Israel Moiseevich Gelfand, Part I

Israel Moiseevich Gelfand, Part I

Vladimir Retakh, Coordinating Editor

Israel Moiseevich Gelfand, a age twelve Gelfand understood that some problems mathematician compared by in geometry cannot be solved algebraically and Henri Cartan to Poincaré and drew a table of ratios of the length of the chord to Hilbert, was born on Septem- the length of the arc. Much later it became clear ber 2, 1913, in the small town for him that in fact he was drawing trigonometric of Okny (later Red Okny) near tables. Odessa in the Ukraine and died From this period came his Mozartean style in New Brunswick, New Jersey, and his belief in the unity and harmony of USA, on October 5, 2009. mathematics (including applied mathematics)— Nobody guided Gelfand in the unity determined not by rigid and loudly his studies. He attended the proclaimed programs, but rather by invisible only school in town, and his and sometimes hidden ties connecting seemingly mathematics teacher could different areas. Gelfand described his school years offer him nothing except and mathematical studies in an interview published I. M. Gelfand encouragement—and this was in Quantum, a science magazine for high school very important. In Gelfand’s students [1]. In Gelfand’s own words: “It is my deep own words: “Offering encouragement is a teacher’s conviction that mathematical ability in most future most important job.” In 1923 the family moved professional mathematicians appears…when they to another place and Gelfand entered a vocational are 13 to 16 years old…. This period formed my school for chemistry lab technicians. However, style of doing mathematics. I studied different he was expelled in the ninth grade as a son of subjects, but the artistic form of mathematics that a “bourgeois element” (“netrudovoi element” in took root at this time became the basis of my taste Soviet parlance)—his father was a mill manager. in choosing problems that continue to attract me After that Gelfand (he was sixteen and a half at to this day. Without understanding this motivation, that time) decided to go to Moscow, where he had I think it is impossible to make heads or tails of some distant relatives. the seeming illogicality of my ways in working Until his move to Moscow in 1930, Gelfand lived and the choice of themes in my work. Because of in total mathematical isolation. The only books this motivating force, however, they actually come available to him were secondary school texts and together sequentially and logically.” several community college textbooks. The most ad- The interview also shows how a small provincial vanced of these books claimed that there are three boy was jumping over centuries in his mathematical kinds of functions: analytical, defined by formulas; discoveries. At the age of fifteen Gelfand learned empirical, defined by tables; and correlational. Like of a series for calculating the sine. He described Ramanujan, he was experimenting a lot. Around this moment in the “Quantum” interview: “Before this I thought there were two types of mathematics, Vladimir Retakh is professor of mathematics at Rutgers Uni- algebraic and geometric…When I discovered that versity. His email address is [email protected]. the sine can be expressed algebraically as a series, He thanks Mark Saul for his help in the preparation of this the barriers came tumbling down, and mathematics collection. became one. To this day I see various branches of The AMS wants to thank M. E. Bronstein, Elena Ermakova, mathematics, together with mathematical physics, Tatiana I. Gelfand, Tatiana V. Gelfand, and Carol Tate for as parts of a united whole.” their help with photographs for this article. After arriving in Moscow, Gelfand did not DOI: http://dx.doi.org/10.1090/noti937 have steady work and lived on earnings from

24 Notices of the AMS Volume 60, Number 1 occasional odd jobs. At some point he had the stabilized, anti-Semitism be- good fortune to work at the checkout counter at came part of the Soviet the State (Lenin’s) Library. This gave Gelfand a rare system, and Gelfand be- opportunity to talk with mathematics students came a full member of from Moscow University. He also started to attend the Soviet Academy only in university seminars, where he found himself under 1984 after being elected to intense psychological stress: new breezes were leading foreign academies. blowing in mathematics with the new demands In 1989 Gelfand moved for rigorous proofs. It was so different from his to the United States. Af- “homemade” experiments and his romantic views ter spending some time of mathematics. He also learned that none of at Harvard and MIT, he his discoveries were new. Neither this nor other became a professor at Rut- circumstances of his life deterred him, and his gers University, where he interest in mathematics continued to grow. worked until his death. Gelfand’s parents. Just as his abrupt expulsion from school, the The articles in the Notices next twist in Gelfand’s fate was also one of many present various descriptions paradoxes of life in the Soviet Union. On one hand, of Gelfand’s multifaceted re- as a son of a “bourgeois element” he could not be a search, his way of doing university student. On the other hand, at eighteen mathematics, and interaction he was able to obtain a teaching position at one with people. A leading Amer- of many newly created technical colleges and at ican expert once told me that nineteen to enter the Ph.D. program at Moscow after reading all definitions University. The reasons were simple: the Soviet in Gelfand’s papers, he could state needed knowledgeable instructors to educate easily prove all his theorems. its future engineers and scientists of the proper Well, this easiness was based “proletarian origin”. But at that time the system on long computations and a was not rigid enough to purge or even strictly thorough consideration of a regulate graduate schools. As a result, a talented variety of carefully selected boy was able to enter a Ph.D. program without a examples. Gelfand himself college or even a high school diploma. liked to repeat a statement At the beginning of his career Gelfand was by a Moscow mathematician: influenced by several Moscow mathematicians, “Gelfand cannot prove hard especially his thesis adviser, A. N. Kolmogorov. theorems. He just turns any Gelfand at age 3, 1916. In the Quantum interview Gelfand said that from theorem into an easy one.” Kolmogorov he learned “that a true mathematician Gelfand always was surrounded by numerous must be a philosopher of nature.” Another influence collaborators attracted by his legendary intuition was the brilliant L. G. Shnirelman. In 1935 Gelfand and the permanent flow of new ideas: in every defended his “candidate” (Ph.D.) thesis and in 1940 decade he was establishing a new area of research. obtained the higher degree of Doctor of Science. He had completely different approaches to different In 1933 he began teaching at Moscow University, people, as described by A. Vershik, A. Zelevinsky, where he became a full professor in 1943 and and me in our recollections. His students and started his influential seminar. He lost this position collaborators represent an unusual variety of styles temporarily in 1952 during the infamous “anti- and interests. The only similarity for members of cosmopolitans” (in fact, anti-Semitic) campaign but the Gelfand school was their inherited passion for was allowed to continue the seminar. Gelfand also mathematics. worked at the Steklov Institute and for many years One cannot write about Gelfand without men- at the Institute for Applied Mathematics. There tioning his legendary seminar, which started in he took part in the secret program related to the 1943 and continued until his death. Gelfand con- Soviet version of the Manhattan Project and its sidered the seminar one of his most important extensions. Andrei Sakharov mentioned his work creations. It is hard to describe the seminar in a few with Gelfand in [1]. words: it was a “mathematical stock exchange”, a In 1953 Gelfand was elected a Corresponding breeding ground for young scientists, a demonstra- Member of the Academy of Science (an important tion of how to think about mathematics, a one-man title in the Soviet hierarchy). This happened right show, and much more. It was not about functional after Stalin’s death and at the end of the anti- analysis or geometry, it was about mathematics. cosmopolitans campaign. According to Gelfand, Some would come to the seminar just to hear the political uncertainty of the times made his Gelfand’s jokes and paradoxes. For example, after election possible. Later the situation in the USSR his visit to the U.S. he stated, “[The] Mathematical

January 2013 Notices of the AMS 25 world is not a metric space: recollections by Sergei Tabachnikov in the second The distance from Harvard to part of this article). Gelfand founded, ran, and MIT is greater than the sum wrote several textbooks for the school. His uni- of distances from Harvard to versity textbooks Linear Algebra and Calculus of Moscow and from Moscow to MIT.” Variations (written with S. Fomin) also bear the One should add that sometimes imprint of his style and personality. Gelfand’s jokes were rather sharp, It is hard to describe all of Gelfand’s achieve- but for a young person to become ments in mathematics. He left his unique and a subject of Gelfand’s joke meant powerful imprint everywhere (excluding, probably, to be noticed, to be knighted. The mathematical logic). Some (but far from all) of seminar was also the right place to his breakthroughs are described here by Simon find out about fresh preprints com- Gindikin, David Kazhdan, Bertram Kostant, Peter ing from the inaccessible West. But Lax, Isadore Singer, and Anatoly Vershik. But Gelfand in 1934. most participants were attracted Gelfand’s interests spread far beyond pure and by the power and originality of applied mathematics. He left a number of papers Gelfand’s approach to mathematics. in biology, physiology, medicine, and other fields. He used the simplest grass-roots Gelfand was the first to obtain the Wolf Prize, examples, but he could turn them in 1978 (together with C. L. Siegel), and had many around in a totally unexpected way. other awards, including the Kyoto Prize (1989) and Gelfand always paid special at- the MacArthur Fellowship (1994). He was elected tention to students, who formed the to all leading academies and had honorary degrees majority of the seminar audience. from many universities. From time to time he would re- This collection of articles about Gelfand also peat: “My seminar is for high school contains an impression of the Gelfand seminar students, decent undergraduates, by a foreigner (Dusa McDuff), a look at the bright graduates, and outstanding Gelfand school from inside (Andrei Zelevinsky and professors.” One of the best descrip- Vladimir Retakh), a description of Gelfand’s School tions of the seminar was given by S. by Correspondence (Sergei Tabachnikov), and an Gindikin [3]. You may also see the essay, “Gelfand at 92” by Mark Saul. Gelfand, circa 1950. notes by Zelevinsky and me in this collection (my notes will appear in References the next issue). Dusa McDuff described the seminar [1] A talk with professor I. M. Gelfand, recorded by V. Re- impressions of a young foreigner. takh and A. Sosinsky, Kvant (1989), no. 1, 3–12 (in The seminar also served as a constant supply of Russian). English translation in: Quantum (1991), no. 1, Gelfand’s collaborators, who were already familiar 20–26. [2] A. Sakharov, Memoirs, Alfred Knopf, New York, 1990. with Gelfand’s style and his way of thinking. Their [3] S. Gindikin, Foreword to I. M. Gelfand Seminar, Ad- roles were quite different. Sometimes they would vances in Soviet Mathematics, vol. 16, Part 1, Amer. discuss specific examples, sometimes very vague Math. Soc., Providence, RI, 1993. ideas; sometimes they would bring their own suggestions that would be ridiculed, torn apart, I. M. Singer turned upside down, and then transformed into something exquisite. Only a few could bear the I. M. Gelfand task, but the pool of mathematicians in Moscow Israel Gelfand was one of the most influential was enormous. mathematicians of the twentieth century—I dare The seminar was a reflection of Gelfand’s passion say, the most outstanding in the last sixty years. to teach, as he tried to teach everyone and every- Unfortunately, our society neither understands where. Among his former students are F. Berezin, nor appreciates mathematics. Despite its many J. Bernstein, E. Dynkin, A. Goncharov, D. Kazh- applications, despite its intellectual power which dan, A. Kirillov, M. Kontsevich, and A. Zelevinsky. has changed the way we do science, mathematicians The number of his informal students is hard to are undervalued and ignored. Its practitioners, estimate. its leaders go unrecognized. They have neither From the early period of his life also comes power nor influence. Watching the negative effects Gelfand’s interest in education, especially in edu- cation for school students living far from research I. M. Singer is Emeritus Institute Professor at the Mas- centers. He was among the founders of the Moscow sachusetts Institute of Technology. His email address is Mathematical Olympiads and later organized his [email protected]. famous Mathematics School by Correspondence Adapted from: I. M. Singer, “Tribute to I. M. Gelfand”, for middle and high school students (see the Progress in Mathematics, vol. 132, Birkhäuser Boston, 1995.

26 Notices of the AMS Volume 60, Number 1 popularity causes in other fields and looking at the few superficial articles about mathematics, I think it is just as well. Faced constantly with problems we can’t solve, most mathematicians tend to be modest about themselves and their accomplishments. Perhaps that is why we have failed to recognize a giant in our midst. I won’t compare Gelfand with other outstanding mathematicians or scientists of the twentieth century; if I did, you would start checking for yourselves whether you agree with me. But focus on my point—we had a giant in our midst. I turn to other fields to find comparable achievements: After receiving honorary degree at Oxford in Balanchine in dance or Thomas Mann in literature 1973. Left with Gian-Carlo Rota. or Stravinsky, better still, Mozart in music, but for me, a better comparison is with artists like Cezanne and Matisse. I commend to you the great 10. Combinatorial characteristic classes (begin- poet Paul Rilke’s letter on Cezanne. He said, “Paul ning with MacPherson) Cezanne has been my supreme example, because 11. Dilogarithms, discriminants, hypergeometric he has remained in the innermost center of his functions work for forty years…which explains something 12. The Gelfand seminar beyond the freshness and purity of his paintings” It is impossible to review his enormous contri- (of course, much longer for Gelfand). butions in a short note. I will just comment on a Evoking Matisse is perhaps more apt. A Matisse few results that affected me. is breathtaking. No matter what his personal As a graduate student, one of the first strong circumstance, he turns to new frontiers with joy influences on me was Gelfand’s normed ring paper. and energy. Particularly outstanding is his later Marshall Stone had already taught us that points work: Jazz and the remarkable “papier-decouples”, could be recaptured in Boolean algebras as maximal efforts done in the early 1880s. ideals. But Gelfand combined analysis with algebra Gelfand always dazzled us with new and pro- in a simple and beautiful way. Using the maximal found ideas. One of his latest works, the book with ideal in a complex commutative Banach algebra, he Kapranov and Zelevinsky, is a major effort that represented such algebras as algebras of functions. maps out new directions for decades to come. Thus began the theory of commutative Banach In preparing this article, I asked many people for algebras. The spectral theorem and the Wiener’s topics I should emphasize. You will be interested Tauberian theorem were elementary consequences. in what happened. First, there was little inter- I was greatly influenced by the revolutionary view section in the subjects my correspondents chose. begun there. Second, everyone gave me a five- to twenty-minute A natural next step for Gelfand was the study of enthusiastic lecture on the essence of Gelfand’s noncommutative C*-algebras. He represented such contribution—simple and profound. algebras as operator algebras using the famous GNS Reviewing Gelfand’s contributions to mathemat- construction. It seemed inevitable to find unitary ics is an education. Let me remind you of some of representations of locally compact groups using his main work. their convolution algebras. The representation 1. Normed rings theory of complex and real semisimple Lie groups 2. C*-Algebras (with Raikov)—the GNS Con- followed quickly. What struck me most was the struction geometric approach Gelfand and his coworkers 3. Representations of complex and real semisim- took. Only recently it appears this subject has ple groups (with Naimark and Graev) become geometric again. 4. Integral Geometry—Generalizations of the In 1963 twenty American experts in PDE were Radon Transform on their way to Novosibirsk for the first visit of 5. Inverse scattering of Sturm-Liouville systems foreign scientists to the academic city there. It (with Levitan) was in the midst of Khrushchev’s thaw. When I 6. Gelfand-Dickey on Lax operators and KdV learned about it, I asked whether I could be added 7. The treatises on generalized functions to the list of visitors, citing the index theorem 8. Elliptic equations Atiyah and I had just proved. After reading his 9. The cohomology of infinite dimensional Lie early papers, I wanted to meet Gelfand. Each day of algebras (with Fuchs) my two-week stay in Novosibirsk I asked Gelfand’s

January 2013 Notices of the AMS 27 students when knew more about the construction of fountain he was com- pens than I ever cared to know and had ever ing. The response believed possible! Gelfand’s infinite curiosity and was always “to- focused energy on details were unbelievable; those, morrow”. Gelfand coupled with his profound intuition of essential never came. I features, are rare among human beings. He was sadly returned to beyond category. Moscow. When I Talking about Oxford, let me emphasize got to my room Gelfand’s paper on elliptic equations. In 1962 at the famous Ho- Atiyah and I had found the Dirac operator on tel Ukraine, the spin manifolds and already had the index formula telephone rang for geometric operators coupled to any vector and someone said bundle, although it took another nine months to With M. Atiyah, 1973. Gelfand wanted to prove our theorem. Gelfand’s paper was brought meet me, could to our attention by Smale. It enlarged our view I come down- considerably, as Gelfand always did, and we quickly stairs. There was realized, using essentially the Bott periodicity Gelfand. He in- theorem, that we could prove the index theorem vited Peter Lax and for any elliptic operator. me for a walk. I should also mention the application of During this walk, Gelfand’s work to physics: Gelfand-Fuchs, for Peter tried to tell example, on vector fields on the circle, the so- Gelfand about his called Virasoro algebra, which Virasoro did not in work on SL(2,R) fact define. Although I mentioned Gelfand-Dickey, with Ralph Phillips. I have to stress its influence on matrix model Gelfand tried to ex- theory. And I have to describe how encouraging plain his own view he was and how far ahead of his time he was in With I. M. Singer, 1973. of SL(2,R) to Pe- understanding the implication of a paper which ter, but his English seemed obscure at the time. was inadequate. (He was rusty; within two days his Claude Itzykson told me that his famous pa- English was fluent.) I interrupted and explained per with Brezin, Parisi, and Zuber that led to Gelfand’s program to Peter. At the corner Gelfand triangulating moduli space at the beginning went stopped, turned to me and said, “But you are my unnoticed by scientists. The authors then received student.” I replied, “Indeed, I am your student.” one request for a reprint—from Gelfand. (By the way, Gelfand told me he didn’t come to Ray and I were very excited about our definition Novosibirsk, because he hated long conferences.) of determinants for Laplacian-like operators and Although it was an honor to be Gelfand’s student, its use in obtaining manifold invariants-analytic it was also a burden. We tried to imitate the depth torsion. The early response in the United States was and unity Gelfand brought to mathematics. He silence; Gelfand sent us a congratulatory telegram. made us think harder than we believed possible. In conclusion, I want to mention one special Gelfand and I became close friends in a matter of quality of Gelfand. He was a magician. It is not very minutes, and we remained so ever since. I was ill difficult—not very difficult at all—for any of us in Moscow, and Gelfand took care of me. mere mortals to keep the difference in our ages a I didn’t see him again for ten years. He was constant function of time. But with Gelfand, when I scheduled to receive an honorary degree at Oxford, met him at his fiftieth, and in his sixtieth, I thought where I was visiting. It was unclear that he would be he was older than I. Ten years later, I felt that we allowed to leave the Soviet Union to visit the West. were the same age. Later it became clear to me I decided not to wait and returned home. A week that Gelfand was, in fact, much younger than most later, I received a telegram from Atiyah: Gelfand mathematicians. was coming—the queen had asked the Russian ambassador to intercede. I flew back to England and accompanied Gelfand during his visit—a glorious time. Many things stand out, but I’ll mention only one: our visit to a Parker fountain pen store. Those of you who have ever shopped with Gelfand will smile; it was always an unforgettable experience. Within fifteen minutes, he had every salesperson scrambling for different pens. Within an hour I

28 Notices of the AMS Volume 60, Number 1 David Kazhdan measures on G with compact support Works of I. Gelfand on the Theory of and on an appli- Representations cation of Gelfand’s theory of normed The theory of group representations was the center rings. of interest of I. Gelfand. I think this is related to the nature of this domain, which combines Next, in the analysis, algebra, and topology in a very intricate late forties, there fashion. But this richness of the representation was a stream of theory should not be taken as self-evident. To a papers (most of them jointly with great extent we owe this understanding to works of Photograph by Carol Tate. I. Gelfand, to his unique way of seeing mathematics M. Naimark) that On Raoul Bott’s porch, 1976. Left to as a unity of different points of view. developed the main right, I. Gelfand, R. MacPherson, In the late thirties, when Gelfand started his concepts of the rep- R. Bott, D. Kazhdan. mathematical career, the theory of representations resentation theory of compact groups and the general principles of of noncompact classical groups G. It would be a harmonic analysis on compact groups were well much simpler task to describe the concepts that understood due to works of Hermann Weyl. Har- appeared later than to describe the richness of this monic analysis on locally compact abelian groups work. was developed by Lev Pontryagin. The general Gelfand believed that the space Gˆ of irreducible structure of operator algebras was clarified by representations of G is a reasonable “classical” Murray and von Neumann. But the representation space. If I understand correctly, the first indication theory of noncompact noncommutative groups of the correctness of this intuition came from the was almost nonexistent. The only result I know of theory of spherical functions, developed in the is the work of Eugene Wigner on representations early forties but published only in 1950. Let K ⊂ G of the inhomogeneous Lorentz group. Wigner has be the maximal compact subgroup and Gˆ0 ⊂ Gˆ be shown that the study of physically interesting the subset of irreducible representations of class 1 irreducible representations of this group can be (that is, the representations (π, V ) of G such that K reduced to the study of irreducible representations V ≠ 0). Gelfand observed that the subset Gˆ0 is of its compact subgroups. equal to the set of irreducible representations of It was not at all clear whether the theory of the subalgebra of two-sided K-invariant functions representations of real semisimple noncompact on G, proved the commutativity of this algebra, groups is “good”, i.e., whether the set of irreducible and identified the space of its maximal ideals with representations could be parameterized by points the quotient LT /W where LT is the torus dual of a “reasonable” set, and whether the unitary to the maximal split torus T ⊂ G and W is the representations can be uniquely decomposed into Weyl group. The generalization of this approach irreducible ones. The conventional wisdom was developed in the fifties by Harish-Chandra and to expect that the beautiful theory of Murray-von Godement led to the proof of the uniqueness of Neumann factors is necessary for the description the decomposition of any representation of the of representations of real semisimple noncompact group G into irreducibles. groups. On the other hand, Gelfand, for whom Given such a nice classification of irreducible Gauss and Riemann were the heroes, expected representations of class 1, it was natural to that the theory of representations of such groups guess that the total space Gˆ is also an algebraic should possess classical beauty. variety. But for this purpose one had to find a Gelfand’s first result in 1942 with D. Raikov way to construct irreducible representations of in the theory of representations of groups is G. Gelfand introduced the notion of parabolic a proof of the existence of “sufficiently many” induction (for classical groups) and, in particular, unitary representations for any locally compact studied representations πξ of G induced from group G. In other words, any unitary irreducible a character ξ of a Borel subgroup B ⊂ G. He representation of G is a direct integral of irreducible showed that, for generic character ξ of T = B/U, ones. The proof of this result is based on the very the representation πξ is irreducible and that the important observation that the representation representations πξ, πξ0 are equivalent if and only theory of the group G is identical to the theory if the characters ξ, ξ0 of T are conjugate under of representations of the convolution algebra of the action of the Weyl group W . The proof is based on the decomposition G = S BwB for David Kazhdan is professor of mathematics at the Ein- w∈W stein Institute of Mathematics. His email address is classical groups. This decomposition was extended [email protected]. by Harish-Chandra to the case of an arbitrary

January 2013 Notices of the AMS 29 30

Photograph by Elena Ermakova. n onnpi oihv elztosi pcsof spaces in realizations have tori nonsplit correspond- to series ing the Moreover, tori. maximal of called group the SL(n, of representations irreducible generic many for years. theory representation the of influenced which development cases) particular answer in to only able them was (he questions of number a posed work. difficult prove for calculations—a to reward explicit great intricate and very groups by 1 complex equality classical of algebraic measure case beautiful Plancherel a the guess for expression to able was Gelfand the of of representation decomposition knowledge regular explicit the the and to equivalent exists), is it (if unique is (1) of measure a for distribution the defined by and class trace of is operator the function support, smooth compact any for to that was showed he Gelfand of characters idea define ingenious operator The the defined. of trace the group representations for But G salna obnto ftecharacters the of combination linear a as T .I on okwt .Gav efn classified Gelfand Graev, M. with work joint In 1. Gelfand works, of series this of continuation a As Naimark, M. with works joint of series a In measure a such that see to difficult not is It rπ( ) tr(π)(f uhthat such W R series G ). ivratmaue(aldthe (called measure -invariant hc r yial infinite-dimensional, typically are which , ) 0hbirthday. 60th hyfudthat found They µ X hc orsodt ojgc classes conjugacy to correspond which π ntespace the on δ : = ξ e f) (f = rπf)) tr(π(f Z tr(π) : = ξ Z ∈ L (g)π f 2 Xtr(π eiipegopadis and group semisimple h et function delta the write can one that show to sufficient is it complete, represen- of tations list see a to that Therefore, representa- tion. regular the of all G representations that known the well group is of compact case a the is In much missing? not to that how show But sentations. repre- irreducible many decomposition. Bruhat the as now known (G) π ) V (π, as o n ol look could one Now . X G hscntuto gives construction This ˆ r osiunsof constituents are distributions π(g) fuiaycharacters unitary of noirdcbeones. irreducible into sauino pieces of union a is ξ (g)dg ξ )µ fanoncompact a of π oie fteASVolume AMS the of Notices , a X , g . a ∈ ∈ Plancherel G A µ (g) f htis, That . X snot is , of , tr(π nthe in tr(π) δ G e with G µ µ on a is it X X ) . a nirdcbefiiedmninlrepresenta- finite-dimensional irreducible on was netnino hsrsl oabtaysc pairs such arbitrary to result this of extension An h rtpr ftecnetr a utfidby justified was conjecture the of part first The prily nltcfntos efn conjectured Gelfand functions. analytic (partially) .Atu n1983. in Arthur J. ersnainof representation groups for theorem SL Paley-Wiener the of analog the bundles). sections such in of realization the only considered Gelfand bundles vector holomorphic cohomologies is torus of there compact if only maximal and a of if exist realization (which the series discrete suggested who modified was Langlands, part by second The subgroups. Levi series of discrete from induced uni- representations on tary concentrated measure and Plancherel the groups found semisimple real for representations of series discrete constructed who Harish-Chandra, functions. analytic discrete groups of realize spaces semisimple to appropriate in possible real series be should all it space for that and the true of be description should analogous the that o ofidabssin basis a “good” find a to find how to words, other In how representations. these physics: of realization in interest influenced his partially by question, new a asked Gelfand representations such groups classical of tions polynomials). Koornwinder and Macdonald, Askey-Wilson, of interpretation theoretic repre- on sentation others and Sugitani, ex- Stokman, Noumi, Rosengren, for Koelink, Tsuchiya-Kanie, (see, of analogs papers quantum ample, their representations as or of groups interpreted traces of be or nineteenth coefficients can the matrix centuries in twentieth (almost) studied that and functions now clear special is all It functions. these equations for differential and functional immediately the functions explains special This of representations. interpretation irreducible Ja- matrix of as and coefficients appear functions polynomials, Legendre Whittaker and and cobi Bessel as such 2002). Congress, ICM the at Delorme P. H) (G, H group the of of decomposition representations the about question the asked and group of the representations of decomposition the structed to result by obtained this was generalization of This extension groups. other the the raised about and question support) compact with functions ⊂ 2 .I on okwt re,Gladconstructed Gelfand Graev, with work joint In 2. h etsre fGladswr wt .Tsetlin) M. (with work Gelfand’s of series next The functions, special many that showed Gelfand 4. con- Gelfand Graev, with work joint in Also 3. ( C G ) a civdol eety(e h akby talk the (see recently only achieved was stesto xdpit fa involution. an of points fixed of set the is and SL SL 2 ( 2 C ( ) R G ntespace the on ) H ta s eopsto fthe of decomposition a is, (that ntespace the on i (G/T (π V λ G λ c V , , h lsicto of classification The . T F hc losoeto one allows which G λ c ) ) F L ⊂ on fhomogeneous of 2 a nw,but known, was C (SL G on c ∞ L 60 ntespace the in ) (G) 2 (G/H) 2 G/T Number , ( C fsmooth of )/SL c (while where 2 ( R G 1 ˆ ) compute matrix coefficients of πλ(g), g ∈ G, in irreducible represen- these bases. Such a basis (Gelfand-Tsetlin basis) tations of groups was constructed for irreducible representations of over local fields groups SLn and SOn and became the core of many and number theory works in representation theory and combinatorics. was greatly clarified Gelfand and Graev found the expression for by Langlands. On the matrix coefficients of representations πλ in the other hand, a terms of discrete versions of Γ -functions. This generalization of al- realization of finite-dimensional representations gebraic formulas for has an important analog for infinite-dimensional the Mellin transform representations of groups over local fields. of characters and As part of the theory of finite-dimensional for the Plancherel representations, Gelfand studied the Clebsch- measure was never Gordan coefficients, which give a decomposition of found. tensor products of irreducible representations into In other work, irreducible components. He noticed (at least for Gelfand and Graev Talking at 90th birthday conference, G = SL2) that Clebsch-Gordan coefficients of G are found a description 2003. discrete analogs of Jacobi polynomials which are of the irreducible representations of the multiplica- matrix coefficients of irreducible representations tive group D∗ of quaternions over F as induced of G. Possibly an explanation of this can be given from 1-dimensional representations of appropriate by using the theory of quantum groups, where subgroups. This was of constructions of irreducible multiplication and comultiplication are almost representations of D∗ that were later generalized symmetric to each other. by R. Howe to other p-adic groups. The next series of Gelfand’s work was on The description of representations of the groups representations of groups over finite and local SL2(F) and GL2(F) for local fields is presented in fields F. The basic results here are the proof of the the book Generalized Functions, Volume 6 written uniqueness of a Whittaker vector, the existence with Graev and I. Piatetski-Shapiro. In the same of a Whittaker vector for cuspidal representations book, Gelfand developed the theory of representa- of GLn(F), the construction of an analog of the tions of semisimple adelic groups G(AK ) for global Gelfand-Tsetlin basis for such representations, fields K. He defined the cuspidal part and the description of cuspidal representations 2 2 L0(G(AK )/G(K)) ⊂ L (G(AK )/G(K)) of GLn(F) in terms of Γ -functions (joint work with M. Graev and D. Kazhdan). But I think of the space of automorphic forms, proved that 2 that his most important work in this area is the the representation of G(AK ) on L0(G(AK )/G(K)) complete description of irreducible representations is a direct sum of irreducible representations, and of groups SL2(F) and GL2(F) for local fields with developed a representation-theoretical interpreta- the residue characteristic different from 2 (joint tion of the theory of modular forms. The work of work with M. Graev and A. Kirillov). They showed Langlands is very much influenced by these results that irreducible representations of GL2(F) are of Gelfand. essentially parameterized by conjugacy classes It became clear that generic representations of of pairs (T , ξ) where T ⊂ GL2(F) is a maximal any semisimple Lie group or Lie algebra almost torus and ξ : T → C is a character. Moreover, do not depend on a choice of a particular group, they found a formula for the characters trTξ (g) of so Gelfand tried to find a way to express this these representations and an explicit expression similarity in an intrinsic way. In a series of papers for the Plancherel measure. A striking and until with A. Kirillov he studied the structure of the now unexplained feature of these formulas is that skew-field F(g) of fractions for the universal they are essentially algebraic. For example, the enveloping algebra of a Lie algebra g. He found

Mellin transform L(g, t) in ξ of trTξ (g), which is a that the skew-fields F(G) are almost defined by function of GL2(F) × T , is given by the transcendence degree of the center Z(g) (equal to the rank r(g) of g) and by their Gelfand-Kirillov L(g, t) = δ(det(g), Nm(t))T dimension (equal to (dim(g) − r(g))/2). These × − | − | (tr(g) tr(t))/ tr(g) tr(t) . results are the foundation of works of A. Joseph Here T = E∗ where E is a quadratic extension on the structure of the category of Harish-Chandra ∗ of F, T : F → ± is the quadratic character modules. corresponding to E, tr is the matrix trace, and tr The last series of work of Gelfand on represen- and Nm are the trace and norm maps from E to F. tation theory was on category O of representations The understanding of the existence of an of a semisimple Lie algebra. This category of intrinsic connection between the structure of representations was defined by Verma, but the

January 2013 Notices of the AMS 31 basic results are due to J. Bernstein, I. Gelfand, foundation for geometric representation theory of and S. Gelfand. They constructed a resolution Lie algebras and quantum groups. of finite-dimensional representations by Verma modules Vw , w ∈ W (known as the BGG-resolution), Anatoly Vershik discovered the duality between irreducible and pro- jective modules in the category O, and found the Gelfand, My Inspiration relation between the category O and the category The achievement of Israel Moiseevich Gelfand of Harish-Chandra modules. These results form was an unusual phenomenon of twentieth-century the cornerstone of the theory of representations of mathematics. His name must be included on any semisimple Lie algebras and their affine analogs. short list of those who formed the mathematics of However, their main discovery was the existence that century. He pioneered many new ideas and of a strong connection between algebraic geometry created whole new provinces of knowledge. An ex- of the flag space B of a semisimple group and ceptional intuition, a depth and breadth of thought, the structure of the category O. For example, a liveliness in comprehending mathematics—these they showed that there is an embedding of Vw were his chief qualities. into Vw 0 if and only if the Bruhat cell BwB ⊂ B But the most important quality of this wonderful belongs to the closure of Bw 0B. This connection mathematician was his ability to inspire others. between algebraic geometry and the category of Many other mathematicians, both in Russia and representations is the basis for the recent geometric abroad, felt an amazing sense of inspiration in theory of representations. talking to or corresponding with Gelfand. His ideas There are other important works of Gelfand and his advice stimulated many discoveries and on representation theory (such as indecompos- much research done by others. able representations of semisimple Lie groups, models of representations, and representations Gelfand and Leningrad Mathematics of infinite-dimensional groups), but I want to I was a student in Leningrad in the 1950s, when mention two series of works that originated in Gelfand’s name was well known to all Leningrad representation theory but have an independent mathematicians, including students interested in life. The appearance of such works is very natural, the subject. This is not surprising. The interests since for Gelfand representation theory was a part of most Leningrad mathematicians (including of a much broader structure of analysis. L. V. Kantorovich, V. I. Smirnov, G. M. Fikhtengoltz, Integral geometry is an offshoot of representa- and others) centered on functional analysis and tion theory. The proof of the Plancherel theorem its applications, and this “Leningrad approach” for complex groups is equivalent to the construc- was one of the branches of functional analysis tion of the inversion formula, which gives the that developed in the Soviet Union in the years value of a function in terms of its integrals over 1930–1950. I. M. Gelfand was an iconic figure in horocycles. Gelfand (in a series of joint works this development. with Graev, Z. Shapiro, S. Gindikin, and others) The Leningrad approach differed from that found inversion formulas for reconstruction of taken in Moscow (by A. N. Kolmogorov, I. M. the value of a function on a manifold in terms Gelfand, L. A. Lyusternik and A. I. Plessner) and of its integrals over an appropriate family of in the Ukraine (by M. G. Krein and N. I. Akhiezer). submanifolds. The existence of such inversion It was centered around the theory of functions, formulas found applications in such areas as operator theory, and its applications to differential symplectic geometry, multi-dimensional complex equations and methods of computation. The main analysis, algebraic analysis, nonlinear differential difference between the Moscow approach and the equations, and Riemannian geometry, as well as other two was the study advocated by Gelfand of in applied mathematics (tomography). (A more infinite-dimensional representations of groups and detailed description of Gelfand’s work on integral algebras and an interest not so much in Banach geometry is given by S. Gindikin in the current spaces as in general problems of noncommutative issue of the Notices.) Fourier analysis and, more broadly, in a synthesis Analogously, the work of Gelfand with V. Pono- of algebra, classical analysis, and the theory of functions. These ideas were not present in marev and, later, with J. Bernstein on quivers Leningrad for many years. was motivated by the problems in representation theory—the description of indecomposable repre- Anatoly Vershik is professor of mathematics at the St. sentations for the Lorentz group. But the inner Petersburg Department of the Steklov Mathematical Institute development of this subject led to a beautiful and RAS. His email address is [email protected]. deep theory which later made a full circle in works This segment of the article was translated from the Russian of Ringel, Lusztig, and Nakajima and became the by Vladimir Retakh and Mark Saul.

32 Notices of the AMS Volume 60, Number 1 Gelfand’s fame and authority went unchallenged politics, not just at Leningrad seminars. While still young, Gelfand mathematics. I had broken into the elite club of Moscow mathe- would like to maticians of the highest level, amazing everyone write here about with his theory of maximal ideals of commutative the area of Banach algebras (the so-called “Gelfand trans- his mathemati- form”), and he remained a leader of this club until cal activity that his death. was closest to Among Leningrad mathematicians Gelfand es- my interests and pecially valued L. V. Kantorovich, D. K. Faddeev, about the prob- and V. A. Rokhlin, whom he remembered from his lems on which I was fortunate to student days. Like Kantorovich, Gelfand played an Lab of Function Theory at Moscow State active role in the atomic research project and gen- work with him. I would go so University, circa 1958. Left to right sitting, erally knew the work of Kantorovich on functional far as to say that I. Gelfand, Polyakov, D. E. Menshov, analysis. Both these outstanding mathematicians the main legacy N. K. Bari, G. P. Tolstov; standing, did much work on applied problems. of the work of P. L. Ulyanov, A. G. Kostychenko, In 1986, at the funeral of Kantorovich, Gelfand Gelfand’s first F. A. Berezin, G. E. Shilov, R. A. Minlos. told me that of all of Kantorovich’s results, the one period was his foundation (along with the group of that impressed him the most was his pioneering students he led) of the theory of normed commu- work on linear programming. That was my view as tative rings (commutative Banach algebras, as they well. On the other hand, when I asked Kantorovich are now called) and, most importantly, the theory to submit my paper to Doklady in 1971, he noted of unitary infinite-dimensional representations of that the paper really represented the work of locally compact groups. These papers became Gelfand and that it was disgraceful that Gelfand classics of mathematics despite the youth of the was not still a full member of the Soviet Academy authors. of Science. Gelfand had direct contact with only a As a byproduct of this theory, Gelfand and few of the younger Leningrad mathematicians in Naimark gave a definition of general C∗-algebras addition to me—in particular, with L. D. Faddeev that is now in common usage. Once, as early and M. S. Birman. as the 1970s, Gelfand said to me, “If I could only talk to von Neumann, I would explain to My First Acquaintance with Gelfand’s Work him why our C∗-algebras are more important The first significant mathematical papers that I than W ∗-algebras (von Neumann algebras).” I can studied were the papers of I. M. Gelfand, D. A. add that Gelfand was always more interested Raikov, and G. E. Shilov (Gimdargesh, as I called in topological and smooth problems (which are ∗ them) on commutative normed rings and their pa- connected with C -algebras) than in theoretical pers that followed on generalized Fourier analysis. metric problems (which are formulated in terms ∗ This theory was a mathematical awakening for me. of W -algebras). The representation theory that I was enchanted by its beauty and simplicity, its will forever be connected with his name always generality and depth. was a favorite subject of Gelfand’s, but his ideas Before this I had been hesitant. I could join about relationships and details within this theory the Department of Algebra, where I attended kept changing. Among his wonderful traits was a fearless irony towards his own earlier results. lectures by D. K. Faddeev. Or I could work in the When I told him that as a student I was impressed Department of Analysis, where my first mentor was by his work on Banach algebras, Gelfand replied, G. P. Akilov and where I might choose to specialize “But why did we consider only maximal and not in complex analysis (V. I. Smirnov, N. A. Lebedev) prime ideals?” This remark is similar to one he or real and functional analysis (G. M. Fikhtengoltz, made in a talk in the 1970s, when I mentioned to L. V. Kantorovich, G. P. Akilov). Now functional him my interest in his early papers. “I am always analysis was my only choice. And my interest was praised for my work that was done five or more mostly in the analysis of the Moscow school— years ago, and the same people are not satisfied Gelfand’s functional analysis, as I described earlier. with what I am doing right now,” he explained Since that time the work of Gelfand and his school about his not uncommon critics. In fact, he was in various areas have become my mathematical always ahead of his time, he was always creating a guide. new fashion. One could say a lot about the great variety I remember being strongly influenced by of Gelfand’s interests and works. In fact, he Gelfand’s talk in 1956 about problems in func- was interested in everything: biology, music, and tional analysis (he was probably following the

January 2013 Notices of the AMS 33 example of a similar talk by von Neumann at the However, the hope that arose at that time that International Congress of Mathematicians in 1954). this reformulation would give essentially new In this address he posed several general problems results did not exactly come true, and we cannot and talked about their relationship to various talk about a serious change from the theory of mathematical phenomena. One of these problems Banach spaces to the theory of locally convex was a conjecture about a possible connection spaces, despite the expectations of that time. between Wiener measures and von Neumann The celebrated six-volume series of books Gen- factors and of its possible utility in future work. eralized Functions by Gelfand and his coauthors “Such beauty must not perish unused,” noted demonstrated their unparalleled depth of under- Gelfand. I was a student at that time and I was standing of the subject. The breadth of the area deeply impressed by this strong assertion of the covered, from partial differential equations to rep- primacy of aesthetic estimation of mathematical resentations and number theory, is characteristic theories above other criteria. of Gelfand’s style. In particular, Gelfand was among the first to un- A Few Comments on Gelfand’s Work of the derstand the importance of turning from Banach’s 1950s and 1960s methods in functional analysis to more general Perhaps Gelfand’s highest achievement was the linear topological (nuclear) spaces and their value theory of infinite-dimensional unitary representa- for the spectral theory of operators (the Gelfand- tions of complex semisimple Lie groups, created Kostyuchenko theorem) and the theory of measure together with M. A. Naimark, M. I. Graev, and in linear spaces (the Minlos theorem). Gelfand others. also immediately understood the usefulness of It is hardly possible to comment on this devel- generalized stochastic processes (Gelfand-Ito pro- opment completely here, but it is worth noting cesses). This stimulated the construction of the that Gelfand faced many difficulties common to all theory of measures in linear spaces started by pioneers: many details had to be clarified later or A. N. Kolmogorov in the 1930s. The famous “holy proofs rewritten, and so it would happen that attri- trinity” (triples of Hilbert spaces), generalized butions of priority or authorship were given to other spectral decompositions, an original approach to researchers. But one must remember that Gelfand Levi processes and to Gelfand-Segal constructions, was the first to initiate and develop this huge quasi-invariant measures, and so on were studied area of mathematics, which is so useful in physics. and developed by hundreds of mathematicians. There was no theory of infinite-dimensional rep- resentations before Gelfand’s. From reading the My First Meeting with Gelfand recently published diaries of A. N. Kolmogorov, Not counting a few brief discussions with Gelfand one can understand how impressed the world of during the Mathematical Congress in 1966 or in the mathematics was by these works in the 1940s. It is late 1960s or on my rare visits to his seminar when remarkable how confident Gelfand was in starting I happened to be in Moscow, our first close meeting up completely new areas. took place in the spring of 1972 when I came to Gelfand’s next great passion in the late 1950s Moscow for two months. After the seminar I walked and 1960s—an interest common to many centers to his home with several participants. I started of mathematics—was the theory of distributions talking about the work I had recently started of L. Schwartz (called in Russia “generalized on asymptotic statistics of the lengths of cycles functions”). Several Russian mathematicians had of random permutations. This paper initiated a keen interest in the subject but with a touch of a long series written by me and my students bitterness and perhaps chagrin. The reason was on what I later called the asymptotic theory of simple: Gelfand, and before him N. M. Gyunter group representations. Gelfand was interested and (a famous Petersburg mathematician) and L. V. invited me to his home the next day to continue the Kantorovich and certainly S. L. Sobolev in fact conversation. I remember that Dima Kazhdan, who not just made use of but created the theory of was also there, understood me sooner than Gelfand, generalized functions. In one of his lectures in who often asked me to repeat what I had said. Leningrad in the early1960s, Gelfand stated directly His active and sometimes aggressive questioning that he and Naimark used distributions in their was very helpful to the development of the theme work on representation theory. However, the merit and the improvement of the talk, if you could of L. Schwartz’s work is that he understood the ignore the form of the criticism. But I had long importance of the general theory and substantiated ago heard about Gelfand’s manner in such things. it with many examples. It is important to add that, In talking about mathematics he would openly as often happens in the theory of functions, this express his displeasure to his listener. One of his reformulation actually gave a new language and expressions, which is useful in the upbringing of new understanding to many concepts. young mathematicians, was “Keep your work and

34 Notices of the AMS Volume 60, Number 1 your self-esteem separate.” In other words, do not and generally take even harsh intellectual criticism personally. knew how the I told Gelfand about my plans to study the discussion would symmetric group and its representations. My turn out. There motivation for this research lay not only in the study are many ex- of the subject by itself but also in its applications amples of this, to optimization theory, linear programming, and the most striking combinatorics. (At that time I worked in the was his mathe- Department of Computational Mathematics and matical intuition, Operational Research.) There was also a way which allowed of connecting asymptotic theory with ergodic him to guess the theory. In our discussions, Gelfand remarked result (often with- that everything seemed to be clear with regard out calculations) to representations of finite groups, then started and to foresee With M. I. Graev, 1999. talking enthusiastically about symmetric functions. what to expect He recommended that I look at a mysterious paper from a given method or even from the fur- of E. Thoma about characteristics of the infinite ther work of a mathematician or a whole team. symmetric group, which were of special interest to During that discussion Gelfand approved of my me. ideas (later called asymptotic representations the- Many things about the symmetric group were ory). Many of these plans were then acted on by S. unclear to me, and I was not satisfied with the Kerov, G. Olshanski, A. Okounkov, A. Borodin, and classical exposition of the theory of its representa- a number of Western mathematicians. tions despite the fact that it had been developed At Gelfand’s seminar in 1977 I talked with by Frobenius, Schur, and Young, and also by such S. Kerov about my results on the limiting form giants as von Neumann and H. Weyl. It seemed to of Young diagrams and about our approximation me then that the foundations of this theory and approach to representations of the infinite symmet- its relations with combinatorics were not yet clear. ric group. At that time Gelfand, Graev, and I had Gelfand gave his final conclusion many years later, already started our work on the representations of after he had gotten to know and appreciate my current groups. I. M. always commented on every papers with S. Kerov and later with A. Okounkov. talk at his seminar. During my talk he said (and He said something like, “It is all clear now.” But repeated this many times) that combinatorics is that was all years later. In fact, the main idea of my becoming a central part of mathematics of the approach was, roughly speaking, an application of future. As a permanent seminar participant said the Gelfand-Tsetlin method, created in the early to me, “He was excited.” 1950s for representation of compact Lie groups, to the theory of symmetric groups. It is worth noting Our Collaboration that a dissatisfaction with established theory often Let me return to 1972 and to the history of our begins with those studying the theory in their collaboration. After our conversation I said good- mature years. That is what happened in my case. I bye and intended to leave for Leningrad. In Moscow started to study representations theory later than I always stayed with my old friend, the virologist usual and got a certain advantage from this. N. V. Kaverin. He once attended a Gelfand seminar In Moscow, during our first conversation at in biology. Gelfand remembered him, but they had his home, Gelfand once again surprised me. He no other contact. I mentioned to Gelfand that I was repeated part of our discussion to Dima Kazhdan, staying with Kaverin. Suddenly, on the day of my who had come in later, and said, as if it had already departure, Gelfand called me at Kaverin’s (having been done, that I was studying the asymptotics of found the phone number with some difficulty1) Young diagrams. However, I had not mentioned and asked me to come over immediately. He also this and was just planning to do it. When I invited M. I. Graev, and during our long walk corrected him, Gelfand said, “Yeah, yeah, but you together started to talk about a construction of will do it later.” This unique ability of Gelfand noncommutative integrals of representations of to see “three meters under the ground where semisimple groups, primarily for SL(2,R). He said his interlocutor was standing” was one of his that he had been incubating this problem for a long most remarkable qualities. It instilled trust in his time and had suggested it to his other students, judgment by many who got to know him, but but he was confident that the problem was perfect also drove off many who feared his ability to for me. see through them. He immediately and (usually) correctly guessed what his companions were about 1There was no “official” way to get someone’s telephone to say or what they had in mind but did not express, number in Moscow at the time.

January 2013 Notices of the AMS 35 I was a bit Our first paper was published in Uspekhi in surprised that 1973 in an issue honoring the seventieth birthday Gelfand was of Kolmogorov, and this was the beginning of aware of my my collaboration with Gelfand and Graev, which knowledge of continued with some breaks over ten years. (At the represen- some point I will write more about this.) The first tation theory paper in this series (in Gelfand’s opinion and mine, of Lie groups, the best one) touched on many subjects that were in particular important at that time. In particular, we discussed of SL(2,R). We cohomology with coefficients in irreducible rep- had not talked resentations. In the next paper we described the about this. But At the time of the ICM, 1966. Second cohomology of all semisimple groups without the Gelfand was from left, I. Gelfand; third from left, A. Kazhdan property and gave explicit formulas for right: the prob- Kolmogorov; fifth from left, S. V. Fomin; semisimple groups of rank 1. We also constructed lem was posed far right, O. A. Oleinik. irreducible nonlocal representations of the cor- at just the right responding current groups. The next paper was time for me. In the early 1970s, independent of my devoted to representations of groups of diffeo- other work, I was lecturing on representations of groups, C∗-algebras, and factors. Perhaps Gelfand morphisms and the start of the development of had heard about this, but not from me. More the geometry of configurations and its application likely, this was the result of his ability to foresee to representation theory. We had no doubts (and things that I mentioned earlier. Gelfand said that it was later confirmed and reconfirmed) that this in order to construct the multiplicative integral series of papers would have various applications of representations we have to study “infinitesi- and that the work would continue. In recent years mal” representations, i.e., a neighborhood of the M. I. Graev and I have found new constructions identity representation. In the summer of 1972 and new representations of functional groups, and Gelfand came to Leningrad for the thesis defense this idea certainly has a future. of M. Gromov, and I told him about my preliminary I would like to recall here one particular story. experiments with the Heisenberg group, where a Our first paper had a natural continuation related construction of the integral by other methods was to representations of groups of smooth functions already known. In December of 1972 we found on a manifold with values in a simple compact Lie a solution, a spherical function of the required group. The properties of these so-called “energy” representation of SL(2,R), or the “canonical state”, representations depend on the dimension of the as Gelfand suggested we call it. I came to Moscow manifold. and we drafted the text. The problem was solved This is not the place for details, but we were in a few months. The next spring Gelfand came to able to prove that the representations are reducible Leningrad with M. I. Graev and stayed with us. My when the dimension is three or higher and when domestic helpers prepared food as instructed by the dimension is two under some conditions on 2 Zorya Yakovlevna. While we were working on the the length of a simple root (see our papers and also paper with Graev, I. M. talked with my wife, Rita, papers by R. Ismagilov, R. Hoeg-Krohn, S. Albaverio, about his favorite music and paintings. Of course, and N. Wallach). Dimension one is special, and we went to the Hermitage Museum, where Gelfand even at the beginning of our work, Gelfand said talked a lot about a painting of El Greco (Saints that we would probably have to deal with a von Peter and Paul) and where we also discussed how Neumann factor-representation. There were no to continue our work. obvious reasons or even preliminary estimates I recall next Gelfand’s last visit to Leningrad in for this at that time; we had just started to 1984. I invited him to talk about his own life at a meeting of the Leningrad Mathematical Society (he work on the subject. After some years Gelfand’s had turned seventy-six months earlier). It was an prediction was confirmed: in dimension one the interesting talk with many details about his first energy representation is a factor-representation of steps in mathematics, including his discovery at a type III1, and it turned out to be connected with young age of the Euler-Maclaurin formula (recall a Wiener measure (not with the scalar one, but that he taught himself almost entirely and never with a measure generated by Brownian motion on attended a high school). a compact group). Do you remember the mystic prediction by Gelfand fifty years ago that factors 2Zorya Yakovlevna Shapiro was then Gelfand’s wife, with should connect with Wiener measures? Today the whom he wrote several important papers. subject is actively being developed.

36 Notices of the AMS Volume 60, Number 1 The Seminar His telegrams on my birthday were always very Gelfand left us an enormous number of statements complimentary. on different mathematical subjects. Someone with patience should collect them. Similarly, someone Life in the USSR should have kept a diary of the Gelfand Seminar. It is more and more difficult nowadays to explain Unfortunately, no one took the responsibility. Now to my Western colleagues, as well as young people we can pick up only bits and pieces. The seminars in Russia, what academic life was like in the USSR, were like one-man shows, sometimes successful, or even what other parts of life were like. For sometimes rough, with salty humor, often relevant example, why did Gelfand, with all his achievements and instructive. I gave several talks at Gelfand in science and contributions to classified state Seminars in Moscow and later at Rutgers. I have projects (which were considered exceptionally already described my first very successful talk. important), become a full member of the Soviet The second was shorter and less successful but Academy of Science only after a shameful delay? Of also distinctive. When I mentioned a not very course, one of the reasons is an official academic interesting but very new fact, Gelfand called a and even governmental anti-Semitism. But, for graduate student to the blackboard and asked him example, this feeling was not so strong among to give a proof right away. Should I take this as an physicists. There were other reasons as well. offense? Certainly not. Once Gelfand told me, “The situation was quite I attended the seminar for the first time when I simple for me in the beginning of the 1950s [the was a graduate student. A French mathematician time of the infamous “fight with cosmopolitism”,4 was talking about a paper by von Neumann. The talk when Gelfand lost his position in Moscow State was not very clear. Then Gelfand said, “Next time so- University]: only those who were really interested and-so will give a talk on this paper” and mentioned in mathematics became my students.” It was several names of professors and graduate students true at that time (and also later) that it was and immediately started to evaluate the future better to have another advisor for one’s career. talks. He said that one professor’s talk would give But he still had quite a number of students. an illusion of understanding, but in reality neither And here we face one of the most important the speaker nor the listeners would have any idea and, in some ways, transcendental qualities of of what was going on. Another one would speak Gelfand: he could attract very different kinds of about something else, including his own results, people. He surrounded himself with a whirlpool and so on. All this added some drama to the of established mathematicians, newcomers to seminar. One could take it seriously or with a smile. the field, biologists, and others. His seminar Much later Gelfand told me that L. D. Landau ran a was a place for meetings, corridor discussions, seminar in the style of Pauli (which bore a surface exchanges of news, opinions, and so on. It was resemblance to Gelfand’s but was somewhat more not by chance that the famous “letter from 99 rude). Gelfand himself attended Landau’s seminar mathematicians” defending A. Esenin-Volpin5 was and once repeated to me a joke made by M. Migdal: openly distributed for signatures at his seminar. It “Gelfand goes to the physicists as an intellectual to was met by authorities with resentment and fear the peasantry.”3 at the highest levels. In a totalitarian state such as At the same time, we must note that the Gelfand the USSR, only those who were trusted and tested style was not for everyone. It drove some away by the state could be a magnet for people. For this and even complicated their lives. For example, reason, officials, including official mathematicians, one of his first, and most beloved, students was did not like Gelfand and his circle, not just out F. A. Berezin. At one point they parted ways of consideration of their origins but also by the completely. In the 1970s I tried unsuccessfully to principle of “either with us or against us”. get them to meet and talk. Nowadays, Berezin’s “Of course, we live in a prison,” Gelfand once work, especially in supermathematics, has gotten told me during a conference outside Moscow in worldwide recognition, but he did not live to see the late 1970s during lively discussions between this. Because I was not nearby and did not talk our scientists and others visiting from the West. to Gelfand very often, I was able to avoid the Still, he always refused to take any Samizdat books frustration of some of his coauthors who worked from me. People should write more about this. with him more closely. On the other hand, with Before any of his few visits abroad, Gelfand me he was always extremely polite and friendly. had to pass a number of unpleasant procedures, as did almost all Soviet people. My friends and I, 3A reference to the political movement in Russia in the 4 1870s known as “Going to the People” (Hodenie k nar- A euphemism for an anti-Semitic government program. odu), during which intellectuals went to the countryside in 5In 1969 Esenin-Volpin was thrown into a psychiatric expectations of “enlightening” the peasants. “hospital” for political reasons.

January 2013 Notices of the AMS 37 who were openly denied the right to travel abroad of papers. No doubt many of these papers were for any purpose, were for that reason spared outgrowths of these seminars. He seemed to have the examinations by party commissions of our a very distinctive style of inspiring research by behavior and of our knowledge of party politics. posing probing questions to potential collaborators These seemed to us unfortunate, but at the same and insisting on not letting go until there was time as rather a prestigious distinction. some sort of resolution. I went to Budapest At the end of the 1980s and the beginning of carrying a recently written paper on the spherical the 1990s the desire of many scientists to leave principal series. Gelfand requested that I submit Russia (quite understandably, especially at that the paper to the proceedings (Lie Groups and time) touched Gelfand as well. His Moscow seminar Their Representations) of the Budapest conference. died soon after that. It was impossible to replace its Even though this paper was later (1990) to win leader. A relatively modest version of the seminar the Steele Prize, it was probably a mistake for me was reestablished at Rutgers. One can only guess to publish the paper in the proceedings of the what might have happened had Gelfand stayed in Budapest conference. The initial publisher, Halsted Moscow. Press, went out of business, and the book did not I met Gelfand in the U.S. several times. While appear until 1974. When it finally did appear, many mathematical life was different in the U.S., Gelfand people reported that it was very hard to find. found his place. The celebration of his ninetieth In June 1972, responding to an invitation, I went birthday at Harvard was perfectly organized and to Moscow, accompanied by my wife, Ann, for a became a real scientific event with his brilliant talk. three-week stay. I have to say that I was deeply I have never had any doubts that the name of I. M. touched and happy to be the recipient of Gelfand’s Gelfand will be one of the icons of mathematics warmth, friendship, and respect. In his 1970 ICM in the twentieth century, because the century was report he had included me in a very small (5) group not just a century of outstanding achievements of people whom he said had made outstanding but also of new conceptions. I. M. Gelfand valued contributions to representation theory. In Moscow, and created exactly this sort of mathematics. Gelfand made arrangements for me to address a meeting of the Moscow Mathematics Society, Bertram Kostant chaired by Shafarevich, with whom I had a pleasant lunch. Gelfand also made arrangements for me I. M. Gelfand to meet on a regular basis with a number of his students, including Kazhdan, Bernstein, Kirillov, I first heard of I. M. Gelfand when I was a graduate Gindikin, and his son, Sergei. I was also introduced student in Chicago in the early 1950s. At that time, to his colleagues Manin, Novikov, and Graev, and Gelfand’s paper “Normed rings” played a major I spent an afternoon with Berezin in Gorky Park role in the area of modern harmonic analysis, which talking about (from my perspective, geometric) was then very popular with students. My thesis quantization. advisor, Irving Segal, had arranged for me to spend One of the mathematical topics that came up the years 1953–1955 at the Institute for Advanced during my conversations with Gelfand was the Study in Princeton. I already knew Chern and results in the first of the BGG, [BGG-71] papers. Weil from my years in Chicago, and at Princeton Somewhat earlier I had solved a problem of Bott I became friendly with Lefschetz, Hermann Weyl, which asked for a determination of the polynomial von Neumann, and Einstein. Certainly I considered functions on the dual of a Cartan subalgebra Gelfand to be in the same class as these twentieth- which mapped, via Borel transgression, onto the century math and physics luminaries, and I looked dual basis of the Schubert classes of a general flag forward to meeting him as well. manifold. Acknowledging my precedence in solving My chance for this meeting came about when I this problem, [BGG-71] made a penetrating further was invited to a 1971 summer school in Budapest. development in the solution of this problem by The activities of the school were organized by introducing the very important BGG operators. Gelfand himself, and I believe this was the first What was apparent to me during my stay time he had been given permission to attend was that Gelfand seemed to have created an a conference outside the Soviet Union. A large environment where he was involved with both the number of his students and colleagues came with personal as well as the professional lives of the him. It was an unforgettable experience for me to many people around him. Gelfand’s apartment was be a participant in one of his famous multihour like Grand Central Station, with any number of seminars. Gelfand is a coauthor in a vast number people going in and out. He seemed to be carrying Bertram Kostant is professor emeritus at the Mas- on n conversations. I can’t speak for the people sachusetts Institute of Technology. His email address is orbiting around him, but as an outsider looking [email protected]. in I was charmed. Doing mathematics for me has

38 Notices of the AMS Volume 60, Number 1 had many lonely moments, but in this exciting for his flight back to Rus- environment I sensed that mathematics (like much sia, he started to recite of physics) could be done in a communal, socially poems of Ossip Mandel- satisfying way. stam, Anna Akhmatova, There are any number of unforgettable incidents and others. I remember which occurred during my Moscow visit. The recording him on tape, following was very embarrassing for me (but not but unfortunately I seem without its comic aspects). One day Gelfand invited to have misplaced the Ann and me to have some borscht with him at the tape. restaurant of the Soviet Academy. All the tables in Several years later my the room were filled with people, except for the mathematical interests table opposite us. That table was occupied by only intersected with those With Serre and MacPherson at one man. Suddenly Gelfand started whispering to of Gelfand in the area Monday night seminar, Moscow me, but I couldn’t make out what he was saying. of completely integrable State University, 1984. Then he repeated it somewhat louder. Finally, when systems. In 1979 I had I heard what he was saying, I foolishly blurted written a paper show- out loud the name Lysenko. The room became ing that the complete still, and then I realized what I had done. “Oh my integrability of the open God,” I thought. “I have gotten Gelfand into trouble Toda lattice arises from with the authorities.” For a moment visions of him a consideration of a being dragged off to the Gulag went through my certain coadjoint orbit head. But all turned out well. By the early seventies of the Borel subgroup. Lysenko had apparently been defanged. Gelfand brilliantly put Much later Ann and I saw a great deal of Gelfand this result in an infinite- when he was invited to Cambridge, Massachusetts, dimensional context and in the United States to receive an honorary degree applied it to the theory of pseudodifferential op- from Harvard. A number of scenes during that Lecture at MIT, 1990. erators. I believe a visit still stick in my head. One was at my house similar observation was made by Mark Adler. during a party as I watched him on the floor with a I would like to end here by citing a mathemat- bunch of young kids, somehow managing to keep ically philosophical statement of Gelfand which their interest by telling them some mathematical I think deserves considerable attention. It also gems. Another was the scene at the breakfast table opens a little window, presenting us with a view of after Gelfand spent the night in our house. Gelfand the way Gelfand’s mind sometimes worked. One of was complaining about the absence of good bread my first papers gave a formula for the multiplicity in the U.S. and the difficulty of finding healthy of a weight in finite-dimensional (Cartan-Weyl) rep- food. I countered by pointing to the Swiss muesli resentation theory. A key ingredient of the formula in my breakfast bowl. Exhibiting a sample of his was the introduction of a partition function on delicious (no pun intended) sense of humor, he the positive part of the root lattice. The partition then proceeded to eat the fruits and raisins in my function was very easy to define combinatorially, bowl, all the while cautioning me not to have that but giving an expression for its value at a particular for breakfast because it wasn’t good for me. lattice point was altogether a different matter. Sometime during his visit I met him in New Gelfand was very interested in this partition func- York City and introduced him to my son, Steven, tion and mentioned it on many occasions. He then engaged in filmmaking. Gelfand gave a long finally convinced himself that no algebraic formula discourse to Steven and his friends on Stanislavski. existed which would give its values everywhere. Apparently method acting was one of Gelfand’s He dealt with this realization as follows. One many artistic interests. I also took him to see Jean day he said to me that in any good mathematical Renoir’s famous film Le Grande Illusion, whose theory there should be at least one “transcendental” main theme was the disintegration of European element and this transcendental element should aristocracy in the wake of World War I. I was account for many of the subtleties of the theory. In totally surprised by his reaction to the film. Since the Cartan-Weyl theory, he said that my partition it was made in the late thirties, he thought it function was the transcendental element. was unconscionable that a prominent film was produced on such a topic at a time when Hitler was planning his march across Europe. Another scene that sticks in my head was when, while driving Gelfand to the airport

January 2013 Notices of the AMS 39 Simon Gindikin Now a word about Gelfand’s mathematical mo- tivation. He considered that the main problem of 50 Years of Gelfand’s Integral Geometry the theory of representations was the decompo- sition of certain reducible representations—most As I begin this paper, the entire fifty-year history of importantly, regular representations on homoge- integral geometry in Gelfand’s life unfolds before neous spaces—into a sum (maybe continuous) of my eyes. I was fortunate enough to collaborate irreducible representations (Plancherel formulas). with him during some key moments. I would like The class of groups and homogeneous spaces was to discuss here those points that appear to me to not specified in the talk. It was only clear that be the most important in this wonderful endeavor. complex semisimple Lie groups, together with their homogeneous spaces with maximally compact and First Presentation Cartan isotropy subgroups, were in this class. The I remember well a meeting of Dynkin’s seminar on first two spaces are symmetric, but the third is Lie groups, most likely in the spring of 1959. The not. This is of principal importance—spaces that seminar, which had been only for undergraduate are not symmetric are considered. From the very students at first, at this point combined both beginning the decomposition into irreducible rep- undergraduate students (Kirillov, Vinberg, me) and resentations is interpreted as a noncommutative well-known mathematicians (Karpelevich, Berezin, analog of the Fourier integral: under the conditions Piatetski-Shapiro). Suddenly, Gelfand appeared of a continuous simple spectrum, we can talk (late as usual and accompanied by Graev), and with about the “projections” of a function on a homo- enormous enthusiasm he began talking about their geneous space onto irreducible representations. new work [1]. This was the first time their work on The usual Fourier transform on Rn has a geomet- integral geometry was presented. ric twin in the Radon transform: integration on It may appear strange that Gelfand did not hyperplanes. They are connected with each other present this in his own seminar (in fact, he almost via the one-dimensional Fourier transform. The never presented his new results there) and that marvelous discovery of Gelfand-Graev was that, he selected what was mostly a student seminar. in the case of semisimple Lie groups, there is an Without a doubt, this was no accident, since Gelfand analogous geometric twin: on homogeneous spaces was always very precise in selecting venues for his we consider horospheres—the set of orbits of all talks, and this fit well into traditions of Moscow’s maximally unipotent subgroups—and the horo- mathematical life. My mind’s eye does not see spherical transform (the operator of integration on any specifics of the presentation but rather recalls horospheres). The generalized Fourier transform Gelfand’s excitement, obvious to the listeners, and and the horospherical transform are connected by his certainty that something significant had opened the (commutative) Mellin transform, equivalently, before him—a new direction of geometric analysis, with the commutative Fourier transform. For this which he proposed to call “integral geometry”, as reason, problems about the Plancherel formula it was equaled in importance only by differential and inversion of the horospherical transform on a geometry. He commented that Blaschke used this homogeneous space are trivially reduced to each term for a certain class of problems connected other. Horospheres in hyperbolic geometry were with calculations of geometric measures but that well known (and highly valued) already by their this narrow area did not deserve such an ambitious creators: they are spheres of infinite radius with title, and so he felt justified in appropriating it for centers at infinity and different from hyperbolic this new field. The contentious nature of such a hyperplanes. For other homogeneous spaces (in- position is obvious. Such things worried Gelfand cluding symmetric spaces), they had remained only a little, and I will not venture an opinion on the unnoticed before the work of Gelfand-Graev. subject. I think this is important for understanding A posteriori, the idea of the horospherical Gelfand’s emotional state. He reminded us of his transform seems almost obvious. Let G be a frequent saying that “representation theory is all of complex semisimple Lie group and let B = MAN mathematics.” (I heard this many times, and Manin be a Borel subgroup; here N is the nilpotent radical, once recalled that he also heard from Gelfand that MA is a Levi subgroup (which is commutative in “all mathematics is representation theory,” noting this case) with M and A its compact and vector the delicate difference between these aphorisms.) subgroups, respectively. Then the manifold of From now on, Gelfand said, he considered that horospheres is Ξ = G/N. Gelfand and Graev called “integral geometry is all of mathematics.” it the “principal affine space”. On this space there are two commuting actions: that of the group G Simon Gindikin is Board of Governors Professor at Rutgers acting by left shifts and that of the group MA University. His email address is [email protected]. acting by right shifts; the latter action is well edu. defined because MA normalizes N.

40 Notices of the AMS Volume 60, Number 1 The decomposition of the regular representation considerations of on Ξ relative to the action of MA gives precisely the von Neumann fac- irreducible representations of group G realized in tors. Doubtless the the spaces of sections of line bundles over the flag biggest surprise manifold F = G/B (the “principal projective space” for the authors in Gelfand-Graev’s terminology). This is simply a was how explicitly rephrasing of the conventional realization of the and simply all ir- principal series representations. Therefore, if the reducible unitary horospherical transform (which is an intertwining representations of operator) is injective, the decomposition of the the Lorentz group regular representation is reduced to the decompo- can be described. Lecture at Collège de France, 1978. sition relative to the action of MA on the space Bargmann also of horospheres and the above-mentioned Mellin considered representations of the “real” Lorentz transform along the abelian subgroup A. group SL(2; R) and discovered representations of the discrete series, realized in holomor- Prehistory phic functions on the disk. The central result Needless to say, the real road to the discovery just is the completeness of the constructed unitary outlined was entirely different. I was lucky to hear representations, although Bargmann and Harish- about it from Gelfand, and I want to share it here. Chandra proved only an infinitesimal version of Three papers appeared in 1947—by Bargmann, the completeness. Bargmann also proved a cer- Gelfand-Naimark, and Harish-Chandra—in which tain statement of the completeness of matrix unitary representations of the Lorentz group were coefficients. found. I believe that I once heard Gelfand’s ob- At the same time, Gelfand and Naimark [2] servation that important things in mathematics went much further, establishing essentially all spring forward independently from three places the principal concepts of the theory of unitary at once (the history of mathematics indeed de- representations. They introduced characters of livers striking examples of this, beginning with irreducible unitary representations as distributions, non-Euclidean geometry). The Lorentz group is showed that characters define representations up to locally isomorphic to group SL(2; C). Not sur- equivalence, and computed explicitly the character prisingly, the initiative in this problem belonged values on regular elements, thus obtaining an exact to the physicists, and the authors were target- analog of the classical Weyl character formula. ing physical applications. (Bargmann was solving The paper contains a complicated analytical proof Pauli’s problem and discussed it with Wigner and of the fact that their list of irreducible unitary von Neumann; the problem was suggested to representations is complete; however, the focus of Harish-Chandra by Dirac; the first publication of the work is an analog of the Plancherel formula. Gelfand-Naimark was in a physics journal.) Because This provides the (continuous) decomposition into representations of the rotation group play a key irreducibles of the two-sided regular representation role in nonrelativistic quantum mechanics, it was on the L2 space of the group relative to the invariant natural to expect representations of the Lorentz measure. group to play an analogous role in the relativistic It turned out that in this decomposition only case. The difficulty lay in the fact that, while the some unitary representations appear; they were rotation group is compact and all its irreducible called representatives of the principal series (the unitary representations are finite dimensional, the rest of the unitary representations make up the Lorentz group is not compact and its only finite- complementary series). It is remarkable that the dimensional unitary representation is the trivial Plancherel measure, which appears in the expres- one-dimensional representation. (It is precisely sion of a norm of a function on a group through the the unitary representations that have a physical norms of its projections on irreducible components, interpretation.) was computed explicitly. However, this remarkable In the 1930s Dirac and Wigner considered certain formula, reminiscent of the classical Weyl formula infinite-dimensional unitary representations of the for finite-dimensional representations, was a result Lorentz group. From the other direction, in the of almost ten pages of dense computations without early 1940s Gelfand-Raikov constructed an analog any visible attempts to uncover the conceptual to the theory of Peter-Weyl for general locally structure. Later this was pointed out by Mautner compact groups using infinite-dimensional unitary while refereeing a more conceptual proof found by representations. It seems likely that there was Harish-Chandra. This fitted the style of Gelfand, a consensus that the description of all unitary who at the beginning was more concerned with the representations of the Lorentz group could not be beauty and clarity of the final result than with a done in an explicit form and had to include difficult simplification of the proof.

January 2013 Notices of the AMS 41 42

Photograph by Carol Tate. akn ihRn hmat Thom René with Talking a odrv ttruhapiaino h result the of application through it derive to way very from problems on coauthors several with 93h n re ffrda xetoal elegant exceptionally an offered Graev and he 1953 o nesod n15 udnyh a in saw he suddenly left 1958 was In important and something understood. groups felt not other he to which generalized in been which not group, had Lorentz the for formula Plancherel a of powers of form. regularization quadratic about Riesz M. of he In time formula. to Plancherel the time to from returning and continued satisfied, being also from and groups Lie for semisimple formula complex Plancherel all the for of proof Harish-Chandra conceptual a Soon gave collaborator. primary became Graev his of years many theory for the and abandon representations, not did He areas. different simultaneously worked he now moment, every at representations) (Banach of theory direction algebras, one Banach before, spaces, some If on work. concentrated his he of decisively he organization that, the than he changed More all, Naimark. collabora- with of fruitful tion First extraordinarily his life. concluded mathematical Gelfand’s in elliptic the of coordinates. generalization prized certain a especially bril- is in calculations Gelfand book many inventions. the contains in liant and given difficult proof exceptionally the the an about group; paper generalize the Lorentz to from proof unable difficult were Plancherel already authors the stands especially The of theory, out. analog the of an peak the of formula, proof The obsta- cles. many overcoming attacks, analytic intense efn etcmn akt i ro fthe of proof his to back coming kept Gelfand place took changes significant 1950 Around SL( 2; HS 1974. IHÉS, R ) oee,smtigkp Gelfand kept something However, . ad epritnl re to tried persistently he wards h tl scer derivation clear: is style The fepii omlsthrough formulas explicit of heroic. of short nothing be to appears me to which work, of amount incredible an of sult about n paper the the and of cover, color the of “Blue” because book the call to liked (Gelfand groups classical the of representations unitary to and Gelfand dedicated appeared of Naimark book big a of group case the the to gen- eralizations announcing appeared publica- the [ of of tion time the at Already situation. the after- clarify years that Gelfand = oee,i stpclof typical is it However, Rd) hsbo sare- a is book This “Red”). 2 2 eea oe had notes several ] SL(n oie fteASVolume AMS the of Notices ; C ) n1950 In . h uhr f[ of authors The i h elsne nalcmlxlnsintersecting lines complex all on sense) real the (in nescigtehproa h ouint the to [ in solution problem The inversion hyperbola. lines the into transform intersecting points) into project that those coordinates δ) with β, (α, plane the onto hyperboloid the as considered group, hyperboloid the (on a in family lines (complex) three-parameter all the of exactly be to matrices of shifts subgroup two-sided unipotent the the group: this in horocycles to group have Lorentz lines the these do connection What integrals. λ hyperbola the function this Integrate variables. complex of three of solution function of a the Consider problem analysis: to geometrical part elementary-sounding dedicated large following was a the that proof namely, the years, of ten hidden remained almost had for that something proof this pcsbtwihas hwdwihmodifications which showed also which but odd-dimensional spaces in formula inversion the Radon resembled the which produced, was formula remark- able A from formula. Plancherel derived already-known was the spaces and homogeneous groups certain Lie semisimple horospherical complex the for of transform inversion opposite: the the for essentially formula is the situation uncovered, the not [1] was in way and a such formulas. 1959 Plancherel in derive However, to way be natural to more had a invert there and to transforms, methods horospherical the including direct transforms, be idea Radon generalized the to certain had have Gelfand there beginning, that very representations? the of From theory the to what question: contribute the it raise does to fair its is and it transform but significance, horospherical the about of doubt beauty no the is There begin. mathe- problems everyday matical prosaic and concludes story the formulas). the of changes any substantial to any addition without the words in only certain paper of 1947 the of text the from the of in volume geometry integral fifth of problem this corresponding about (the text calculations their they in closely it how at followed marveling stop the can’t one at paper, looking but interpretation, geometrical this case. and general horospheres the in define transform for to horospherical the straightforward Graev was and After Gelfand it transform. remains Mellin interpretation, it usual this the formula), invert (Plancherel to only group transform the Fourier generalized on the For of formula. inverse inversion the Radon the of reminiscent ∈ oee,a hspittermni ieof side romantic the point this at However, C hnreconstruct Then . hnams l h ooyls(excluding horocycles the all almost then ,  1 0 1 u  αδ α eeaie Functions Generalized with 2 paetyddntko of know not did apparently ] = − SL( βγ λ 1 u , 2; savr ipeformula, simple very a is ] N = δ ∈ C fui pe triangular upper unit of f ) = in 1 C osdrtestof set the Consider ? hog l fthese of all through hssttrsout turns set This . λ − C 1 , 4 fw project we If . 60 β = Number , [ 4 ,where 0, α ,δ) β, (α, f differs ] C 1 ) are required for spaces of rank greater than 1: one significant part of the theory of representations needed to average (over the set of horospheres of the Lorentz group is connected not with the passing through a point) the result of applying an group structure, but rather with some sort of explicit differential operator (of the order equal more general geometrical structure. This was the to the rank) that acts along parallel horospheres. first supporting evidence for Gelfand’s project to It remained to develop direct methods of integral embed the theory of representations into a wider geometry and to understand what advantages area of geometrical analysis. this approach had to harmonic analysis. This was The next step was again evident: understand accomplished, partially, only ten years later [5]. the nature of the condition on a family of lines “to Even so, some important supporting obser- intersect a fixed curve”. The family of all lines in C3 vations slowly accumulated. Gelfand noticed depends on four (complex) parameters. For integral in the very beginning that the symmetric Rie- geometry it is natural to consider families with mannian space for the group SL(2; C) is the three parameters (complexes of lines in classical three-dimensional hyperbolic space, with horo- terminology), because then integration transforms spheres in the classical sense. It turned out that a function of three variables into a function of the inversion formula for this horospherical trans- three variables. For which complexes of lines are form, with the appropriate definitions, exactly there inversion formulas of Radon’s type? Gelfand matches the formula for the inversion of the Radon and Graev [4] showed that this is possible not only transform in three-dimensional Euclidean space. for complexes of lines intersecting a fixed curve By contrast, the Plancherel formulas for the Fourier but also for complexes of lines tangent to a fixed transform in Euclidean and hyperbolic spaces surface and not for any others. They called such differ dramatically. In a sense, the horospherical complexes admissible. It is worthwhile to say a few transform is independent of curvature! There was words about the proof of this astounding result. no doubt of the importance of this observation, but F. John discovered that in the real case the for a surprisingly long time there were no attempts image of the operator of integration along lines to incorporate this into some general result. Today in a three-dimensional space is described by it is clear that if one considers for a Riemannian the ultrahyperbolic differential equation. In the symmetric space of negative curvature its “flat” complex case one has to consider the holomorphic twin (sending curvature to zero), then the inversion and antiholomorphic versions of this differential formulas for the horospherical transforms will equation. It turns out that existence of the required be identical for these two spaces. It seems to me inversion formula is equivalent to solvability of the that this explains the nature of the simple explicit Goursat problem and therefore coincides with the formulas for the representations of semisimple characteristic condition for this operator. This is a Lie groups (which surprised their creators): from nonlinear equation that can be integrated using the the point of view of the horospherical approach, method of Hamilton-Jacobi. The consideration of the problems turn out to be equivalent to flat ones. bicharacteristics gives the description of admissible There exists another important circumstance which complexes. These complexes had already appeared doubtlessly was tacitly understood, even though in classical differential geometry. They are, in a I’ve never encountered a discussion of it. Namely, natural sense, maximally degenerate: the lines of in the standard approach to the Plancherel formula, the complex that intersect one fixed line of the integrals on conjugacy classes are regularized so complex lie (infinitesimally) in one plane. as to be meaningful as “integrals” on singular This result is generalized to complexes of lines conjugacy classes, eventually, on the unit class. in general position in spaces of any dimension. The Horospheres, in a certain sense, are generators of natural next step is to consider the possibility of conjugacy classes; bringing them into the analysis the replacement of lines by curves. The final result illuminates the consideration of singular classes. in this direction was obtained by Bernstein and me [12] (after Gelfand, Shapiro, and I previously found Integral Geometry of Lines and Curves a universal structure of inversion formulas for The biggest success in the following period was complexes of curves [6]). Admissible complexes of connected with the interpretation of horocycles on curves turned out to be exactly infinitesimally full the Lorentz group as lines intersecting a hyperbola. families of rational curves (infinitesimally, they are Everything began with a natural question: What spaces of all sections of a certain vector bundle happens if the hyperbola is replaced by another on the projected line). Here the most significant curve? Kirillov proved that under such a replace- condition is for the curves to be simultaneously ment the inversion formula remains unchanged. rational. This is a quite effective condition, allow- This meant something very important: with a ing the construction of many explicit examples replacement of the hyperbola by another curve, of families of curves with inversion formulas of the group disappeared, and it became clear that a Radon type, for example, curves of the second

January 2013 Notices of the AMS 43 order. In these constructions it is natural to give the set of planes of the complex only under very up the requirement that the family of curves be a strong conditions. Let us call complexes satisfying complex (the number of parameters of the family these conditions admissible. There again arises equals the dimension of the manifold). In this a certain condition involving characteristics. We case, the possibility exists to consider a certain can describe the situation slightly differently. The generalization of a problem of integral geometry, image of the operator of integration over k-planes but more instructive are the possibilities of applica- is described via a certain system of differential tions beyond the theory of representations. There equations of the second order, generalizing the exists, for example, a connection with the twistor John equation, and admissible complexes must be theory of Penrose: the four-parameter admissi- characteristic for this system. In some sense, the ble family of curves corresponds to conformal admissible complexes are maximally degenerate. right-flat four-metrics (the self-dual part of Weyl The nonlinear problem of the description of curvature equals zero). This allowed us to develop admissible complexes is unlikely to be integrable a method of construction of explicit solutions of for k > 1. However, Gelfand and Graev [5] checked the Einstein self-dual equation. It seems to me that, directly that the complex of horospheres in SL(n; C) in the case of curves, integral geometry has been is admissible. As a result, we get an inversion of successfully fully developed. It appears to be some the horospherical transform and as a consequence part of nonlinear analysis, with a focus on explicit the Plancherel formula. I am certain that there is integrable problems, and it is connected with the something very significant in the degeneracy of the method of the inverse problem for the case of one manifold of horospheres. I think that the geometric spectral parameter. background of the theory of representations of semisimple Lie groups lies in this phenomenon. Complexes of Planes Clearly, this result makes substantial use of the The situation with submanifolds of dimension possibility of considering horospheres on SL(n; C) higher than one is significantly more complicated, as planes. For other semisimple Lie groups this and there may be no opportunity in this case is impossible, but I showed that the construction for work to advance as far. It appears that it is described above may be generalized for these similar to the possibility of the integrability in groups by considering admissible complexes of inverse problems with several spectral parameters. arbitrary submanifolds [8]. However, a few essential results have been achieved. Let us recall that beginning with the first work Nonlocal Problems and Integral Geometry for on integral geometry [1], the grand challenge Discrete Series was to find a proof of the Plancherel formula In the case of complex groups the structure of through inversion of the horospherical transform. inversion formulas with averaging of a differential Ten years passed before Gelfand, Graev, and operator means that, in particular, these formulas Shapiro did this [6], [5] for the group SL(n; C). are local: for reconstruction of the function at a We have already discussed the fact that when point, it is sufficient to know the integrals on the n = 2, horospheres can be considered as lines. For horospheres close to this point. In the case of arbitrary n, horospheres for this group can also be the Radon transform, this is the case only in odd- considered as planes of dimension k = n(n−1)/2 in dimensional spaces. In even-dimensional spaces CN , where N = n2 −1. The dimension of this family the inversion formula is nonlocal: in this situation of planes coincides with N, so we have a complex a pseudodifferential operator is averaged. If we of k-planes. The key idea is to study the problem were to go from complex groups to real ones and of reconstructing a function through its integrals their homogeneous spaces, the first new difficulty on planes from an arbitrary complex, not limiting would be connected with the inevitable appearance ourselves to a complex of horospheres. Let us of nonlocal formulas. It is natural to begin with note that the problem of reconstructing a function Riemannian symmetric spaces of noncompact through its integrals on all planes is extremely type X = G/K, where G is a real semisimple Lie overdetermined and the transition to complexes is group and K is a maximally compact subgroup a natural way to make the problem well defined. of G. Then the horospherical transform is always It turns out that for reconstruction at a point it injective, and the inversion formula is local if and is always possible to write a formula reminiscent only if the multiplicities of all roots are even. In of the Radon inversion formula. This formula this case the inversion formula can be derived involves averaging a certain explicit differential using the method described above for complex operator (of order 2k) along the set of planes of groups. However, the general case must involve the complex which pass through a point. This nonlocal inversion formulas and requires new ideas. operator is defined on the set of all k-planes; After Karpelevich and I computed the Plancherel however, we can compute it using its restriction to measure on these spaces through the product

44 Notices of the AMS Volume 60, Number 1 formula for the Harish-Chandra c-function, we connected with certain very special circumstances, found [13] in this case the analog of the approach and there was no chance of direct generalization. of Gelfand-Graev to complex groups in their first The case of SL(2; R) re- paper [1]: we computed the kernel of the inverse mained the first call to horospherical transform as the inverse Mellin action. In 1977 Gelfand and transform of the Plancherel density. We made this I attempted to understand calculation for classical groups. Later, Beerends it [10]. The philosophy of made it in the general case. The next step should our approach was the fact be a direct derivation of these inversion formulas that the Plancherel formula using methods of geometric analysis (as in the case has to be derived in two of even multiplicities), but so far this has not been stages. First, one has to find done. We tried with Gelfand [9] to move in this projections into subspaces direction, developing a certain symmetric analog corresponding to the repre- of differential forms on real Grassmannians, which sentation series. Series of doubtless has its own interest. However, we did representations correspond not achieve significant progress in this problem. It to equivalence classes of seems to me that today many points look clearer, Cartan subgroups. The de- and this problem looks realistically solvable. composition of each series Finally, we can begin discussing the main into irreducible subspaces obstacle to the development of the theory of is reduced to a commu- tative Fourier transform representations on the basis of the horospherical Budapest swimming pool, transform, which was clear from the very beginning. (continuous or discrete) corresponding to the asso- 1966. First row, P. Dirac, For real semisimple Lie groups the horospherical ciated Cartan subgroup. So I. Gelfand; second row, transform as a rule has a kernel, which consists the first stage is the princi- B. Bollobas, M. Arato. of all of its representation series except for the pal one. Two of the main connected problems are maximally continuous ones. This is already true the finding of projections into the series and the for the group SL(2; R): if we decompose its regular internal analytic characterization of subspaces for representation into the sum of subspaces L ,L c d the series. We solved these problems for SL(2; R). corresponding to continuous and discrete series Subspaces corresponding to holomorphic and an- respectively, then the kernel of the horospherical tiholomorphic discrete series can be characterized transform coincides with L and the image is d as boundary values on SL(2; R) of holomorphic isomorphic to L . There does not appear to be c functions in certain tube domains in the complex a way to invert the horospherical transform or group SL(2; C). Projections may be interpreted as to derive in this way the Plancherel formula. It certain analogs of the Cauchy integral formula. is easy to interpret this as a dramatic limitation I remember how happy Gelfand was with this of the power of integral geometry in the case of result. He often said that new, significant things real groups, where discrete series play a central in representations have to be already nontrivial role. This appears to be the reason that Gelfand’s for SL(2). We expected that in the general case as idea of applying integral geometry to the theory of well the series would be connected with certain representations has not evoked much enthusiasm tube domains, which may not be Stein manifolds, amongst experts on representation theory. and then it would be required to consider ∂¯ I can be a witness to the fact that Gelfand cohomology on these domains. Later this was never believed that the area of applications of called the Gelfand-Gindikin program. Significant horospheres is bounded by continuous series. progress was achieved in this program, but only In his opinion it was necessary to understand for holomorphic discrete series. what corresponds to horospheres in the case of It could have been expected that the projections discrete series, and this is perfectly possible. This into the series had to be somehow connected resonated with his faith in the aesthetic harmony of with integral geometry, but at the time it proved mathematics. In his first talk, which it befell me to impossible to find an appropriate generalization of hear (in 1955), he said about von Neumann factors the horospheres. Much later I discovered a certain of the second type (which did not have any known natural construction [14]. On G = SL(2; C) there applications at that time): “Such beauty must not are two classes of horospheres, each of which al- vanish!” Only in one case—an imaginary three- lows the possibility of constructing a horospherical dimensional hyperbolic space—did Gelfand and transform. We have discussed the one-dimensional Graev manage [7] to find an appropriate expansion: horocycles. However, it is also possible to consider the discrete series there was connected with two-dimensional horospheres—orbits of maximal certain degenerate horospheres. However, this was unipotent groups in SL(2; C) × SL(2; C) under the

January 2013 Notices of the AMS 45 two-sided action. Their geometrical characteriza- References tion consists of sections of the group, considered as [1] I. M. Gelfand, M. I. Graev, Geometry of homogeneous a hyperboloid, by isotropic planes. Horocycles are spaces, representations of groups in homogeneous linear generators of two-dimensional horospheres; spaces and related questions of integral geometry, Amer. Math. Soc. Transl. (2), 37 (1964), pp. 351–429. for this reason, both versions of the horospherical [2] I. M. Gelfand, M. A. Naimark, Unitary representations transform are equivalent. For SL(2; R) both trans- of the Lorentz group, Izv. Acad. Nauk SSSR, Ser. Mat., forms have kernels. The idea is that since there are 11 (1947), pp. 411–504. not enough real horospheres, we must consider [3] I. M. Gelfand, Integral geometry and its relation to certain complex ones. the theory of group representations, Russ. Math. Surv., Let us consider those complex horospheres 15 (1960), no. 2, pp. 143–151. [4] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Gen- which do not intersect the real subgroup SL(2; R). eralized Functions, Vol. 5. Integral Geometry and There are three types of such horospheres. Instead Representation Theory, Academic Press, 1966. of integration on real horospheres, we consider the [5] I. M. Gelfand, M. I. Graev, Complexes of k- convolution (on the real group) of Cauchy kernels dimensional planes in the space Cn and Plancherel’s with singularities on the complex horospheres formula for the group GL(n, C), Soviet Math. Dokl., 9 without real points. As a result, the horospherical (1968), pp. 394–398. transform of their three components is defined. [6] I. M. Gelfand, M. I. Graev, Z. Ya. Shapiro, Integral geometry on k-dimensional planes, Funct. Anal. Appl., It already does not have a kernel, and images of 1 (1967), no. 1, pp. 14–27. the various components can be decomposed using [7] I. M. Gelfand, M. I. Graev, An application of the horo- representations of various series. The inversion of spheres method to the spectral analysis of functions the horospherical representation gives projectors in real and imaginary Lobachevsky space, Trudi Mosk. onto series and then also the Plancherel formula. Math. Soc., 11 (1962), pp. 243–308. At the same time, the continuation of functions [8] S. G. Gindikin, Integral geometry on symmetric man- ifolds, Amer. Math. Soc. Transl. (2), 148 (1991), pp. from the continuous series is obtained as a 29–37. ¯ one-dimensional ∂-cohomology in a certain tube [9] I. M. Gelfand, S. G. Gindikin, Nonlocal inversion for- domain, in agreement with our hypothesis. It mulas in real integral geometry, Funct. Anal. Appl., 11 is my hope that the correct development of (1977), pp. 173–179. the understanding of complex horospheres for [10] , Complex manifolds whose skeletons are real semisimple Lie groups is sufficient for the semisimple real Lie groups and holomorphic discrete series of representations, Funct. Anal. Appl., 11 (1977), construction of the integral-geometrical equivalent pp. 258–265. of the theory of representations. Interestingly, this [11] I. M. Gelfand, S. G. Gindikin, Z. Ya. Shapiro, The can be done even for compact Lie groups, for which local problem of integral geometry in the space of there are no real horospheres at all. curves, Funct. Anal. Appl., 13 (1979), pp. 87–102. Gelfand (in collaboration with Graev and me) [12] J. Bernstein, S. Gindikin, Notes on integral geometry worked on many other aspects of integral geometry. for manifolds of curves, Amer. Math. Soc. Transl. (2), 210 (2003), pp. 57–80. I have not discussed these results because I [13] S. G. Gindikin, F. I. Karpelevich, A problem of wanted to concentrate here on integral geometry integral geometry, Selecta Math. Sov., 1 (1981), pp. connecting with the theory of representations, 160–184. which seems to me to be the most important part [14] S. Gindikin, Integral geometry on SL(2; R), Math. Res. of this project. It is intriguing that Gelfand was Lett., 7 (2000), pp. 417–432. interested in the Radon transform long before its connection with the theory of representations was Peter Lax discovered. He liked to offer problems concerning the Radon transform to students, though he never I. M. Gelfand worked in this direction himself. This was almost Like everyone else, I first heard Gelfand’s name a premonition of future connections, an ability as the author of the famous theorem on maximal often shown by exceptional mathematicians and ideals in commutative Banach algebras and its typical for him. Integral geometry was not one application to prove Wiener’s theorem about of his most successful projects. It did not gain functions with absolutely convergent Fourier series. him broad recognition or a multitude of followers. The following story describes the reception of this Yet we find in it, in a very pronounced way, his result in the U.S. inimitable approach to mathematics. Like every Shortly after its publication, Ralph Phillips other mathematician lucky enough to have worked presented this result at Harvard. It created such a with Gelfand, I have gotten used to trusting his extraordinary intuition, and I am certain that the Peter Lax is professor emeritus at the Courant Insti- story is not over. Perhaps I will see the realization tute of Mathematical Sciences. His email address is of his fantastic project yet. [email protected].

46 Notices of the AMS Volume 60, Number 1 stir that he was asked to repeat the lecture, this mathematics into the lives time to the whole faculty. And then he was asked of schoolchildren all around to present it a third time to G. D. Birkhoff alone. the Soviet Union). I. M., who Other early basic results were the Gelfand for- as usual was simultaneously mula for the spectral radius and the Gelfand-Levitan involved in a myriad of inverse spectral theory for ordinary differential various projects both sci- operators. If you have a completely integrable entific and pedagogical, had system, you are supposed to be able to integrate it decided to organize and run completely. For the KdV equation, the landmark a special mathematical class result of Gelfand and Levitan on the inverse spec- of seventh-graders within tral problem for second-order ODEs turned out to the famous Moscow School be the key to perform the integration. No. 2, which already had It is a measure of Gelfand’s lifetime achieve- a decade-long tradition of ments that these spectacular early results are such classes. To help him viewed today as merely a small part of his total run this program, I. M. asked work. Victor to find him several Photograph by Carol Tate. Here is a story that sheds light on Gelfand’s young assistants who had Math in the kitchen, 1982. modesty. I was one of three members of a committee themselves passed through to award the first Wolf Prize. We all agreed that such a class during their school years. I was very the prize should be given to the greatest living fortunate to become one of four such assistants. mathematician. That person, in my opinion, was (Another was my old friend and classmate since Carl Ludwig Siegel, but another member of the the seventh grade Borya Feigin, who is now a committee insisted that it was Israel Moiseevich distinguished mathematician.) All four of us had Gelfand. So we argued back and forth fruitlessly graduated a year earlier from the same School until we decided to divide the prize. Some time No. 2, all were math undergraduates beginning our later Gelfand remarked in a chance conversation, second year at Moscow State University, and all “It was a great honor for me to share a prize with felt a little lost in our mathematical studies. Siegel.” The organizational matters could have been The world shall not see the like of Israel M. resolved within a few minutes, but our first Gelfand for a long, long time. meeting with I. M. lasted for several hours. The four of us (plus poor Victor) walked with I. M. Andrei Zelevinsky for hours, and he talked to us about all kinds of things, mathematical and not. He asked us Remembering I. M. Gelfand what we most loved about mathematics and what seminars and elective courses we had attended On December 6, 2009, two months after I. M. during our freshman year. Of course, he declared Gelfand passed away, Rutgers University, his last that we had done everything wrong and were place of work, held a Gelfand Memorial. This almost lost for mathematics, but there was still event brought together the crème de la crème of some little hope for us if we started to attend his several generations of Moscow mathematicians seminar at once. He explained to us how to study (now scattered all around the world) whose life a new mathematical subject: focus on the most in mathematics was to a large extent shaped basic things at the foundation and dwell upon by I. M.’s influence. Despite the sad occasion, it them until you reach full understanding; then the was a pleasure to see many old friends and to technicalities of the subject would be understood share memories of our student years when we all very quickly and effortlessly. I remember vividly attended the famous Gelfand Seminar at Moscow how he illustrated this by explaining to us the State University. And of course to share stories foundations of linear algebra: a subspace in a about I. M. He was such a huge presence in so vector space is characterized by one integer, its many lives, and his passing left a gap which will dimension; a pair of subspaces is characterized by be impossible to fill. three integers (dimensions of the two subspaces I first met I. M. in the early fall of 1970. The and of their intersection). What about a triple of meeting was arranged by Victor Gutenmacher, who subspaces? and what about a quadruple?6 at the time worked at the School by Correspondence organized by I. M. (with the purpose of bringing 6Triples of subspaces demonstrate the limitations of a fruitful analogy between subspaces of a finite-dimensional Andrei Zelevinsky is professor of mathematics at Northeast- vector space and subsets of a finite set: the subspace lattice is ern University. His email address is [email protected]. only modular but not distributive. Thinking about this anal- Zelevinsky is grateful to his daughter, Katya, for helpful ogy led me to my first two published notes, which appeared editorial suggestions. within the next couple of years. As for the quadruples, as I

January 2013 Notices of the AMS 47 This meeting was definitely one of my most life- 1994 book with Misha Kapranov). It would take changing experiences. I have never met a person volumes to give a comprehensive account of I. M.’s with such personal magnetism and such an ability mathematical contributions, so let me just share to ignite enthusiasm about mathematics as I. M. my personal impressions of some of his unique I remember returning home late in the evening, features as a mathematician and as a teacher.8 totally exhausted but happy, with the definite Working with him could be a very frustrating feeling that my mathematical fate had been sealed experience. Just looking at his incredibly prolific that day. I could not sleep and spent most of scientific output, one would imagine him as a model the night thinking about mysteries of triples of of efficiency, never wasting a moment of his time. subspaces! I attended the Gelfand Seminar for But most of the time during our daily meetings almost twenty years. A lot has been said about its was filled with numerous distractions, jumps from unique character, including horror stories about one topic to another, his long phone conversations I. M.’s rough treatment of speakers and participants. with an amazing variety of people on an amazing I had my share of humbling experiences there, both variety of topics, etc. Quite often after long hours as a speaker and as a “control listener” who was spent like this I felt totally exhausted and had sent by I. M. to the blackboard in the middle of a talk a depressing feeling that we had just wasted a to explain what the speaker was trying to say. Many perfectly good working day. But almost without people, including some excellent mathematicians, exception there came a moment (sometimes when could not stand this style and stopped attending I was already saying good-bye at the door) of the seminar. Those who stayed (myself included) his total concentration on our project that led to decided that such a great learning experience extremely rapid progress, completely justifying was worth a little suffering. Equally important, all the torturous hours leading to this moment. or maybe even more important than the talks It seemed as if his subconscious mind never themselves, was Gelfand’s choice of topics; his stopped working on our project (and probably on comments and monologues, which often deviated a multitude of other things at the same time), and a lot from the original topic; and of course his it just took I. M. a long time to become ready to famous “pedagogical” jokes and stories. spell out the results of this work. The official starting time of the seminar was This “nonlinearity” of I. M.’s thinking process 7 p.m. (or was it 6:30?) on Mondays, but it almost was also one of the many features that made his always started with much delay, sometimes after seminar so unique. He would spend an inordinate up to two hours! I am sure I. M. did this on purpose, amount of time asking everybody to explain to him because these weekly get-togethers before the some basic definitions and facts, and just when seminar with numerous friends coming to the most of the participants (starting with the speaker, university from all around the city were also a big of course) would get totally frustrated, I. M. would suddenly switch gears and say something very attraction. Sometimes even after his arrival, I. M. 9 did not immediately start the seminar but stayed illuminating, making it all worthwhile. I have never met any other mathematician with in the corridor for some time and chatted with such an ability to see the “big picture” and always people just like everybody else. go to the heart of the matter, ignoring unnecessary I think of myself as I. M.’s “mathematical technicalities. He had an uncanny ability to ask grandson”: my first real teacher and de facto Ph.D. the “right” questions and to find unexpected advisor was Joseph Bernstein, one of Gelfand’s best connections between different mathematical fields. students.7 However, at some point after Joseph’s I. M. was fully aware of this gift and liked to emigration (in 1980, I believe), I. M. approached illustrate it by one of his numerous “pedagogical” me and suggested that we start working together. stories: “An old plumber comes to repair a heater. Our close mathematical collaboration lasted about He goes around it, thinks for a moment, and hits a decade (from the first joint note in 1984 to our it once with a hammer. The heater immediately starts working. The plumber charges 200 rubles realized much later, I. M. had just finished his remarkable paper with V. A. Ponomarev, “Problems of linear algebra for his service. The owner says, ‘But you just spent and classification of quadruples of subspaces in a finite- two minutes and didn’t do anything.’ The plumber dimensional vector space”, so he was talking to us about his replies, ‘I am charging you 3 rubles for hitting your own cutting-edge research! 8 7Joseph could not be my official advisor, since he was never For I. M. these two occupations were inseparable. His way affiliated with the mathematics department at Moscow State, of doing mathematics always involved close personal inter- which was also the case for many other first-rate mathe- action with his innumerable students and collaborators. maticians from the “Gelfand circle”. A. A. Kirillov Sr., also 9I. M. often interrupted speakers with the words “May I ask a student of Gelfand, kindly agreed to serve as my official a stupid question?” The best reply to this was given by Y. I. advisor. Of course, I also learned a lot from him and from Manin during one of his rare appearances as a speaker: “No, attending his seminar. I. M., I don’t think you are capable of such a thing!”

48 Notices of the AMS Volume 60, Number 1 heater with a hammer and the rest for knowing where to hit, which took me forty years to learn!’10 I. M. had such an enormous wealth of ideas and plans that he needed many collaborators to help him realize even a small part of them. I was always Enhancement and Partnership Program very impressed with his great intuitive grasp of people. A few minutes of penetrating questioning The Clay Mathematics Institute invites proposals under its of a new acquaintance allowed him to take the new program, “Enhancement and Partnership”. The aim full measure of a person, understand his or her is to enhance activities that are already planned, particu- larly by funding international participation. The program is scientific potential, strengths and weaknesses, even broadly defined, but subject to general principles: how much pressure the person could withstand. In my own case, it seems that I. M. detected my • CMI funding will be used in accordance with the soft spot for algebraic combinatorics (quite rare Institute's mission and its status as an operating in Moscow at that time, I must say) even before foundation to enhance mathematical activities I realized it myself. Very soon after I joined his organised by or planned in partnership with other seminar, he asked me to study (and then give organisations. a talk on) the old book The Theory of Group • It will not be used to meet expenses that could be Characters and Matrix Representations of Groups readily covered from local or national sources. by D. E. Littlewood, where the representation theory • All proposals will be judged by the CMI's Scientific was treated with a strong combinatorial flavor. Advisory Board. This was truly a sniper’s shot: the facts and ideas that I learned from this book continue to serve me Examples include: to this day. Another truly inspired suggestion by • Funding a distinguished international speaker at a I. M. was to bring together Borya Feigin and Dmitry local or regional meeting. Borisovich Fuchs, which led to many years of very • Partnership in the organisation of conferences and fruitful collaboration. Like so many of my friends workshops. and colleagues, I feel very fortunate for having known Israel Moiseevich and for being given a • Funding a short visit by a distinguished mathemati- chance to be close to this gigantic, complex, wise, cian to participate in a focused topical research inspiring, and infinitely fascinating personality. program at an institute or university. • Funding international participation in summer schools (lecturers and students) or repeating a successful summer school in another country. • Funding a special lecture at a summer school or during a research institute program. • Funding an extension of stay in the host country or neighbouring countries of a conference speaker. Applications will only be received from institutions or from organisers of conferences, workshops, and summer schools. 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 president@ claymath.org. Applicants should set out in a brief letter a 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 be obtained from other local or national sources. Funds requested should not be out of proportion to those ob- tained from other sources. The CMI may request indepen- 10 I. M.’s amazing record of initiating new fruitful directions dent letters of support. of mathematical research provides plenty of examples of “knowing where to hit.” Let me just mention one example: the Applications should be sent to [email protected]. theory of general hypergeometric functions initiated by I. M. There is no deadline, but the call will be closed when (where a significant part of my collaboration with him took the current year’s budget has been committed. place) grew from his insight that hypergeometric functions should live on the Grassmannians.

January 2013 Notices of the AMS 49