Memories of Vladimir Arnold - American Mathematical · PDF fileMemories of Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors Vladimir Arnold, an eminent mathematician

Memories ofVladimir ArnoldBoris Khesin and Serge Tabachnikov,

Coordinating Editors

Vladimir Arnold, an eminent mathematician ofour time, passed away on June 3, 2010, nine daysbefore his seventy-third birthday. This article,along with one in the previous issue of the Notices,touches on his outstanding personality and hisgreat contribution to mathematics.

Dmitry Fuchs

Dima Arnold in My LifeUnfortunately, I have never been Arnold’s student,although as a mathematician, I owe him a lot. Hewas just two years older than I, and according tothe University records, the time distance betweenus was still less: when I was admitted to theMoscow State university as a freshman, he was asophomore. We knew each other but did not com-municate much. Once, I invited him to participatein a ski hiking trip (we used to travel during thewinter breaks in the almost unpopulated northernRussia), but he said that Kolmogorov wanted himto stay in Moscow during the break: they weregoing to work together. I decided that he wasarrogant and never repeated the invitation.

Then he became very famous. Kolmogorov an-nounced that his nineteen year-old student DimaArnold had completed the solution of Hilbert’s13th problem: every continuous function of threeor more variables is a superposition of continuous

Boris Khesin is professor of mathematics at the University

of Toronto. His email address is [email protected].

edu.

Serge Tabachnikov is professor of mathematics at Penn-

sylvania State University. His email address is tabachni@

math.psu.edu.

Dmitry Fuchs is professor of mathematics at the Univer-

sity of California, Davis. His email address is fuchs@math.

ucdavis.edu.

DOI: http://dx.doi.org/10.1090/noti813

functions of two variables. Dima presented a two-hour talk at a weekly meeting of the MoscowMathematical Society; it was very uncommonfor the society to have such a young speaker.

V. Arnold, drawing, 1968.

Everybody ad-mired him, andhe certainly de-served that. Stillthere was some-thing that keptme at a distancefrom him.

I belonged toa tiny group ofstudents, led bySergei Novikov,which studied al-gebraic topology.Just a decadebefore, Pontrya-gin’s seminar inMoscow was atrue center of theworld of topol-ogy, but then

Cartan’s seminar in Paris claimed the leadership,algebraic topology became more algebraic, andthe rulers of Moscow mathematics pronouncedtopology dead. Our friends tried to convince us todrop all these exact sequences and commutativediagrams and do something reasonable, like func-tional analysis or PDE or probability. However, wewere stubborn. We even tried to create somethinglike a topological school, and, already being agraduate student, I delivered a course of lecturesin algebraic topology. The lectures were attendedby several undergraduates, and we were happy toplay this game.

Then something incredible happened. One dayI found the lecture room filled beyond capacity; Ieven had to look for a bigger room. My audience

482 Notices of the AMS Volume 59, Number 4

At Otepya, Estonia.

had become diverse: undergraduates, graduatestudents, professors. This change had a very clearreason: the Atiyah-Singer index theorem.

The problem of finding a topological formulafor the index of an elliptic operator belongedto Gelfand. Our PDE people studied indexes alot, and they had good results. It was not adisaster for them that the final formula was foundby somebody else: their works were respectfullycited by Atiyah, Singer, and their followers. Thetrouble was that the formula stated, “the index isequal to” and then something which they couldnot understand. People rushed to study topology,and my modest course turned out to be the onlyplace to do that.

And to my great surprise, I noticed Dima Arnoldin the crowd.

I must say that Dima never belonged to anycrowd. Certainly the reason for his presence didnot lie in any particular formula. Simply, he hadnever dismissed topology as nonsense, but neitherhad he been aware of my lectures. When he learnedof their existence, he appeared. That was all. Henever missed a lecture.

One day we met in a long line at the studentcanteen. “Listen,” he said, “can you explain to mewhat a spectral sequence is?” I began uttering theusual words: a complex, a filtration, differentials,adjoint groups, etc. He frowned and then said,“Thus, there is something invariant [‘invariant’ inhis language meant ‘deserving of consideration’]in all this stuff, and this is the spectral sequence,right?” I thought for a moment and said, yes. At thismoment we got our meals, and our conversationchanged its direction.

Evidently spectral sequences were not forArnold. Nonetheless, there is such a thing asArnold’s spectral sequence [9], a humble objectin the world of his discoveries, resembling theasteroid Vladarnolda in the solar system (thestability of which he proved approximately at thetime of our conversation in the canteen), namedafter him. When I say that he could not appreciate

spectral sequences, I mean that he in general hada strong dislike for unnecessary technicalities,and technicalities were often unnecessary to himbecause of his extremely deep understanding.By the way, this attitude toward impressive butunnecessary tricks extended beyond mathematics.Years later we spent a week or so with friends ata ski resort in Armenia. We showed each otherdifferent turns and slidings, but Dima obviouslywas not interested. He said that the slope was nottoo steep, and he simply went straight from thetop to the bottom, where he somehow managedto stop. I was surprised: there was a stone hedgein the middle of the slope that you needed togo around. Dima said modestly, “You know, atthis place my speed is so high that I simply pullmy legs up and jump over the hedge.” I couldnot believe it, so I waited at a safe distance fromthe hedge and watched him doing that. It wasmore impressive than all our maneuvers takentogether. Whatever he did—mathematics, skiing,biking—he preferred not to learn how to do it butjust to do it in the most natural way, and he dideverything superlatively well.

I do not remember how it came about that Ibegan attending his Tuesday seminar. Probably heasked me to explain some topological work there,then I had to participate in some discussion, andthen I could not imagine my life without spendingtwo hours every Tuesday evening in a small roomon the fourteenth floor of the main building of theMSU. Works of Arnold, his numerous students,and other selected people were presented at theseminar, and Dima insisted that every word ofevery talk be clear to everybody in the audience.My role there was well established: I had to re-solve any topology-related difficulty. Some of myfriends said that at Arnold’s seminar I was a “coldtopologist”. Certainly, a non-Russian-speakercannot understand this, so let me explain. In manyRussian cities there were “cold shoemakers” inthe streets who could provide an urgent repair toyour footwear. They sat in their booths, usuallywith no heating (this is why they were “cold”),and shouted, “Heels!…Soles!….” So I appeared asif sitting in a cold booth and yelling, “Cohomol-ogy rings!…Homotopy groups!…Characteristicclasses!….”

In my capacity as cold topologist, I even hadto publish two short articles. One was called“On the Maslov–Arnold characteristic classes,”and the other one had an amusing history. Oneday Dima approached me before a talk at theMoscow Mathematical Society and asked whetherI could compute the cohomology of the purebraid group (“colored braid group” in Russian);he needed it urgently. I requested a descriptionof the classifying space, and the calculation wasready at the end of the talk. It turned out thatthe (integral) cohomology ring was isomorphic

April 2012 Notices of the AMS 483

to a subring of the ring of differential formson the classifying manifold. He suggested thatI write a note, but I refused: for a topologist itwas just an exercise; it could be interesting onlyin conjunction with an application to somethingelse. (I knew that Dima was thinking of Hilbert’s13th problem in its algebraic form: the possibilityof solving a general equation of degree 7 notin radicals, but in algebraic functions of twovariables.) I suggested that he write an article andmention my modest contribution in an appropriateplace. He did [2]. But, a couple of months later,he needed the cohomology of the classical Artin’sbraid group. This was more difficult and took meseveral days to complete the calculation. I did itonly modulo 2, but I calculated a full ring structureand also the action of the Steenrod squares.(The integral cohomology was later calculatedindependently by F. Vainshtein, V. Goryunov, andF. Cohen and still later Graeme Segal provedthat the classifying space of the infinite braidgroup was homologically equivalent to Ω2S3.) Iphoned Dima and explained the results. Firsthe requested that I give a talk at the seminar(Next Tuesday! That is tomorrow!), and then hedecidedly refused to do what we had agreed uponfor pure braids: to write an article and mentionmy participation where appropriate. After a briefargument, we arrived at a compromise: I publishan article about the cohomology of the braid groupwithout any mentioning of Hilbert’s problem, andhe publishes an article where this cohomology isapplied to superpositions of algebraic functions.When we met the next day, his article was fullywritten and mine had not even been started. Buthis article contained a reference to mine andhence the title of the latter. I could delay nolonger, and the two articles were published in thesame volume of Functional Analysis [5], [14]. Sincethe articles in Functional Analysis were arrangedalphabetically, his article was the first, and minewas the last. But this was not the end of thestory. A cover-to-cover translation of FunctionalAnalysis was published by an American publisher.The braid group in Russian is called gruppa kos;the word kos is simply the genitive of kosa, abraid, but the American translators thought thatKOS was a Russian equivalent of COS, and theEnglish translation of my article was attributedto a mysterious cosine group. I do not know howmany English-speaking readers of the journal triedto guess what the cosine group was.

As a permanent participant of Arnold’s sem-inar, I had an opportunity to give talks on myworks not explicitly related to the main direc-tions of the seminar. I gave a brief account ofmy work with Gelfand on the cohomology ofinfinite-dimensional Lie algebras, of characteristicclasses of foliations. These things did not interestDima much, although he himself had a work on

Summer School “Contemporary Mathematics”at Dubna, near Moscow, 2006.

similar things [3]. He always considered algebraand topology as something auxiliary. Once I heardhim saying respectfully, “Siegel’s case, this is atrue analysis,” and this sounded like “true math-ematics”. Whatever he did, his unbelievably deepunderstanding of analysis was always his maininstrument.

One more story of a similar kind. In 1982 JohnMilnor, who briefly visited Moscow, delivered atalk at Arnold’s seminar on a very recent (andnot yet published) work of D. Bennequin on anew invariant in the theory of Legendrian knots incontact 3-manifolds. The main result of Bennequinstated that the “Bennequin number” (now justlycalled the “Thurston-Bennequin number”) of atopologically unknotted Legendrian knot in thestandard contact space must be strictly negative.For an illustration, Milnor showed an example of aLegendrian trefoil with the Bennequin number+1.Arnold said that at last he had seen a convincingproof that the trefoil is a topologically nontrivialknot. Certainly, this was a joke: Bennequin’s proofat that time did not look convincing, and thenontriviality of the trefoil has a popular proofunderstandable to middle school students (viathe tricolorability invariant). But for Dima only ananalytic proof could be fully convincing.

When I joined the Arnold seminar, it had justacquired the name of “the seminar on singularitiesof smooth maps”. In the mid-1960s, Arnold wasfascinated by work of John Mather on singularities.People could not understand this. Allegedly, Pon-tryagin said: “We can always remove complicatedsingularities of a smooth map by a small perturba-tion; it is sufficient to study the generic case.” Butsingularities appear in families of smooth maps;you cannot remove them, insisted Dima. Somepeople mocked his affection for singularity the-ory. There is a short story of Stanislav Lem (a Polishscience fiction writer) in which robots that couldexperience human emotions were manufactured.One of these robots felt an immense joy when hesolved quadratic equations—just like you, Dima!


Dima smiled at such jokes but continued studyingsingularities.

The results of Arnold and his students in thisarea were very deep and diverse. He classifiedall singularities that appear in generic familiesdepending on no more than 14 parameters andstudied their moduli varieties and discriminants.He discovered the relations of the theory to sym-plectic, contact, and differential geometry. It haddeep applications in topology (Vassiliev’s invari-ants of knots), differential equations, and classicalmechanics.

More or less at the same time, a widely popu-larized version of the singularity theory emergedunder the colorful name of the theory of catas-trophes. It was promoted by two remarkabletopologists, R. Thom and E. C. Zeeman. “Themost catastrophic feature of the theory of catas-trophes is a full absence of references to the worksof H. Whitney,” Dima wrote in one of his books. In-deed, mathematically, the theory of catastropheswas based on a classification of singularities ofgeneric smooth maps of a plane onto a plane. Theclassification was fully done in 1955 by Whitney[15], but the founding fathers of catastrophe the-ory preferred to pretend that the works of Whitneynever existed. Still, Dima made his contributionto the popularization of catastrophes: he wrotea short popular book under the title “Theory ofCatastrophes”. It was written in 1983 and thentranslated into a dozen languages.

In 1990 I moved to a different country, andwe met only four or five times after that. Thelast time that I saw him was in spring 2007,when he visited California. We travelled togetherthrough the Napa and Sonoma Valleys; he was es-pecially interested in visiting Jack London’s grave.He spoke endlessly of his new (was it new?) pas-sion for continued fractions, numerical functions,and numerical experimentation. I boasted that Itaught a course of history of mathematics, and heimmediately began testing my knowledge of thesubject: Who proved the Euler theorem of poly-hedra? Who proved the Stokes theorem? To hisapparent displeasure I passed the exam. (He wasespecially surprised that I knew that Descartesproved the Euler theorem more than one hundredyears before Euler. Why do you know that? I saidthat Efremovich told me this some thirty yearsbefore.) More than that, I knew something that hedid not know: the Stokes theorem as it is stated inmodern books,

∫C dϕ =

∫∂Cϕ, was first proved and

published by the French mathematician E. Goursat(1917). We discussed a bit our further plans, andDima said that whatever he plans, he always adds,as Leo Tolstoy did, EBЖ = esli budu жiv,“If I am alive.” I said that I also never forget to addthis, but apparently neither of us took it seriously.Anyhow, we never met again.

In a cavern, 2008.

My tale of Dima Arnoldis becoming lengthy, al-though I feel that what Ihave said is a small fractionof what I could say aboutthis tremendous personal-ity. Still, the story wouldbe incomplete if I did notmention something knownto everybody who hasever communicated withhim, if only occasionally:his universal knowledge ofeverything. Whatever thesubject was—Chinese his-tory, African geography,French literature, the skyfull of stars (especially this:he could speak endlesslyon every star in every constellation)—he demon-strated without effort a familiarity with the subjectwhich exceeded and dwarfed everybody else’s, andthis, combined with his natural talent as a story-teller, made every meeting with him a memorableevent. Some friends recollect a sight-seeing tour inParis he gave a couple of months before his death.Obviously, no tourist agency ever had a guide ofthis quality. Instead of adding my own recollec-tions, I finish my account with a translation ofa letter I received from him a year after our lastmeeting and two years before his death.

Paris, March 26, 2008Dear Mitya,

I have recently returned toParis from Italy where I wan-dered, for three months, in karsticmountains working at ICPT (theInternational Center for Theo-retical Physics) at Miramare, theestate of the Austrian princeMaximilian who was persuadedby Napoleon III to become theEmperor of Mexico (for which hewas shot around 1867 as shownin the famous and blood-drenchedpicture of Edouard Manet).

I lived in the village of Sistiana,some 10 kilometers from Miramarein the direction of Venice. It wasfounded by the pope Sixtus, thesame one who gave names to boththe chapel and Madonna. Passingthe POKOPALIŠCE1 (the cemetery)some 3 versts2 to the North, Ireached a deer path in a moun-tain pine grove. These deer do

1This word has a notable similarity to Russian KOPATЬ,

to dig.2VERSTA is an old Russian measure of length, ≈ 1.1 km.


not pay much attention to a smalltin sign, DERŽAVNAYA MEŽA3 (thestate border). After that it is Slove-nia to which I ran, following thedeer. But at the next sign, PERI-COLO, the deer refused to go anyfarther. The local people (whoselanguage is closer to Russian thanUkrainian or Bulgarian) explainedto me that the sign is a warn-ing that the nearby caves havenot been demined. And they weremined during the FIRST world warwhen my deer path was called SEN-TIERA DIGUERRA and was a frontline (described by Hemingway in“A Farewell to Arms”).

I did not go down to theseparticular caves, but every dayI visited tens of them, of whichsome (but not all) were shown ona map (where they were calledYAMA,4 GROTTA, CAVA, CAV-ERNA, ABISSA, dependingly of thedifficulty of the descent). All thesecaves look pretty much the same(a colorful scheme is provided):there is a hole on the mountain, ameter in size, and down go walls,of not even vertical but rather anegative slope. The depth of themine is usually around 10 or 20meters (but I descended to YAMAFIGOVICHEVA with the officiallydeclared depth of 24 meters and tothe half of the height, or rather thedepth, of GROTTA TERNOVIZZAwhose depth is marked as 32meters and to which one cannotdescend without a rope). At thebottom of the YAMA a diverginglabyrinth of passages starts, of thelengths on the order of 100 meters.They go to lakes, stalactites, etc.Sometimes there is even a descentto the Timavo river (which flowsabout 50 kilometers at the depth100 or 200 meters, depending onthe height of hills above). Beforethis 50 kilometers it is a forestriver resembling Moscow River atNikolina Gora5 with a charmingRoman name of REKA.6

3Both words belong to old Russian.4Russian MA means a gap.5A village some 30 kilometers from Moscow where many

remarkable Russian people (including Dima) used to

spend their vacations.6REKA is the Russian for a river.

Ya. Eliashberg and V. Arnold, 1997.

This was a part of Jason’s expe-dition (with argonauts). On his wayback from Colchis (with the goldenfleece) he sailed his ship Argo up-stream Ister (Danube) and its trib-utaries to the Croatian peninsulanamed Capudistria (which is visi-ble from my window at Sistiana),then they dragged the ship toREKA and, following Timavo, theyreached to northernmost point ofthe Adriatic, where the Roman cityof Aquileia was later built.

Near Aquileia, I discovered agoddess Methe, new to me, butthis is a separate story. (She savesany drinker of drunkenness, how-ever much he drank. Allegedly, shewas the mother of Athena, andJupiter ate her, since he was afraidthat she would give birth to a son,and that this son would dethronehim, precisely as he himself haddethroned his father.) Aquileia isa Roman port of the first century,preserved as well as Pompeii, with-out any Vesuvius: simply Attilawho destroyed the city left the portintact, including the canals, ships(which survive to our time), quays,knechts and basilicas (which be-came Christian in the IV century)with mosaics of 50 m × 100 m insize, and absolutely everything asin Pompeii. No room to describeeverything, I am just sending mybest (Easter) wishes.

On June 3, I go to Moscow, therewill be a conference dedicated tothe centenary of LSP.7

Dima

7Lev Semenovich Pontryagin


Yakov Eliashberg

My Encounters with Vladimir IgorevichArnoldMy formation as a mathematician was greatlyinfluenced by Vladimir Igorevich Arnold, though Inever was his student and even lived in a differentcity. When I entered Leningrad University in 1964as an undergraduate math student, Arnold wasalready a famous mathematician. By that timehe had solved Hilbert’s 13th problem and hadwritten a series of papers which made him the “A”in the KAM theory. Arnold was also working asan editor of the publishing house Mir, where heorganized and edited translations of several booksand collections of papers not readily accessiblein the USSR. One of these books, a collection ofpapers on singularities of differentiable mappings,was an eye opener for me.

The first time I met Arnold was in January 1969at a Winter Mathematical School at Tsakhkadzor inArmenia. I was eager to tell him about some of myrecently proved results concerning the topology ofsingularities. Later that year he invited me to givea talk at his famous Moscow seminar. I rememberbeing extremely nervous going there. I could notsleep at all in the night train from Leningrad toMoscow, and I do not remember anything aboutthe talk itself.

In 1972 Vladimir Igorevich was one of myPh.D. dissertation referees or, as it was called, an“official opponent”. I remember that on the dayof my defense, I met him at 5 a.m. at the MoscowTrain Station in Leningrad. He immediately toldme that one of the lemmas in my thesis was wrong.It was a local lemma about the normal form ofsingularities, and I thought (and, frankly, still do)that the claim is obvious. I spent the next twohours trying to convince Vladimir Igorevich, andhe finally conceded that probably the claim iscorrect, but still insisted that I did not really havethe proof. A year later he wrote a paper devotedto the proof of that lemma and sent me a preprintwith a note that now my dissertation is on firmground.

After my Ph.D. defense I was sent to work ata newly organized university in Syktyvkar, thecapital of Komi Republic in the north of Rus-sia. In 1977 we organized there a conference onglobal analysis which attracted a stellar list ofparticipants, including V. I. Arnold. During thisconference I asked Arnold to give a lecture forour undergraduate students. He readily agreedand gave an extremely interesting lecture aboutstability of the inverse pendulum, and even madea demonstration prepared with the help of one of

Yakov Eliashberg is professor of mathematics at Stan-

ford University. His email address is eliash@math.

stanford.edu.

A talk at Syktyvkar, 1976.

our professors, Alesha Zhubr. Arnold had certainpedagogical methods to keep the audience awake.During his lectures he liked to make small mis-takes, expecting students to notice and correcthim. Apparently, this method worked quite well atthe Moscow University. Following the same routineduring his Syktyvkar lecture, he made an obviouscomputational error—something like forgettingthe minus sign in the formula (cosx)′ = − sinx—and expected somebody in the audience to correcthim. No one did, and he had to continue withthe computation, which, of course, went astray:the terms which were supposed to cancel did not.Very irritated, Arnold erased the blackboard andstarted the computation all over again, this timewithout any mistakes. After the lecture, he told methat the undergraduate students at Syktyvkar Uni-versity are very bad. The next day, after my regularclass, a few students came to me and asked howis it possible that such a famous mathematician ismaking mistakes in differentiating cosx?

Whenever I happened to be in Moscow, whichwas not very often, Arnold usually invited meto visit the hospitable home he shared with hiswife, Elya. When he moved to a new apartment inYasenevo on the outskirts of Moscow, he told meover the phone how to get there. In particular, Iwas instructed to walk south when I got out of themetro station. When I got to that point it was adark gray late winter afternoon, and it was quite achallenge to figure out in which direction I shouldgo.

Once he ran a psychological test on me todetermine which of my brain hemispheres is thedominant one. To his satisfaction, the test showedthat it was the right one, which, according toArnold, meant that I have a geometric rather thanan algebraic way of thinking. During another visit, Iwas deeply honored when he told me that while hefiles most preprints systematically, I was amongthe few people who were assigned a personalfolder.

Over the years I gave a number of talks at hisseminar with variable success. The mostdisastrouswas my last talk in 1985. Shortly before one


of my trips to Moscow, Misha Gromov sent mea preliminary version of his now very famouspaper “Pseudoholomorphic curves in symplecticgeometry”, which is one of the major foundationalmilestones of symplectic topology. I was extremelyexcited about this paper and thus volunteered totalk about it at Arnold’s seminar. I think that I wasat this moment the only person in the Soviet Unionwho had the paper. Arnold heard about Gromov’sbreakthrough but had not seen the paper yet. Aftera few minutes of my talk, Arnold interrupted meand requested that before continuing I shouldexplain what is the main idea of the paper. Thispaper is full of new ideas and, in my opinion, it isquite subjective to say which one is the main one.I made several attempts to start from differentpoints, but Arnold was never satisfied. Finally,towards the end of the two-hour long seminar, Isaid something which Arnold liked. “Why did youwaste our time and did not start with this fromthe very beginning?”, he demanded.

Vladimir Igorevich made two long visits to Stan-ford. During his first quarter-long visit Arnold wasgiving a lecture course, but he made it a rule forhimself to go every morning for a long bike rideinto the hills (called the Santa Cruz Mountains)surrounding Stanford. I have heard a lot of sto-ries about Arnold’s superhuman endurance andhis extremely risky adventures, especially in hisyounger years. I can testify that at almost sixtyyears old, Arnold at Stanford was also very im-pressive. On a windy day after swimming in ourcold Pacific Ocean, where the water temperatureis usually around 13C, he refused a towel. Hehad a very poor bike which was not especiallysuited for mountain biking. Yet he went with iteverywhere, even over the roads whose parts weredestroyed by a mudslide and where he had toclimb clutching the tree roots, hauling his bikeon his back. During one of these trips, VladimirIgorevich met a mountain lion. He described thisencounter in one of his short stories. Both Arnoldand the lion were apparently equally impressedwith the meeting. Many years later, during hissecond visit to Stanford, Arnold again went tothe same place hoping to meet the mountain lion.Amazingly, the lion waited for him there! I am alsofond of hiking in those hills, yet neither I nor anyof people I know ever met a mountain lion there.

When he was leaving Stanford, Vladimir Igore-vich gave me a present—a map of the local hillson which he had marked several interesting placesthat he had discovered, such as an abandonedapple farm or a walnut tree grove.

In between the two visits Arnold had a terriblebike accident in Paris which he barely survived. Itwas a great relief to see him active again when Imet him in Paris two years later. He proudly toldme that during this year he had written five books.“One of these books,” he said, “is coauthored with

Vladimir Arnold, 1957.

two presidents. Can you guess with which ones?” Icertainly could not guess that these were VladimirPutin and George W. Bush.

During his last visit to Stanford and Berkeley ayear ago, Arnold gave two series of lectures: onefor “Stanford professors”, as he called it, and theother for the school-age children at Berkeley MathCircles. There is no telling which of these twogroups of listeners Vladimir Igorevich preferred.He spent all his time preparing for his lectures forchildren and even wrote a book for them. Lecturesat Stanford were an obvious distraction from thatmain activity. Each Stanford lecture he wouldusually start with a sentence like “What I am goingto talk about now is known to most kindergartenchildren in Moscow, but for Stanford professors Ido need to explain this.” What followed was alwaysfascinating and very interesting.

It is hard to come to terms that VladimirIgorevich Arnold is no longer with us. It is certainlytrue, though commonplace to say, that Arnold wasa great and extremely influential mathematician,that he created several mathematical schools, andthat his vision and conjectures shaped a large partof modern mathematics. But, besides all that, hewas a catalyst for the mathematical community. Hehated and always fought mediocrity everywhere.With his extreme and sometimes intentionallyoutrageous claims, he kept everybody on guard,not allowing us to comfortably fall asleep.

His departure is also painful to me because thereare several unfulfilled mathematical promiseswhich I made to him but never had time tofinish. Though it is too late, I will do it now as apriority.


Yulij Ilyashenko

V. I. Arnold, As I Have Seen HimA student, visiting his schoolmaster in math, thefamous and severe Morozkin. A radiant slim youth,almost a boy. This was Arnold as I first saw him,more than fifty years ago.

A graduate student (in 1960), conducting tu-torials in honors calculus (taught to freshmen atMekhmat, the Department of Mechanics and Math-ematics of the Moscow State University). There wasa permanent kind of smile on his face, his eyeswere sparkling, and when he looked at you, a waveof good will would come forth.

From 1968 to 1986 I had the privilege of work-ing with Arnold at the same section of Mekhmat,called the “Division of Differential Equations”. Itwas shaped by Petrovski and chaired by him untilhis premature death in 1973. When Arnold joinedthe division, it was full of the best experts in dif-ferential equations, partial and ordinary. BesidesArnold and Petrovski, the faculty of the divi-sion included stars of the elder generation (whowere then in their thirties and forties): Landis,Oleinik, Vishik, as well as brilliant mathemati-cians of Arnold’s generation: Egorov, Kondratiev,Kruzhkov, and others.

The first glorious results of Arnold are de-scribed in other papers in this collection. Let meturn to differential equations, a subject whose de-velopment I have been closely following. Needlessto say, these are personal remarks, not a completehistory.

In 1965 Arnold came back from France, wherehe spent almost a year. From there he brought akeen interest in the newborn singularity theory, ofwhich he became one of the founding fathers. Healso brought the philosophy of general positioninvented by René Thom, which became sort of acompass in Arnold’s investigations in differentialequations and bifurcation theory.

In the form that Arnold gave to it, this phi-losophy claimed that one should first investigateobjects in general position, then the simplestdegenerations, together with their unfoldings. Itmakes no sense to study degenerations of highercodimension until those of smaller codimensionhave been investigated.

In 1970 he published a short paper [6], in whicha strategy for developing any kind of local theorybased on the above philosophy was suggested. Healso defined algebraically solvable local problems.He started to call them “trivial”, but later stoppeddoing that. “Let us forget the overloaded term,” heonce told me about this word. In the same paperhe also stated that the problem of distinguishing

Yulij Ilyashenko is professor of mathematics at Cornell

University and Independent University of Moscow. His

email address is [email protected].

center and focus is trivial. Bruno challenged thisstatement, and I proved that the center-focusproblem is algebraically unsolvable (1972).

Also in 1970 Arnold proved that the problemof Lyapunov stability is algebraically unsolvable.He constructed a 3-parameter family in the spaceof high-order jets, where the boundary of stabilityis nonalgebraic. In the same paper he wrote:“One may expect that the Lyapunov stability,having lost algebraicity and no more restricted byanything, may present some pathologies on theset theoretic-level….” He also suggested that theproblem may be algorithmically unsolvable. Thisconjecture is still open. In the mid-1970s it turnedout that a nonalgebraic boundary of Lyapunovstability occurs in unfoldings of degenerationsof codimension three in the phase spaces ofdimension four. This was discovered by Shnol’ andKhazin, who investigated the stability problem inthe spirit of Arnold and studied all the degeneratecases up to codimension three.

In 1969 Bruno defended his famous doctoralthesis about analytic normal forms of differentialequations near singular points. One of his resultsis the so-called Bruno condition: a sufficient con-dition for the germ of a map to be analyticallyequivalent to its linear part. In dimension one,Yoccoz proved the necessity of this condition(1987); this result was rewarded by a Fields Medal,which he got in 1994. So the problem is still afocus of interest in the mathematics community.But let us get back to the late 1960s. In his reviewof the Bruno thesis, Arnold wrote: “The existingproofs of the divergence [of normalizing series]are based on computations of the growth of coef-ficients and do not explain its nature (in the samesense as the computation of the coefficients ofthe series arctanz does not explain the divergenceof this series for |z| > 1, although it proves thisdivergence).” Following this idea, Arnold tried tofind a geometric explanation of the divergenceof normalizing series when the denominators aretoo small. He predicted an effect which he latercalled “materialization of resonances”. An “almostresonant” germ of a vector field that gives riseto “exceedingly small denominators” is close to acountable number of resonant germs. Under theunfolding of any such germ, an invariant mani-fold bifurcates from a union of coordinate planesand remains in a small neighborhood of the sin-gular point of this almost resonant germ. Theseinvariant manifolds, which constitute a countablenumber of “materialized resonances”, accumulateto the singular point and prevent the linearization.

A. Pyartli, a student of Arnold, justified thisheuristic description in his thesis in the early 1970sfor vector fields with planar saddles. He continuedthe investigation and in 1976 found an invariantcylinder, a materializationof resonances fora germof a planar map. Then he asked Arnold, “Why does


With students of Moscow MathematicsBoarding School, 1960s.

such a cylinder prevent the linearization?” Why,indeed?! Arnold himself started thinking aboutthe problem and came to the theory of normalforms for neighborhoods of embedded ellipticcurves. An overview of this theory is given in hisbook [12]. As usual, this new path was paved bythe followers of Arnold: Pyartli, myself, Saveliev,Sedykh, and others.

Arnold’s approach to the local bifurcation the-ory produced a genuine revolution. In the late1960s he suggested to his students two prob-lems: to prove a reduction principle that excludesexcessive “hyperbolic variables” from any localbifurcation problem and to study the first reallydifficult bifurcation problem in codimension two.The first problem was solved by A. Shoshitaishvili,the second one by R. Bogdanov. “It was not bychance that I launched two different people in twodirections simultaneously,” Arnold later said tome. Arnold was especially proud that Bogdanovproved the uniqueness of the limit cycle that oc-curs under the perturbation of a generic cuspidalsingular point. F. Takens investigated indepen-dently the same codimension two bifurcation asBogdanov; it is now named the “Bogdanov-Takens”bifurcation.

In [8] Arnold described the new approach tothe theory and listed all problems that occur inthe study of local bifurcations of singular pointsof vector fields in codimension two. This wasa long-standing program. J. Guckenheimer andN. Gavrilov made important contributions to itsdevelopment; final solutions were obtained byH. Zoladec (in the mid-1980s), again under the(nonofficial) supervision of Arnold.

In the mid-1970s Arnold himself consideredanother local bifurcation problem in codimensiontwo, the one for periodic orbits. He discoveredstrong resonances in the problem and predicted allpossible unfoldings occurring in generic perturba-tions of the Poincaré maps with these resonances(1977). There were four of them. The first case

was reduced to Bogdanov-Takens; two other caseswere investigated by E. Horosov (1979), a graduatestudent of Arnold, in his Ph.D. thesis. The fourthcase, the famous resonance 1 : 4, was investigatedby A. Neishtadt, F. Berezovskaya, A. Khibnik (in-fluenced by Arnold), and B. Krauskopf, a studentof Takens. The problem that remains unsolved forbifurcations of codimension two is the existenceof very narrow chaotic domains in the parameterand phase spaces.

Later local bifurcations of codimension threewere investigated by Dumortier, Roussarie, So-tomayor, and others. The bifurcation diagramsand the phase portraits became more and morecomplicated. It became clear that it is hopelessto get a complete picture in codimension four.The new part of the bifurcation theory startedby Arnold and his school seems to be completedby now. What is described above is a very smallpart of the new domains that were opened inmathematics by Arnold.

One should not forget that Arnold also inspiredmany discoveries in oral communications, whileno trace of this influence is left in his publications.For instance, he discovered “hidden dynamics” invarious problems of singularity theory. This meansthat a classification problem for singularities oftengives rise, in a nonevident way, to a classificationproblem for special local maps. Thus, he inspiredthe solution by S. Voronin (1982) of the local clas-sification problem for singularities of envelopesfor families of planar curves and the discoveryof quite unexpected Ecalle-Voronin moduli of theanalytic classification of parabolic fixed points(1981).

Arnold suggested a sketch of the proof ofanalytic unsolvability of the Lyapunov stabilityproblem (Ilyashenko, 1976). Only later did I un-derstand that, honestly speaking, it should havebeen a joint work.

In 1980 he pointed out that our joint workwith A. Chetaev on an estimate of the Hausdorffdimension of attractors might be applied to the2D Navier-Stokes equation. This gave rise to anexplicit estimate of the Hausdorff dimension ofthese attractors (Ilyashenko, 1982–83), a first stepin the subject later developed by O. Ladyzhenskayaand M. Vishik with his school.

This is only my personal experience, a minorpart of the great panorama of Arnold’s influenceon contemporary mathematics. He had a verystrong feeling of mathematical beauty, and hismathematics was at the same time poetry and art.From my youth, I considered Arnold as a Pushkinin mathematics. At present, Pushkin is a belovedtreasure of the Russian culture, but during his life,he was not at all treated as a treasure.


V. Arnold and Yu. Ilyashenko, 1997.

The same is true for Arnold. His life in Russiabefore perestroika was in no way a bed of roses.I remember very well how we young admirersof Arnold expected in 1974 that he would beawarded the Fields Medal at the ICM at Vancou-ver. He did not receive it, and the rumor wasthat Pontryagin, the head of the Soviet NationalMathematics Committee, at the discussion of thefuture awards said, “I do not know the works ofsuch a mathematician.” For sure, it could not havebeen the personal attitude of Pontryagin only; itwas actually the position of the Soviet governmentitself. Two medals instead of four were awardedthat year. Much later, Arnold wrote that one of theothers was intended for him, and then awarded tonobody.

In 1984 a very skillful baiting of Arnold wasorganized at Mekhmat. As a result, he had aserious hypertension attack. His election as acorresponding member of the Soviet Academy ofSciences stopped the baiting, but his enemies tried(though unsuccessfully) to renew it five years later.

In 1986 Arnold decided to quit Mekhmat and tomove to the Steklov Institute. Yet he wanted to keepa half-time position of professor at Mekhmat. Onlyafter considerable efforts did he get the desiredhalf-time position. I tried to convince Arnold notto quit Mekhmat. I asked him, “Dima, who may say,following Louis XIV’s ‘L’etat s’est moi,’ Mekhmat isme?” “Well,” he answered, “I guess NN” (he namedan influential party member at the department).“No, Dima, YOU are Mekhmat.” But he did notlisten.

In 1994 he quit Mekhmat completely. He wasoffended. He taught a course and a seminar,and suddenly he was informed that this load wasinsufficient for the half-time position of professor,but only for a quarter-time position (a status that

does not, in fact, exist). He spoke with the headof Mekhmat Human Resources. This was an agedwoman who maintained her position from thecommunist times. “She screamed at me,” saidArnold with a sort of surprise. Then he resignedfrom the Moscow State University.

Needless to say, in such an environment thestudents of Arnold were not hired at Mekhmat.The only exceptions were N. Nekhoroshev andA. Koushnirenko, hired in the early 1970s, andmuch later A. Varchenko. I remember two otherattempts, both unsuccessful. At the same time, thebest of the best Mekhmat students asked Arnoldto be their advisor. So, Mekhmat rejected the bestof the best of its alumni. The same happened withstudents of Manin, Kirillov, Gelfand.…At the end ofthe 1980s, a critical mass of excellent mathemati-cians not involved in the official academic life hadaccumulated. Following a suggestion of N. N. Kon-stantinov, a well-known educator and organizerof mathematical olympiads, these mathematiciansdecided to create their own university. In 1991 agroup of leading Russian mathematicians formeda council and established a new IndependentUniversity of Moscow, IUM. This group includedthe following members of the Russian Academyof Sciences: V. I. Arnold (chairman of the coun-cil), S. P. Novikov, Ya. G. Sinai, L. D. Faddeev;and the following professors: A. A. Beilinson,R. L. Dobrushin, B. A. Dubrovin, A. A. Kirillov,A. N. Rudakov, V. M. Tikhomirov, A. G. Khovanskii,M. A. Shubin. Professors P. Deligne and R. MacPher-son of Princeton and MIT also played crucial rolesin the founding of the Independent University.

Arnold was very enthusiastic about the newuniversity, and in the first years of its existencehe did a lot to shape its spirit and teaching style.Together with the first dean of the College of Math-ematics of the IUM, A. Rudakov, Arnold thoroughlydiscussed the programs, and he himself taught acourse on partial differential equations. Under hisinfluence, the Independent University became oneof the focal centers of Russian mathematical life.

In 1994 another educational institution, theMoscow Center of Continuous Mathematical Ed-ucation (MCCME), was created. From the verybeginning, Arnold was the head of the board oftrustees of this center. The center, headed byI. Yashchenko, the director, became a very influen-tial institution in Russian mathematical educationand a powerful tool in the struggle against modernobscurantism. Arnold was one of the leaders ofthat struggle.

In 2005 Pierre Deligne, together with the IUMfaculty, organized a contest for young Russianmathematicians. This contest was funded byDeligne from his Balzan Prize (and named after


him) with the goal “to support Russian mathe-matics, struggling for survival.” The funds of thecontest were strictly limited. In 2006 Arnold metD. Zimin, the head of “D. B. Zimin’s Charity Foun-dation Dynasty”, and convinced him to establish asimilar “Dynasty contest”. Now the contest has be-come permanent, Lord willing and the creek don’t

rise, as the proverb says. This is only one of theexamples of the long-lasting influence of Arnoldon Russian mathematical life.

Arnold’s talks were always special events.He began giving lectures at Mekhmat in Sep-tember 1961 about the newborn theory laternamed KAM (Kolmogorov–Arnold–Moser). A ru-mor spread among the students that “Arnold hassolved problems that Poincaré failed to solve.”His lectures were very fast and intense, yet theyattracted the best students in the department.He repeated this course twice, in 1962–63 and in1963–64.

After that he gave brilliant courses in theo-retical mechanics, ordinary differential equations,supplementary chapters of ODE, singularity the-ory, geometric theory of PDE, and many others. Allthese courses gave rise to world-famous books,written by Arnold, sometimes with his students.In 1968 Arnold started teaching a course in ODEthat became, in a sense, a course of his life. Hetaught it every year until the late eighties, exceptfor sabbaticals.

Arnold completely changed the face of the dis-cipline. His presentation was coordinate-free: allthe constructions were invariant with respect tocoordinate changes. “When you present materialin coordinates,” he said, “you study your coor-dinate system, not the effect that you want todescribe.” His language was quite different fromthat of the previous textbooks and courses: dif-feomorphisms, phase flows, rectification of vectorfields, exponentials of linear operators.…The lan-guage of pictures was even more important in hiscourse than that of formulas. He always requireda student to present the answer in both ways, aformula and a figure, and to explain the relationbetween them. He drastically renewed the prob-lem sets for the course: propagation of rays innonuniform media and geodesics on surfaces ofrevolution, phase portraits of the Newton equationwith one degree of freedom, images of the unitsquare under linear phase flows—students wereexpected to draft all of these even without ex-plicit calculations of the corresponding solutions.In the first years the course was difficult bothfor students and teaching assistants. Later on itsmoothed out and became one of the highlightsof the Mekhmat curriculum.

All his life V. I. Arnold was like a star thatshines, sparkles, and produces new life around it.

V. Arnold and B. Khesin, Toronto, 1997.

Boris Khesin

On V. I. Arnold and HydrodynamicsBack in the mid-1980s, Vladimir Igorevich oncetold us, his students, how different the notion of“being young” (and in particular, being a youngmathematician) is in different societies. For in-stance, the Moscow Mathematical Society awardsan annual prize to a young mathematician underthirty years of age. The Fields Medal, as is wellknown, recognizes outstanding young mathemati-cians whose age does not exceed forty in the yearof the International Congress. Both of the aboverequirements are strictly enforced.

This can be compared with the Bourbaki group,which is comprised of young French mathemati-cians and which, reportedly, has an age bar offifty. However, as Arnold elaborated the story, thislimit is more flexible: upon reaching this age theBourbaki member undergoes a “coconutizationprocedure”. The term is derived from a traditionof some barbaric tribe that allows its chief to carryout his duties until someone doubts his leadershipabilities. Once the doubt arises, the chief is forcedto climb to the top of a tall palm tree, and the wholetribe starts shaking it. If the chief is strong enoughto get a good grip and survives the challenge, heis allowed to climb down and continue to lead thetribe until the next “reasonable doubt” in his lead-ership crosses someone’s mind. If his grip is weakand he falls down from the 20-meter-tall tree, heobviously needs to be replaced, and so the nexttribe chief is chosen. This tree is usually a coconutpalm, which gave the name to the coconutizationprocedure.

As far as the coconutization in the Bourbakigroup is concerned, according to Arnold’s story,the unsuspecting member who reaches fifty is

Boris Khesin is professor of mathematics at the Uni-

versity of Toronto. His email address is khesin@

math.toronto.edu.


invited, as usual, to the next Bourbaki seminar.Somewhere in the middle of the talk, when mostof the audience is already half asleep, the speaker,who is in on the game for that occasion, in-serts some tedious half-a-page-long definition. Itis at this very moment that the scrutinized (“co-conutized”) member is expected to interrupt thespeaker by exclaiming something like, “But excuseme, only the empty set satisfies your definition!” Ifhe does so, he has successfully passed the test andwill remain a part of Bourbaki. If he misses thischance, nobody says a word, but he will probablynot be invited to the meetings any longer.

Arnold finished this story by quoting someone’sdefinition of youth in mathematics which he likedbest: “A mathematician is young as long as hereads works other than his own!”

Soon after this “storytelling” occasion, Arnold’sfiftieth anniversary was celebrated: in June 1987his whole seminar went for a picnic in a suburb ofMoscow. Among Arnold’s presents were a “Returnto Arnold” stamp to mark the reprints he gave tohis students to work on, a mantle with a nicelydecorated “swallowtail”, one of low-dimensionalsingularities, and such. But, most importantly,he was presented with a poster containing acrossword on various notions from his manyresearch domains. Most of the questions wererather intricate, which predictably did not preventArnold from easily cracking virtually everything.But one question remained unresolved: a five-letterword (in the English translation) for “A simplealternative of life”. None of the ideas workedfor quite some time. After a while, having madeno progress on this question, Arnold pronouncedsadly, “Now I myself have been coconutized.…” Buta second later he perked up, a bright mischievousexpression on his face: “This is a PURSE!” (Inaddition to the pirate’s alternative “Purse or Life”,the crossword authors meant the term “purse”in singularity theory standing for the descriptionof the bifurcation diagram of the real simplesingularity D+

4 , also called hyperbolic umbilic—hence the hint on “simple” alternative.)

Arnold’s interest in fluid dynamics can be tracedback to his “younger years”, whatever definitionone is using for that purpose. His 1966 paper inthe Annales de l’Institut Fourier had the effect ofa bombshell. Now, over forty years later, virtuallyevery paper related to the geometry of the hy-drodynamical Euler equation or diffeomorphismgroups cites Arnold’s work on the starting pages.In the next four or five years Arnold laid outthe foundations for the study of hydrodynamicalstability and for the use of Hamiltonian methodsthere, described the topology of steady flows, etc.

New Haven, 1993.

Apparently Arnold’s interest in hydrodynam-ics is rooted in Kolmogorov’s turbulence studyand started with the program outlined by Kol-mogorov for his seminar in 1958-59. Kolmogorovconjectured stochastization in dynamical sys-tems related to hydrodynamical PDEs as viscosityvanishes, which would imply the practical impos-sibility of long-term weather forecasts. Arnold’stake on hydrodynamics was, however, completelydifferent from Kolmogorov’s and involved groupsand topology.

The Euler equation of an ideal incompressiblefluid filling a domain M in Rn is the evolutionequation

∂tv + (v,∇)v = −∇p

on the fluid velocity field v , where this field isassumed to be divergence-free and tangent to theboundary of M (while the pressure p is defineduniquely modulo an additive constant by theseconditions on v). In 1966 Arnold showed that thisEuler equation can be regarded as the equation ofthe geodesic flow on the group SDiff(M) of volume-preserving diffeomorphisms of the domainM . Thecorresponding metric on this infinite-dimensionalgroup is the right-invariant L2 metric definedby the kinetic energy E(v) =

12‖v‖

2L2(M) of the

fluid. (The analysis of Sobolev spaces related tothis group-theoretic framework in incompressiblefluid dynamics was later furnished by D. Ebin andJ. Marsden.) Arnold’s geometric view on hydrody-namics opened a multitude of different researchdirections:

Other groups and metrics. Many other evolutionequations turned out to fit this universal approachsuggested by Arnold, as they were found to de-scribe geodesic flows on appropriate Lie groupswith respect to one-sided invariant metrics. Thisshed new light on the corresponding configurationspaces and symmetries behind the relevant physi-cal systems, and such geodesic equations are now


Vladimir Arnold with his wife, Elya, 1997.

called the Euler-Arnold equations. Here are severalexamples developed by many authors. The groupSO(3) with a left-invariant metric corresponds tothe Euler top (this example appeared in the originalpaper by Arnold along with the hydrodynamicalEuler equation). Similarly, the Kirchhoff equationsfor a rigid body dynamics in a fluid describegeodesics on the group E(3) = SO(3) ⋉R3 of Eu-clidean motions of R3. In infinite dimensions, thegroup of circle diffeomorphisms Diff(S1) with theright-invariant L2-metric gives the inviscid Burg-ers equation, while the Virasoro group for threedifferent metrics, L2, H1, and H1, produces re-spectively the Korteweg-de Vries, Camassa-Holm,and Hunter-Saxton equations, which are differ-ent integrable hydrodynamical approximations.The self-consistent magnetohydrodynamics de-scribing simultaneous evolution of the fluid andmagnetic field corresponds to dynamics on thesemidirect product group SDiff(M) ⋉ SVect(M)equipped with an L2-type metric. Yet another in-teresting example, known as the Heisenberg chainor Landau–Lifschitz equation, corresponds to thegauge transformation group C∞(S1, SO(3)) andH−1-type metric. Teasing physicists, Arnold usedto say that their gauge groups are too simple toserve as a model for hydrodynamics.

Arnold’s stability and Hamiltonian methods in

hydrodynamics. The geodesic property of the Eu-ler hydrodynamical equation implied that it isHamiltonian when considered on the dual of theLie algebra of divergence-free vector fields. Arnoldproposed using the corresponding Casimir func-tions, which are invariants of the flow vorticity,to study stability of steady fluid flows. Arnold’sstability is now the main tool in the study of non-linear stability of fluid motions and MHD flows.In particular, he proved that planar parallel flowswith no inflection points in their velocity profilesare stable. (One should note that, for Hamiltoniansystems, stability in linear approximation is al-ways neutral and inconclusive about the stabilityin the corresponding nonlinear problem, so the

result on a genuine Lyapunov stability of certainfluid flows was particularly rare and valuable.)

Study of fluid Lagrangian instability and curva-tures of diffeomorphism groups. Negative sectionalcurvature on manifolds implies exponential diver-gence of geodesics on them. In the 1966 Ann. Inst.Fourier paper Arnold launched the first compu-tations of curvatures for diffeomorphism groups.Negativity of most of such curvatures for thegroups of volume diffeomorphisms suggested La-grangian instability of the corresponding fluidflows. By applying this to the the atmosphericflows, he gave a qualitative explanation of un-reliability of long-term weather forecasts (thusanswering in his own way the problem posed byKolmogorov in the 1950s). In particular, Arnoldestimated that, due to exponential divergence ofgeodesics, in order to predict the weather twomonths in advance one must have initial dataon the state of the Earth’s atmosphere with fivemore digits of accuracy than that of the expectedprediction. In practical terms this means that adynamical weather forecast for such a long periodis impossible.

The hydrodynamical Appendix 2 in the famousClassical Mechanics by Arnold,8 where one canfind the details of the above-mentioned calcula-tion for the Earth’s atmosphere, also contains oneof Arnold’s widely cited phrases: “We agree ona simplifying assumption that the earth has theshape of a torus,” which is followed by his cal-culations for the group of area-preserving torusdiffeomorphisms. It is remarkable that the latercurvature calculations for the group of spherediffeomorphisms (performed by A. Lukatskii) gaveexactly the same order of magnitude and quan-titative estimates for the curvature, and hencefor the atmospheric flows, as Arnold’s originalcomputations for the torus!

Topology of steady flows. One of the mostbeautiful observations of Arnold (and one ofthe simplest —it could have belonged to Eu-ler!) was the description of topology of stationarysolutions of the 3D Euler equation. It turns outthat for a “generic” steady solution the flow do-main is fibered (away from a certain hypersurface)into invariant tori or annuli. The correspondingfluid motion on each torus is either periodic orquasiperiodic, while on each annulus it is periodic.This way a steady 3D flow looks like a completely

8Speaking of writing, once I asked Arnold how he man-

aged to make his books so easy to read. He replied: “To

make sure that your books are read fast, you have to

write them fast.” His own writing speed was legendary.

His book on invariants of plane curves in the AMS Univer-

sity Lecture series was reportedly written in less than two

days. Once he pretended to complain: “I tried, but failed,

to write more than 30 pages a day.…I mean to write in

English; of course, in Russian, I can write much more!”


integrable Hamiltonian system with two degreesof freedom.

The nongeneric steady flows include Beltramifields (those collinear with their vorticity) and, inparticular, the eigenfields for the cur l operatoron manifolds. The latter include the so-called ABCflows (for Arnold–Beltrami–Childress), the curleigenfields on the 3D torus, which happen to havebecome a great model for various fast dynamoconstructions.

Fast dynamo and magnetohydrodymanics.Arnold’s interest in magnetohydrodynamics wasto a large extent related to his acquaintancewith Ya. Zeldovich and A. Sakharov. One of theresults of their interaction at the seminars wasthe Arnold-Ruzmaikin-Sokolov-Zeldovich modelof the fast dynamo on a 3D Riemannian manifoldconstructed from Arnold’s cat map on a 2Dtorus. For a long time this was the only dynamoconstruction allowing complete analytical studyfor both zero and positive magnetic dissipation.

The asymptotic Hopf invariant. Finally, one ofthe gems of topological hydrodynamics is Arnold’s1974 study of the asymptotic Hopf invariant for avector field. He proved that, for a divergence-freevector field v in a 3D simply connected manifoldM , the field’s helicity, H(v) :=

∫M(cur l

−1v, v)d3x,is equal to the average linking number of all pairsof trajectories of v . This theorem simultaneouslygeneralized the Hopf invariant from maps S3 → S2

to arbitrary divergence-free vector fields in S3,enriched K. Moffatt’s result on the helicity oflinked solid tori, described the topology behindthe conservation law of the 3D Euler equation, andprovided the topological obstruction to the energyrelaxation of magnetic vector fields. This eleganttheorem stimulated a tide of generalizations tohigher-dimensional manifolds, to linking of folia-tions, to higher linkings, and to energy estimatesvia crossing numbers. In particular, there was sub-stantial progress in the two directions suggestedin the original 1974 paper: the topological invari-ance of the asymptotic Hopf numbers for a largeclass of systems was proved by J.-M. Gambaudoand É. Ghys, while the Sakharov–Zeldovich prob-lem on whether one can make arbitrarily smallthe energy of the rotation field in a 3D ball bya volume-preserving diffeomorphism action wasaffirmatively solved by M. Freedman.

Virtually single-handedly Arnold spawned anew domain, now called topological fluid dynam-ics. His contribution to this area changed thewhole paradigm of theoretical hydrodynamics byemploying groups to study fluid flows. What dou-bles the awe is that this gem appeared almostat the same time with two other Arnold’s foun-dational contributions—the KAM and singularitytheories.

Vladimir Arnold lecturing.

Victor Vassiliev

Topology in Arnold’s WorkArnold worked comparatively little on topologyfor topology’s sake. His topological studies wereusually motivated by specific problems from otherareas of mathematics and physics: algebraic ge-ometry, dynamical systems, symplectic geometry,hydrodynamics, geometric and quantum optics. Sothe (very significant) place of topological studiesin his work is well balanced with the (equally verysignificant) place and applications of topology inthe entirety of contemporary mathematics.

The main achievement in a number of hisworks is a proper recognition and formulation ofa topological result, allowing topologists to enterthe area with their strong methods. A huge part ofArnold’s work is contained not in his own articlesbut in well-formulated problems and hints thathe gave to his students and other researchers; seeespecially [13]. So I will discuss below such Arnoldhints as well and what followed from them.

Superpositions of Functions

The case of real functions: Kolmogorov-Arnold’stheorem and Hilbert’s 13th problem. This theoremstates that every continuous function of n > 2variables can be represented by a superpositionof functions in 2 variables (and the superpositioncan be taken in a particular form). The firstapproach to this problem (based on the notionof the Kronrod tree of connected componentsof level sets) was found by Kolmogorov (1956),

Victor Vassiliev is professor of mathematics at the Steklov

Institute of Mathematics Higher School of Economics and

at the Independent University of Moscow. His email ad-

dress is [email protected].


Vladimir Arnold.

who did not, however, overcome some technicallow-dimensional difficulties and proved only thesame theorem with 2 replaced by 3. The finaleffort was made by (then-19-year-old) Arnold.

This theorem gives a negative solution to (prob-ably the most natural exact understanding of) thefollowing Hilbert 13th problem:

…it is probable that the root of theequation of the seventh degree isa function of its coefficients whichdoes not belong to this class offunctions capable of nomographicconstruction, i.e., that it cannotbe constructed by a finite numberof insertions of functions of twoarguments. In order to prove this,the proof would be necessary thatthe equation of the seventh degree

(1) t7 + xt3 + yt2 + zt + 1 = 0

is not solvable with the help of anycontinuous functions of only twoarguments.

A widespread belief concerning this problemis as follows: “with the help of functions” in itslast sentence means that a continuous solutiont(x, y, z)of (1) should indeed be given by a functionof the form described in the first one, i.e., by asuperposition of continuous functions of twoarguments. In this case the Kolmogorov-Arnoldtheorem would give a direct negative answer tothis problem. Nevertheless, this understanding ofHilbert’s question is probably erroneous, because(1) does not define any continuous function atall: the multivalued function t(x, y, z) defined by

(1) does not have any continuous cross-section on

the whole of R3(x,y,z) . Indeed, such negative-valued

cross-sections do not already exist in a small

neighborhood of the polynomial

t7 − 14t3 − 21t2 − 7t + 1

≡ (t + 1)3(t4 − 3t3 + 6t2 − 10t + 1).

Such a neighborhood admits two positive-valued

cross-sections, but they obviously cannot be con-tinued to the polynomial t7 + 1. So this directunderstanding of the Hilbert problem could becorrect only under the (quite improbable) con-jecture that Hilbert has included in this problemthe question whether (or was confident that) (1)defines a continuous function on the entire R3;in this case the problem would have a positivesolution.

A more realistic assumption is that “with thehelp of continuous functions of two variables”means something more flexible, for example, thatwe can consider a triple of functions (χ, g1, g2)

in x, y, z, defined by such superpositions, andrepresent our function t(x, y, z) by g1 in the areawhere χ > 0 and by g2 where χ ≤ 0. However, inthis case it is unclear why Hilbert did not believethat the desired representation (maybe with morefunctions χk and gi) does exist for his particularfunction, which is piecewise analytic and certainlycan be stratified by easy conditions into pieces withvery simple behavior. The most realistic conjectureis that (like for many other problems) Hilbertwrote a slightly obscure sentence specifically tolet the readers themselves formulate (and solve)the most interesting and actual exact statements:it is exactly what Kolmogorov and Arnold actuallydid.

Complex algebraic functions and braid cohomol-

ogy. Hilbert’s 13th problem, formally askingsomething about real continuous functions, isnevertheless evidently motivated by the study ofsuperpositions of multivalued algebraic functionsin complex variables. A dream problem in this areais to solve literally the same problem concerningsuch functions. Moreover, this problem was ex-plicitly formulated in one of Hilbert’s consequentworks.

Arnold worked much on this problem, revisingand reformulating the proof of the Ruffini-Abeltheorem in topological terms of ramified cover-ings and their topological invariants and trying toextend it to superpositions of functions in morevariables. Although the exact desired theorem wasnot proved, a byproduct of this attack was huge:among other topics, it contains the topologicaltheory of generalized discriminants, homologi-cal theory of braid groups, and theory of planearrangements. A particular result, the topologi-cal obstruction to the representation by complete

superpositions of functions depending on fewvariables, was expressed in [5] in the terms of


cohomology of braid groups. Indeed, the d-valuedalgebraic function t(x1, . . . , xd) given by

(2) td + x1td−1 + · · · + xd−1t + xd = 0

defines a d-fold covering over the set Cd \ Σ ofnondiscriminant points (x1, . . . , xd) (i.e., of polyno-mials (2) for which all d values t(x) are different).This covering defines (up to homotopy) a map fromits base Cd \ Σ to the classifying space K(S(d),1)of all d-fold coverings, thus also a canonical map

(3) H∗(K(S(d),1) → H∗(Cd \ Σ).If our algebraic function (2) is induced fromanother one, as in the definition of completesuperpositions, then this cohomology map factor-izes through the cohomology ring of some subsetof the argument space of this new algebraic func-tion. Hence the dimension of this space cannot besmaller than the highest dimension in which themap (3) is nontrivial.

This approach has strongly motivated the studyof the cohomology ring of the space Cd \ Σ (whichis the classifying space of the d-braid group) and,much more generally, of the following objects.

Discriminants and Their Complements

Given a space of geometric objects (say, func-tions, varieties, subvarieties, matrices, algebras,etc.), the discriminant subset in it consists of alldegenerate (in some precise sense) objects: itmay be the set of non-Morse functions or self-intersecting spatial curves, or degenerate (anotherversion: having multiple eigenvalues) operators.Usually one studies the complementary space ofnonsingular objects. However, Arnold’s seminalreduction replaces the homological part of thisstudy by that of discriminant spaces. Namely, in[4], Arnold exploits the Alexander isomorphism

(4) H i(Cd \ Σ) ≡ H2d−i−1(Σ),where H∗ means the homology of the one-pointcompactification and Cd is considered to be thespace of all complex polynomials (2) in one vari-able t . This reduction turned out to be extremelyfruitful, because the set of nonsingular objects isusually open and does not carry any natural geo-metric structure. To study its topology, we oftenneed to introduce some artificial structures on it,such as Morse functions, connections, families ofvector fields or plane distributions, etc., which canhave singularities helping us to calculate sometopological invariants. On the other hand, the dis-criminant varieties are genuinely stratified sets(whose stratification corresponds to the hierarchyof singularity types); this stratification allows oneto calculate various topological properties of thesevarieties and hence also of their complementarysets of generic objects. Already in [4] this approachhas brought some progress, although the completecalculation of the group (4) was done only later by

Figure 1. Stabilization of unfoldings.

D. Fuchs for Z2-cohomology [14] and by F. Cohenand F. Vainshtein for integral cohomology.

Using the same approach, Arnold studied latermany other spaces of nondegenerate objects,namely, spaces Pd \ Σk of real degree d poly-nomials R1 → R1 without roots of multiplicity ≥ k,k ≥ 3, spaces of functions R1 → R1 (with a fixedbehavior at infinity) also having no zeros of multi-plicity ≥ k (1989), spaces of Hermitian operatorswith simple spectra (1995), spaces of generic (orgeneric Legendrian) plane curves (1994), etc.

Another very important idea of Arnold’s in thisarea was his favorite stabilization problem, pub-lished first in 1976 and repeated many times inseminars; see problems 1975-19, 1980-15, 1985-7,1985-22 in [13]. Formally speaking, the Alexanderduality theorem is a finite-dimensional result. Also,all spaces of objects in which Arnold’s approachoriginally led to more or less explicit results werefinite-dimensional spaces considered as unfold-ings of some particular objects. For example, thespaceCd of complex polynomials (2) can be consid-ered as an unfolding of the monomial td . When thedegree d grows, the cohomology groups of spacesCd \ Σ of nondiscriminant polynomials stabilize(to the cohomology of the infinite braid group),but it was quite difficult to trace the stabiliza-tion process in terms of the original calculations.Moreover, it was unclear what happens with sim-ilar stabilizations for objects more complex thanjust polynomials in one variable, how to deal withsimilar infinite-dimensional problems, and what is“the mother of all unfoldings”. To attack this set ofphilosophical problems, Arnold formulated a veryexplicit sample problem. First, he noticed that thestabilization of cohomology groups such as (2) isnatural: if we have two singular objects, one ofwhich is “more singular” than the other, then theparameter space of the unfolding of the simplerobject can be embedded into that of the morecomplicated one. This map sends one discrimi-nant into the other, thus inducing the pull-back


Vladimir Igorevich Arnold.

map of cohomology groups of their complements.(For real polynomials t3 and t4 this embedding ofparameter spaces of their unfoldings t3 + at + band t4 + αt2 + βt + γ is shown in Figure 1. Thediscriminants drawn in this picture are the sets ofpolynomials having multiple roots.)

Arnold’s respective problem was to deter-mine the stable (under all such pull-back maps)cohomology groups of such complements of dis-criminants of isolated singularities of holomorphicfunctions in Cn (and to prove that they actuallydo stabilize; i.e., these stable cohomology groupsare realized by such groups for some sufficientlycomplicated singularities). Solving this problem,I found in 1985 a method of calculating homol-ogy groups of discriminants that behaves nicelyunder the embeddings of unfoldings and thusgives an effective calculation of stable groups.Some elaborations and byproducts of this calcu-lation method constitute a majority of my resultson topology of discriminants, including my firstworks on knot theory. In the original problem onstable cohomology of complements of discrimi-nants of holomorphic functions, this calculationgives us the following formula: the desired stablecohomology ring for singularities in n complexvariables is equal to H∗(Ω2nS2n+1), where Ωk isthe k-fold loop space.

Moreover, this Arnold problem not onlydealt with the stabilization of particular finite-dimensional objects, but it also gave an approachto the study of actual infinite-dimensional functionspaces.

Topology of Pure Braid Groups and PlaneArrangements

Together with the cohomology of the usual braidgroups (2), Arnold also investigated the pure braidgroup, i.e., the fundamental group of the set ofordered collections of d distinct points in C1. Theclassifying space of this group is just the space Cd

with all diagonal hyperplanes xi = xj for i ≠ jremoved. Arnold’s calculation of its cohomologygroup [2] became a sample and a starting pointof numerous generalizations and initiated the so-called theory of plane arrangements. The Arnoldidentity

ωij ∧ωjk +ωjk ∧ωki +ωki ∧ωij = 0

for basic classes of this cohomology ring laterbecame one of the main ingredients of Kont-sevich’s construction of the universal finite-typeknot invariant.

Maslov Index, Lagrange and Legendre Cobor-dism

Lagrange manifolds are specific n-dimensionalsubmanifolds of the symplectic spaceR2n (or, moregenerally, of the cotangent bundle of an arbitrarymanifoldMn). They occur in problemsof geometricoptics as the manifolds into which all rays of lightconsidered in such a problem can be lifted withoutintersections, and in quantum optics as a firststep in obtaining an asymptotic approximationof light diffusion. However, further steps of thisasymptotic description impose some consistencycondition: the composition of transition functionsrelating their expressions in neighboring localcharts should define the identity operator whenwe go along a closed chain of such charts. Thiscondition is best formulated in terms of a certain1-cohomology class of the Lagrange manifold, itsMaslov index. If the Lagrange manifold Ln ⊂ T∗Rn

is generic, then this index can be defined as theintersection index with the singular locus of theprojection Ln → Rn to the “physical” configurationspace. It is important for this definition that, forgenericLagrangianmanifolds, this locus has a well-defined transversal orientation (so that crossingit, we can always say whether we are going tothe positive or the negative side) and its singularpoints form a subset of dimension at most n−3 inLn (so that all homologous curves have one and thesame Maslov index). If Ln is orientable, then thisindex is even; the above self-consistency conditionrequires that the value of this index on any closedcurve should be a multiple of 4. Arnold [1] relatedthis index with the topology of the LagrangeGrassmann manifold of all Lagrangian planes inthe symplecticR2n-space, i.e., of all planes that canbe tangent to some Lagrange submanifolds in thisspace. This settles immediately various problemsrelated to the invariance of the definition of theMaslov index, as well as to its stability underdeformations of the Lagrange manifold.

In 1980 Arnold initiated the theory of Lagrangeand Legendre cobordisms [11]. Light distributionin the area defines light distribution on its border:for instance, the reflected light on the wall isdefined by the light in the entire room. This meansthat a Lagrange manifold in the cotangent bundle


The Bowen lectures, Berkeley, 1997.

of the room defines its Lagrange boundary, whichis a Lagrange manifold in the cotangent bundleof the wall. The Legendre manifolds are known tous mainly as resolutions of wave fronts. The wavefront evolving in space defines a wave front ofbigger dimension in the space-time. The fronts inMn corresponding to some instants T1 and T2 areobviously defined by the big front inMn× [T1, T2];the way in which they are obtained from thisbig front can be generalized to the notion of theLegendre boundary. Notice that both Lagrange andLegendre boundaries of manifolds are not theirboundaries and not even the subsets in the usualsense: they are obtained from these boundaries bysymplectic and contact reductions.

Arnold introduced cobordism theories basedon these boundary notions and calculated the1-dimensional Lagrange and Legendre cobordismgroups: they turned out to be isomorphic to Z⊕Rand Z, respectively. The Z-term in both answersis defined by the Maslov index, the R-invariant ofthe Lagrange cobordism is given by

∫pdq. Later,

Ya. Eliashberg and M. Audin, using the Gromov-Lees version of the Smale-Hirsch h-principle forLagrange manifolds, reduced the calculation of Le-gendre cobordism groups in any dimension to thestandard objects of the cobordism theory, namely,to homotopy groups of appropriate Thom spaces(over the stable Lagrange Grassmann manifold).

At the same time, in the beginning of 1980,Arnold asked me whether it is possible to extendthe construction of the Maslov index to cohomol-ogy classes of higher dimensions, dual to moredegenerate singular loci of the Lagrangian pro-jection Ln → Rn than just the entire singular set.The resulting cohomology classes were expectedto be closely related to the higher cohomologyclasses of Lagrange Grassmannians and to giveinvariants of Lagrange and Legendre cobordisms.The answer was found soon: I managed to con-struct the desired characteristic classes in terms ofthe universal complex of singularity types. Later,

this theory was nicely and strongly extended byM. Kazarian in terms of equivariant homology.

On the other hand, the work with 1-dimensionalwave fronts led Arnold to many essential problemsof contact geometry, such as the 4-cusps problem(see the photograph above). Solutions of theseproblems by Chekanov, Eliashberg, Pushkar′andothers resulted in significant development ofthis area.

There are many other topological results inArnold’s works, including major breakthroughs inreal algebraic geometry [7], [10]; Arnold’s conjec-ture in symplectic topology; the asymptotic Hopfinvariant; and the vanishing homology theory ofboundary singularities. These topics are coveredin other articles in this collection.

Helmut Hofer

Arnold and Symplectic GeometryV. I. Arnold was a character and a larger-than-lifefigure. I never knew him extremely well, but webecame closer over the years, and I learned toknow him a little bit more from the private side.He could be very charming.

As a student I read Arnold’s wonderful bookMathematical Methods of Classical Mechanics andwas impressed by the ease with which he was ableto bring across important ideas. I never expectedto meet him in real life.

I met him for the first time when I was a tenure-track professor at Rutgers University and wasvisiting the Courant Institute. This was between1986 and 1987, so around three years before theBerlin Wall and the iron curtain came down. TheCourant Institute had worked hard to make itpossible for Arnold to visit. I attended one ofArnold’s lectures, which was remarkable in twoways: there was great mathematics and somethingone would not expect in a mathematics lecture.At some point he went into a tirade about howWestern mathematicians were not giving propercredit to Russian mathematicians. Most people inthe audience took it with some kind of amusement,but not all. Somebody sitting beside me mumbledsomething along the lines that we should have lefthim in Moscow.

A year or so later he attended parts of thesymplectic year (1988) at MSRI in Berkeley. WhatI remember from his visit was that at some pointhe decided to swim in San Francisco Bay. Onehas to know that the locals do not considerthis the best idea, since the currents are quiteunpredictable. The story which was told at thattime was that he almost drowned fighting the

Helmut Hofer is professor of mathematics at the In-

stitute for Advanced Study. His email address is

[email protected].


currents. I thought to myself, “That is a reallyinteresting multidimensional character pushingthe envelope.” I recently asked about this ofRichard Montgomery, who had an account of thisstory from Arnold himself. He had concluded fromthe description that Arnold had tried to swim fromMarina Green to Marin (linked by the Golden GateBridge) during ebb tide and at some point, in hisown words, “It felt like I hit a wall of current”and “had to turn back.” The maximum ebb outof the San Francisco Bay can be over six knots. Ifhe hadn’t returned, he would have been swept atleast a mile out to sea. Talking to Richard I alsolearned about another story. He and Arnold wentkayaking in the bay. After an involuntary Eskimoroll, Arnold insisted on entering orthogonally intothe path of an ongoing yacht race, with 40-footyachts going full speed being unable to dodge akayak. Richard still remembers his fear of goingdown in mathematical infamy as the guy whokilled Arnold. As I said before, Arnold pushed theenvelope in real life as he did in mathematics.

One year later, in 1989, I became a full pro-fessor at the Ruhr-Universität Bochum. Shortlyafterwards the Berlin Wall came down, with dra-matic changes in Eastern Europe. Soon a completebrain drain of the Soviet Union became a concern,and one day I found myself, together with mycolleagues A. Huckleberry and V. Arnold, presid-ing over some research funding to allow Russianmathematicians to spend longer periods with adecent pay at Bochum. Arnold was very concerned,and I got to know him somewhat better. ProfessorArnold became Dima.

Around 1994 I met him again; this time in Stan-ford. Dima, Yasha (Eliashberg), and I went lookingfor walnuts at the San Andreas Fault. I am sure itwas Dima’s idea. Knowing the “almost drowningversion” of Dima’s swimming expedition in SanFrancisco Bay, I had quite high expectations forthe afternoon. However, there was no earthquake.

Around this point we started talking aboutmathematics, specifically symplectic topology. Hisopening bid was, “Helmut, you are using thewrong methods,” referring to pseudoholomorphiccurves, and I responded with, “I am sure you knowsomething better. Make my day!” He liked to probeand enjoyed seeing people’s reactions. I think Idid well that day.

In 1998 he introduced my plenary lecture atthe ICM in Berlin, and we had a friendly chatbefore the talk. The year before I had movedto the Courant Institute. He said, “Helmut, youshould come back to Europe.” I answered, “No,Dima, I love New York. But if it makes you feelbetter, consider me the agent of European culturein the U.S.” I saw immediately that he liked thissentence. We talked about some more things whichI rather thought would stay between us. Of course,I should have known better! He made it all part of

his introduction and started by introducing me asthe agent of European culture in the U.S., to thedelight of many, but that was only the beginning;the rest is on video.

Dima had an amazing mathematical intuitionand (which at this point shouldn’t come as asurprise) was daring enough to make conjectureswhen others would not dare to stick their necksout.

There are quite a number of Arnold conjecturesin symplectic geometry. However, there is onewhich even people outside of the field know andwhich was the initial driving force behind thedevelopment of symplectic geometry.

Arnold and Weinstein developed the modernlanguage of symplectic geometry. This could, forexample, be used to prove interesting perturba-tion results. However, there were no global results.Arnold was the one who raised these types of ques-tions, and the Arnold conjecture I describe belowis an example. Surprisingly, the breakthrough dueto Conley and Zehnder came from outside thefield.

In the following, I try to motivate the ArnoldConjecture. One can understand it as an analogy ofthe relationships between the Euler characteristic,the Hopf index formula, and the Lefschetz fixedpoint theorem. I haven’t seen anything in his writ-ings pointing out this analogy, but added it here asan intermediate step which helps to understandthe conjecture better. Arnold describes a way ofreasoning in Appendix 9 of his previously men-tioned book. The Poincaré twist theorem can beseen as a special case of the two-dimensional toruscase of his conjecture. The general case would bethe generalization of the torus case to arbitraryclosed symplectic manifolds. There is quite oftena difference between the original thought processand the didactical cleaned version. From that pointof view I regret that I never asked him how he ar-rived at his conjecture. The discussion below addsanother point of view, constructing an analogy toa reasoning in topology. I very much believe thatArnold was aware of this analogy.

We start with a closed oriented manifoldM anda vector field X. The Euler characteristic χ(M) isa classical topological invariant, which is a gen-eralization of the original concept introduced forpolyhedra by Euler and which was fully general-ized later by Poincaré. If M is a smooth manifold,Hopf’s index formula establishes a relationshipbetween the zeros of a vector field assumed tobe transversal to the zero section and the Eulercharacteristic of M :

χ(M) =∑

m

i(X,m),

where i(X,m) = ±1 is the local index at azero of X.


In Yosemite, California, 1989.

How can we generalize this? First we observethat a diffeomorphism can be viewed as a gener-alization of a vector field. Indeed, the collectionof smooth vector fields can be viewed as the Liealgebra of the (Fréchet)-Lie group Diff(M), so as aninfinitesimal version of the latter. It is, however,not true that the diffeomorphisms close to theidentity are in the image of the group exponentialmap. This is a consequence of being only a FréchetLie group and a universal problem in dealing withvarious sorts of diffeomorphism groups. Let usmake a conjecture, which will come out as firstgoing from the infinitesimal to the local to gainsome confidence. We fix as an auxiliary structurea Riemannian metric with associated Riemannianexponential map exp. Assume that Φ is a diffeo-morphism which is close to the identity. Then wecan write Φ in a unique way in the form

Φ(m) = expm(X(m)),

for a small vector field X. Tranversality of X to thezero-section is equivalent to Φ not having 1 in thespectrum of its linearizations at fixed points. Mostimportantly, the fixed points for Φ correspond tothe zeros for X. Hence a generic diffeomorphismwhich is close to the identity has an algebraicfixed point count χ(M), where the sign is takenaccording toΦ′(m) being orientation preserving ornot. We can now make the “daring conjecture” thatthis should hold for all generic diffeomorphimsisotopic to the identity. That turns out to be correctand is, of course, a special case of the Lefschetzfixed point formula.

What Arnold did in symplectic geometry issuch a daring conjecture in a more complicatedcontext. We start with a closed symplectic manifold(M,ω), and in analogy to the previous discussionwe generalize the theory of functions on M ratherthan the theory of vector fields. If f is a smooth

function with all critical points nondegenerate,then Morse theory says its number of criticalpoints is at least the sum of the Betti numbers(for any coefficient field). Morse theory also tellsus that the algebraic count of critical points isχ(M). Since we have a symplectic structure, wecan associate to f a vector field Xf by the obviousformula

df = iXfω.

This is the so-called Hamiltonian vector field. Ob-viously we are now back to the first discussion.However, with the vector fields being more spe-cial, one would like a stronger statement for acertain class of diffeomorphisms. This particularclass of diffeomorphisms should generalize func-tions as diffeomorphisms isotopic to the identitygeneralize vector fields.

Symplectic diffeomorphisms isotopic to theidentity are not a good guess, since for T 2 withthe standard symplectic form a small transla-tion would give no fixed points at all. We could,however, look at all symplectic diffeomorphismsobtained as time-1 maps for the family of vectorfields Xft for a smooth time-dependent familyf : [0,1] × M → R, with ft(x) := f (t, x). Thisproduces the group of all Hamiltonian diffeo-morphisms Ham(M,ω). Indeed the collection ofsmooth maps can be viewed as the Lie algebra forHam(M,ω).

How can we go from the infinitesimal to thelocal, as we did in the previous discussion? Abasic and not too difficult symplectic result is thatthe neighborhood of a Lagrangian submanifoldof a symplectic manifold is symplectically iso-morphic to a neighborhood of the zero-section inits cotangent bundle with the natural symplecticstructure. Now comes a little trick which replacesthe use of the exponential map associated to anauxiliary metric. We define N = M ×M with theform τ = ω ⊕ (−ω). Then the diagonal ∆M is aLagrange submanifold of N, and an open neigh-borhood of it looks like an open neighborhood of∆M in T∗∆M . Every symplectic map that is suf-ficiently close to the identity has a graph whichwhen viewed as a subset of T∗∆M is a graphover the zero-section, i.e., the graph of a one-formλ. An easy computation shows that the originaldiffeomorphism is symplectic if and only if λ isclosed. It is Hamiltonian if and only if λ is exact:

λ = dg

for some smooth function. Hence the fixed pointsof a Hamiltonian diffeomorphism Φ correspondto the intersection of its graph with the zero-section and hence with the critical points of g.Now we are in the local situation, similarly asin the previous case. We conclude that a genericelement in Ham(M,ω) has at least as many fixedpoints as a smooth function has critical points ifit is close enough to the identity map.


Knowing all this, Arnold makes the follow-ing daring conjecture (nondegenerate case, in mywords).Arnold Conjecture: A nondegenerate Hamiltoniandiffeomorphism has at least as many fixed pointsas a Morse function has critical points.

It wouldn’t be Dima if it actually was thatstraightforward. The most prominent statement“Arnold-style” of this conjecture is in his bookMathematical Methods of Classical Mechanics. Inthe Springer 1978 edition (being a translation ofthe 1974 Russian edition) it reads on page 419 (andthis is a restatement of some published version ofthe conjecture in 1965):

Thus we come to the following gen-eralization of Poincaré’s theorem:

Theorem: Every symplectic diffeo-morphism of a compact symplecticmanifold, homologous to the iden-tity, has at least as many fixedpoints as a smooth function onthis manifold has critical points(at least if this diffeomorphism isnot too far from the identity).

The symplectic community has been tryingsince 1965 to remove the parenthetical part of thestatement. After tough times from 1965 to 1982,an enormously fruitful period started with theConley-Zehnder theorem in 1982–83, proving theArnold conjecture for the standard torus in any(even) dimension using Conley’s index theory (apowerful version of variational methods). This wasfollowed by Gromov’s pseudoholomorphic curvetheory coming from a quite different direction. Atthis point the highly flexible symplectic languagebecomes a real asset in the field. Finally, Floercombines the Conley-Zehnder viewpoint with thatof Gromov, which is the starting point of Floertheory in 1987. As far as the Arnold conjectureis concerned, we understand so far a homologicalversion of the nondegenerate case. A Luisternik-Shnirelman case (also conjectured by Arnold) isstill wide open, though some partial results areknown.

The development of symplectic geometry hasbeen and still is a wonderful journey. Thanks,Dima!

AcknowledgmentsThe photographs are courtesy of the Arnold familyarchive, F. Aicardi, Ya. Eliashberg, E. Ferrand, B. andM. Khesin, J. Pöschel, M. Ratner, S. Tretyakova, andI. Zakharevich.

References[1] V. Arnold, On a characteristic class participating

in the quantization conditions, Funct. Anal. Appl. 1,no. 1 (1967), 1–14.

V. Arnold, 2008.

[2] , The cohomology ring of the group ofcolored braids, Mat. Zametki 5, no. 2 (1969),227–231.

[3] , The one-dimensional cohomology of theLie algebra of divergence-free vector fields, andthe winding numbers of dynamical systems, Funct.

Anal. Appl. 3, no. 4 (1969), 77–78.[4] , On some topological invariants of algebraic

functions, Trudy Moskov. Matem. Obshch. 21 (1970),27–46.

[5] , Topological invariants of algebraic func-tions. II, Funct. Anal. Appl. 4, no. 2 (1970),1–9.

[6] , Local problems of analysis, Vestnik Moskov.

Univ. Ser. I Mat. Meh. 25 (1970), no. 2, 52–56.[7] , Distribution of ovals of real plane algebraic

curves, involutions of 4-dimensional smooth mani-folds, and arithmetics of integral quadratic forms,Funct. Anal. Appl. 5, no. 3 (1971), 1–9.

[8] , Lectures on bifurcations and versal families,Russ. Math. Surveys 27 (1972), no. 5, 54–123.

[9] , A spectral sequence for the reduction offunctions to normal forms, Funct. Anal. Appl. 9

(1975), 81–82.[10] , Index of a singular point of a vector field,

Petrovsky-Oleinik inequality, and mixed Hodgestructures, Funct. Anal. Appl. 12, no. 1 (1978), 1–14.

[11] , Lagrange and Legendre cobordisms, Funct.

Anal. Appl. 14, no. 3 (1980), 1–13, and 14, no. 4(1980), 8–17.

[12] , Geometrical Methods in the Theory of Or-

dinary Differential Equations, Springer-Verlag, NewYork-Berlin, 1983.

[13] Arnold’s Problems, Springer/Phasis, 2004.[14] D. Fuchs, Cohomology of the braid group modulo

2, Funct. Anal. Appl. 4, no. 2 (1970), 62–73.[15] H. Whitney, On singularities of mappings of Eu-

clidean spaces. I, Mappings of the plane into theplane, Ann. of Math. (2) 62 (1955), 374–410.


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Memories of Vladimir Arnold - American Mathematical · PDF fileMemories of Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors Vladimir Arnold, an eminent mathematician