Memories of Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors Vladimir Arnold, an eminent mathematician of functions of two variables. Dima presented a two- our time, passed away on June 3, 2010, nine days hour talk at a weekly meeting of the Moscow before his seventy-third birthday. This article, Mathematical Society; it was very uncommon along with one in the previous issue of the Notices, for the society to have such a young speaker. touches on his outstanding personality and his Everybody ad- great contribution to mathematics. mired him, and he certainly de- Dmitry Fuchs served that. Still there was some- thing that kept Dima Arnold in My Life me at a distance Unfortunately, I have never been Arnold’s student, from him. although as a mathematician, I owe him a lot. He I belonged to was just two years older than I, and according to a tiny group of the University records, the time distance between students, led by us was still less: when I was admitted to the Sergei Novikov, Moscow State university as a freshman, he was a which studied al- sophomore. We knew each other but did not com- gebraic topology. municate much. Once, I invited him to participate Just a decade in a ski hiking trip (we used to travel during the before, Pontrya- winter breaks in the almost unpopulated northern gin’s seminar in Russia), but he said that Kolmogorov wanted him Moscow was a to stay in Moscow during the break: they were true center of the going to work together. I decided that he was V. Arnold, drawing, 1968. world of topol- arrogant and never repeated the invitation. ogy, but then Then he became very famous. Kolmogorov an- Cartan’s seminar in Paris claimed the leadership, nounced that his nineteen year-old student Dima algebraic topology became more algebraic, and Arnold had completed the solution of Hilbert’s the rulers of Moscow mathematics pronounced 13th problem: every continuous function of three topology dead. Our friends tried to convince us to or more variables is a superposition of continuous drop all these exact sequences and commutative diagrams and do something reasonable, like func- Boris Khesin is professor of mathematics at the University tional analysis or PDE or probability. However, we of Toronto. His email address is [email protected]. were stubborn. We even tried to create something edu. like a topological school, and, already being a Serge Tabachnikov is professor of mathematics at Penn- graduate student, I delivered a course of lectures sylvania State University. His email address is tabachni@ in algebraic topology. The lectures were attended math.psu.edu. by several undergraduates, and we were happy to Dmitry Fuchs is professor of mathematics at the Univer- play this game. sity of California, Davis. His email address is fuchs@math. Then something incredible happened. One day ucdavis.edu. I found the lecture room filled beyond capacity; I DOI: http://dx.doi.org/10.1090/noti813 even had to look for a bigger room. My audience 482 Notices of the AMS Volume 59, Number 4 spectral sequences, I mean that he in general had a strong dislike for unnecessary technicalities, and technicalities were often unnecessary to him because of his extremely deep understanding. By the way, this attitude toward impressive but unnecessary tricks extended beyond mathematics. Years later we spent a week or so with friends at a ski resort in Armenia. We showed each other different turns and slidings, but Dima obviously was not interested. He said that the slope was not too steep, and he simply went straight from the top to the bottom, where he somehow managed to stop. I was surprised: there was a stone hedge in the middle of the slope that you needed to go around. Dima said modestly, “You know, at At Otepya, Estonia. this place my speed is so high that I simply pull my legs up and jump over the hedge.” I could had become diverse: undergraduates, graduate not believe it, so I waited at a safe distance from students, professors. This change had a very clear the hedge and watched him doing that. It was reason: the Atiyah-Singer index theorem. more impressive than all our maneuvers taken The problem of finding a topological formula together. Whatever he did—mathematics, skiing, for the index of an elliptic operator belonged biking—he preferred not to learn how to do it but to Gelfand. Our PDE people studied indexes a just to do it in the most natural way, and he did lot, and they had good results. It was not a everything superlatively well. disaster for them that the final formula was found I do not remember how it came about that I by somebody else: their works were respectfully began attending his Tuesday seminar. Probably he cited by Atiyah, Singer, and their followers. The asked me to explain some topological work there, trouble was that the formula stated, “the index is then I had to participate in some discussion, and equal to” and then something which they could then I could not imagine my life without spending not understand. People rushed to study topology, two hours every Tuesday evening in a small room and my modest course turned out to be the only on the fourteenth floor of the main building of the place to do that. MSU. Works of Arnold, his numerous students, And to my great surprise, I noticed Dima Arnold and other selected people were presented at the in the crowd. seminar, and Dima insisted that every word of I must say that Dima never belonged to any every talk be clear to everybody in the audience. crowd. Certainly the reason for his presence did My role there was well established: I had to re- not lie in any particular formula. Simply, he had solve any topology-related difficulty. Some of my never dismissed topology as nonsense, but neither friends said that at Arnold’s seminar I was a “cold had he been aware of my lectures. When he learned topologist”. Certainly, a non-Russian-speaker of their existence, he appeared. That was all. He cannot understand this, so let me explain. In many never missed a lecture. Russian cities there were “cold shoemakers” in One day we met in a long line at the student the streets who could provide an urgent repair to canteen. “Listen,” he said, “can you explain to me your footwear. They sat in their booths, usually what a spectral sequence is?” I began uttering the with no heating (this is why they were “cold”), usual words: a complex, a filtration, differentials, and shouted, “Heels!…Soles!….” So I appeared as adjoint groups, etc. He frowned and then said, if sitting in a cold booth and yelling, “Cohomol- “Thus, there is something invariant [‘invariant’ in ogy rings!…Homotopy groups!…Characteristic his language meant ‘deserving of consideration’] classes!….” in all this stuff, and this is the spectral sequence, In my capacity as cold topologist, I even had right?”Ithought fora momentandsaid,yes.Atthis to publish two short articles. One was called moment we got our meals, and our conversation “On the Maslov–Arnold characteristic classes,” changed its direction. and the other one had an amusing history. One Evidently spectral sequences were not for day Dima approached me before a talk at the Arnold. Nonetheless, there is such a thing as Moscow Mathematical Society and asked whether Arnold’s spectral sequence [9], a humble object I could compute the cohomology of the pure in the world of his discoveries, resembling the braid group (“colored braid group” in Russian); asteroid Vladarnolda in the solar system (the he needed it urgently. I requested a description stability of which he proved approximately at the of the classifying space, and the calculation was time of our conversation in the canteen), named ready at the end of the talk. It turned out that after him. When I say that he could not appreciate the (integral) cohomology ring was isomorphic April 2012 Notices of the AMS 483 to a subring of the ring of differential forms on the classifying manifold. He suggested that I write a note, but I refused: for a topologist it was just an exercise; it could be interesting only in conjunction with an application to something else. (I knew that Dima was thinking of Hilbert’s 13th problem in its algebraic form: the possibility of solving a general equation of degree 7 not in radicals, but in algebraic functions of two variables.) I suggested that he write an article and mention mymodestcontribution in an appropriate place. He did [2]. But, a couple of months later, he needed the cohomology of the classical Artin’s braid group. This was more difficult and took me Summer School “Contemporary Mathematics” several days to complete the calculation. I did it at Dubna, near Moscow, 2006. only modulo 2, but I calculated a full ring structure and also the action of the Steenrod squares. similar things [3]. He always considered algebra (The integral cohomology was later calculated and topology as something auxiliary. Once I heard independently by F. Vainshtein, V. Goryunov, and him saying respectfully, “Siegel’s case, this is a F. Cohen and still later Graeme Segal proved true analysis,” and this sounded like “true math- that the classifying space of the infinite braid ematics”. Whatever he did, his unbelievably deep 2 3 group was homologically equivalent to S .) I understanding of analysis was always his main phoned Dima and explained the results.Ω First instrument. he requested that I give a talk at the seminar One more story of a similar kind.