace of work, held a Gelfand Memorial. Thisevent brought together the crème de la crème ofseveral generations of Moscow mathematicians(now scattered all around the world) whose lifein mathematics was to a large extent shapedby I. M.’s influence. Despite the sad occasion, itwas a pleasure to see many old friends and toshare memories of our student years when we allattended the famous Gelfand Seminar at MoscowState University. And of course to share storiesabout I. M. He was such a huge presence in somany lives, and his passing left a gap which willbe impossible to fill.

I first met I. M. in the early fall of 1970. Themeeting was arranged by Victor Gutenmacher, whoat the time worked at the School by Correspondenceorganized by I. M. (with the purpose of bringing

Andrei Zelevinsky is professor of mathematics at Northeast-ern University. His email address is [email protected].

Zelevinsky is grateful to his daughter, Katya, for helpfuleditorial suggestions.

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Math in the kitchen, 1982.

mathematics into the livesof schoolchildren all aroundthe Soviet Union). I. M., whoas usual was simultaneouslyinvolved in a myriad ofvarious projects both sci-entific and pedagogical, haddecided to organize and runa special mathematical classof seventh-graders withinthe famous Moscow SchoolNo. 2, which already hada decade-long tradition ofsuch classes. To help himrun this program, I. M. askedVictor to find him severalyoung assistants who hadthemselves passed throughsuch a class during their school years. I was veryfortunate to become one of four such assistants.(Another was my old friend and classmate sincethe seventh grade Borya Feigin, who is now adistinguished mathematician.) All four of us hadgraduated a year earlier from the same SchoolNo. 2, all were math undergraduates beginning oursecond year at Moscow State University, and allfelt a little lost in our mathematical studies.

The organizational matters could have beenresolved within a few minutes, but our firstmeeting with I. M. lasted for several hours. Thefour of us (plus poor Victor) walked with I. M.for hours, and he talked to us about all kindsof things, mathematical and not. He asked uswhat we most loved about mathematics and whatseminars and elective courses we had attendedduring our freshman year. Of course, he declaredthat we had done everything wrong and werealmost lost for mathematics, but there was stillsome little hope for us if we started to attend hisseminar at once. He explained to us how to studya new mathematical subject: focus on the mostbasic things at the foundation and dwell uponthem until you reach full understanding; then thetechnicalities of the subject would be understoodvery quickly and effortlessly. I remember vividlyhow he illustrated this by explaining to us thefoundations of linear algebra: a subspace in avector space is characterized by one integer, itsdimension; a pair of subspaces is characterized bythree integers (dimensions of the two subspacesand of their intersection). What about a triple ofsubspaces? and what about a quadruple?6

6Triples of subspaces demonstrate the limitations of afruitful analogy between subspaces of a finite-dimensionalvector space and subsets of a finite set: the subspace lattice isonly modular but not distributive. Thinking about this anal-ogy led me to my first two published notes, which appearedwithin the next couple of years. As for the quadruples, as I


This meeting was definitely one of my most life-changing experiences. I have never met a personwith such personal magnetism and such an abilityto ignite enthusiasm about mathematics as I. M.I remember returning home late in the evening,totally exhausted but happy, with the definitefeeling that my mathematical fate had been sealedthat day. I could not sleep and spent most ofthe night thinking about mysteries of triples ofsubspaces! I attended the Gelfand Seminar foralmost twenty years. A lot has been said about itsunique character, including horror stories aboutI. M.’s rough treatment of speakers and participants.I had my share of humbling experiences there, bothas a speaker and as a “control listener” who wassent by I. M. to the blackboard in the middle of a talkto explain what the speaker was trying to say. Manypeople, including some excellent mathematicians,could not stand this style and stopped attendingthe seminar. Those who stayed (myself included)decided that such a great learning experiencewas worth a little suffering. Equally important,or maybe even more important than the talksthemselves, was Gelfand’s choice of topics; hiscomments and monologues, which often deviateda lot from the original topic; and of course hisfamous “pedagogical” jokes and stories.

The official starting time of the seminar was7 p.m. (or was it 6:30?) on Mondays, but it almostalways started with much delay, sometimes afterup to two hours! I am sure I. M. did this on purpose,because these weekly get-togethers before theseminar with numerous friends coming to theuniversity from all around the city were also a bigattraction. Sometimes even after his arrival, I. M.did not immediately start the seminar but stayedin the corridor for some time and chatted withpeople just like everybody else.

I think of myself as I. M.’s “mathematicalgrandson”: my first real teacher and de facto Ph.D.advisor was Joseph Bernstein, one of Gelfand’s beststudents.7 However, at some point after Joseph’semigration (in 1980, I believe), I. M. approachedme and suggested that we start working together.Our close mathematical collaboration lasted abouta decade (from the first joint note in 1984 to our

realized much later, I. M. had just finished his remarkablepaper with V. A. Ponomarev, “Problems of linear algebraand classification of quadruples of subspaces in a finite-dimensional vector space”, so he was talking to us about hisown cutting-edge research!7Joseph could not be my official advisor, since he was neveraffiliated with the mathematics department at Moscow State,which was also the case for many other first-rate mathe-maticians from the “Gelfand circle”. A. A. Kirillov Sr., alsoa student of Gelfand, kindly agreed to serve as my officialadvisor. Of course, I also learned a lot from him and fromattending his seminar.

1994 book with Misha Kapranov). It would takevolumes to give a comprehensive account of I. M.’smathematical contributions, so let me just sharemy personal impressions of some of his uniquefeatures as a mathematician and as a teacher.8

Working with him could be a very frustratingexperience. Just looking at his incredibly prolificscientific output, one would imagine him as a modelof efficiency, never wasting a moment of his time.But most of the time during our daily meetingswas filled with numerous distractions, jumps fromone topic to another, his long phone conversationswith an amazing variety of people on an amazingvariety of topics, etc. Quite often after long hoursspent like this I felt totally exhausted and hada depressing feeling that we had just wasted aperfectly good working day. But almost withoutexception there came a moment (sometimes whenI was already saying good-bye at the door) ofhis total concentration on our project that led toextremely rapid progress, completely justifyingall the torturous hours leading to this moment.It seemed as if his subconscious mind neverstopped working on our project (and probably ona multitude of other things at the same time), andit just took I. M. a long time to become ready tospell out the results of this work.

This “nonlinearity” of I. M.’s thinking processwas also one of the many features that made hisseminar so unique. He would spend an inordinateamount of time asking everybody to explain to himsome basic definitions and facts, and just whenmost of the participants (starting with the speaker,of course) would get totally frustrated, I. M. wouldsuddenly switch gears and say something veryilluminating, making it all worthwhile.9

I have never met any other mathematician withsuch an ability to see the “big picture” and alwaysgo to the heart of the matter, ignoring unnecessarytechnicalities. He had an uncanny ability to askthe “right” questions and to find unexpectedconnections between different mathematical fields.I. M. was fully aware of this gift and liked toillustrate it by one of his numerous “pedagogical”stories: “An old plumber comes to repair a heater.He goes around it, thinks for a moment, and hitsit once with a hammer. The heater immediatelystarts working. The plumber charges 200 rublesfor his service. The owner says, ‘But you just spenttwo minutes and didn’t do anything.’ The plumberreplies, ‘I am charging you 3 rubles for hitting your

8For I. M. these two occupations were inseparable. His wayof doing mathematics always involved close personal inter-action with his innumerable students and collaborators.9I. M. often interrupted speakers with the words “May I aska stupid question?” The best reply to this was given by Y. I.Manin during one of his rare appearances as a speaker: “No,I. M., I don’t think you are capable of such a thing!”


heater with a hammer and the rest for knowingwhere to hit, which took me forty years to learn!’10

I. M. had such an enormous wealth of ideas andplans that he needed many collaborators to helphim realize even a small part of them. I was alwaysvery impressed with his great intuitive grasp ofpeople. A few minutes of penetrating questioningof a new acquaintance allowed him to take thefull measure of a person, understand his or herscientific potential, strengths and weaknesses, evenhow much pressure the person could withstand.In my own case, it seems that I. M. detected mysoft spot for algebraic combinatorics (quite rarein Moscow at that time, I must say) even beforeI realized it myself. Very soon after I joined hisseminar, he asked me to study (and then givea talk on) the old book The Theory of GroupCharacters and Matrix Representations of Groupsby D. E. Littlewood, where the representation theorywas treated with a strong combinatorial flavor.This was truly a sniper’s shot: the facts and ideasthat I learned from this book continue to serve meto this day. Another truly inspired suggestion byI. M. was to bring together Borya Feigin and DmitryBorisovich Fuchs, which led to many years of veryfruitful collaboration. Like so many of my friendsand colleagues, I feel very fortunate for havingknown Israel Moiseevich and for being given achance to be close to this gigantic, complex, wise,inspiring, and infinitely fascinating personality.

10I. M.’s amazing record of initiating new fruitful directionsof mathematical research provides plenty of examples of“knowing where to hit.” Let me just mention one example: thetheory of general hypergeometric functions initiated by I. M.(where a significant part of my collaboration with him tookplace) grew from his insight that hypergeometric functionsshould live on the Grassmannians.

Enhancement and Partnership ProgramThe Clay Mathematics Institute invites proposals under its new program, “Enhancement and Partnership”. The aim is to enhance activities that are already planned, particu-larly by funding international participation. The program is broadly defined, but subject to general principles:

• CMIfundingwillbeusedinaccordancewiththeInstitute's mission and its status as an operating foundation to enhance mathematical activities organised by or planned in partnership with other organisations.

• Itwillnotbeusedtomeetexpensesthatcouldbereadily covered from local or national sources.

• AllproposalswillbejudgedbytheCMI'sScientificAdvisoryBoard.

Examples include:

• Fundingadistinguishedinternationalspeakeratalocal or regional meeting.

• Partnershipintheorganisationofconferencesandworkshops.

• Fundingashortvisitbyadistinguishedmathemati-cian to participate in a focused topical research program at an institute or university.

• Fundinginternationalparticipationinsummerschools (lecturers and students) or repeating a successful summer school in another country.

• Fundingaspeciallectureatasummerschoolorduring a research institute program.

• Fundinganextensionofstayinthehostcountryorneighbouringcountriesofaconferencespeaker.

Applications will only be received from institutions orfromorganisersofconferences,workshops,andsummerschools. In particular the CMI will not consider applica-tions under this program from individuals for funding to attend conferences or to visit other institutions or to support their personal research in other ways.

Enquiries about eligibility should be sent to [email protected] inabrief lettera description of the planned activity, the way in which this could be enhanced by the CMI, the existing fund-ing, the funds requested and the reason why they cannot beobtainedfromotherlocalornationalsources.Fundsrequested should not be out of proportion to those ob-tained from other sources. The CMI may request indepen-dent letters of support.

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israel moiseevich gelfand, part i

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