On the 80th Birthday of Dmitry Borisovich Fuchs
Alice Fialowski, Ekaterina Fuchs, Elena Fuchs, Boris Khesin, Alexandre Kirillov, Fedor Malikov, Valentin Ovsienko, Alexei Sossinsky, Serge Tabachnikov
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2020), 023, 17 pages On the 80th Birthday of Dmitry Borisovich Fuchs
Alice FIALOWSKI † , Ekaterina FUCHS † , Elena FUCHS † , Boris KHESIN † ,Alexandre KIRILLOV † , Fedor MALIKOV † , Valentin OVSIENKO † ,Alexei SOSSINSKY † and Serge TABACHNIKOV † † Institute of Mathematics, University of P´ecs, P´ecs, Hungary
E-mail: fi[email protected]
URL: http://web.cs.elte.hu/~fialowsk/ † Department of Mathematics, City College of San Francisco, CA 94112, USA
E-mail: [email protected]
URL: † Department of Mathematics, UC Davis, One Shields Ave Davis, CA 95616, USA
E-mail: [email protected]
URL: † Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
E-mail: [email protected]
URL: † Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
E-mail: [email protected]
URL: † Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
E-mail: [email protected]
URL: https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1003489 † CNRS, Laboratoire de Math´ematiques de Reims, 51687 Reims cedex 2, France
E-mail: [email protected]
URL: http://ovsienko.perso.math.cnrs.fr † Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia
E-mail: [email protected]
URL: https://users.mccme.ru/abs/ † Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
E-mail: [email protected]
URL:
Received February 29, 2020; Published online April 07, 2020https://doi.org/10.3842/SIGMA.2020.023
Abstract.
This article is a collection of several memories for a special issue of SIGMAdevoted to Dmitry Borisovich Fuchs.
Key words: cold topologist; bookcase; new chronology; home-schooling; poetry; mountaintourism; Lie algebra cohomology; SQuaREs a r X i v : . [ m a t h . HO ] A p r B. Khesin, F. Malikov, V. Ovsienko, S. Tabachnikov
This special issue of SIGMA is devoted to Dmitry Borisovich Fuchs, an outstanding mathemati-cian and our teacher, on the occasion of his 80th birthday. It comprises many papers of hisfriends and colleagues, reflecting broad mathematical interests of D.B. himself.Below we collected several memories: those of his daughters Katia and Lyalya, of his old-time friends A. Sossinsky and A. Kirillov, and his students and colleagues A. Fialowski andV. Ovsienko.There are many (semi-legendary) stories about Fuchs and, by way of introduction, we presenta few specimen of this lore.
Figure 1.
Fuchs is an avid bicyclist, making very long distance trips, in particular, from Moscow toKazan (about 500 mi). This picture was drawn by Sergei Ivanov, the artist who illustrated the book“Mathematical Omnibus” by D. Fuchs and S. Tabachnikov, AMS, Providence, RI, 2007.
In his memories of Arnold’s seminar in Moscow D.B. Fuchs writes (see [2]) “My role there waswell established: I had to resolve any topology-related difficulty. Some of my friends said that atArnold’s seminar I was a ‘cold topologist’. Certainly, a non-Russian speaker cannot understandthis, so let me explain. In many Russian cities there were ‘cold shoemakers’ in the streets whocould provide an urgent repair to your footwear. They sat in their booths, usually with noheating (this is why they were ‘cold’), and shouted, ‘Heels! . . . Soles! . . . ’ So I appeared as ifsitting in a cold booth and yelling, ‘Cohomology rings! . . . Homotopy groups! . . . Characteristicclasses! . . . ’ ”The topological reputation of Fuchs extended far beyond Arnold’s seminar. There was a fa-mous story where a group of mathematicians, “all the best people” as Fuchs says, helped one ofthem, A.A. Kirillov, to move to his new apartment on the 14th floor of a 16-story building. Itturned out that a huge custom-made bookcase, which would occupy the entire wall of a room,did not fit into the elevator. It had to be carried to the 14th floor via the staircase but, evenmore importantly, it was so large that there was a unique way of moving it through each flight ofstairs, with the only possible turn of the bookcase fitting the stairwell. (One of the participantswas Ya.G. Sinai, who first formulated the problem as “keep this damned bookcase connected”and later “keep the number of its components not greater than two”.) Eventually, when theyn the 80th Birthday of Dmitry Borisovich Fuchs 3got to the apartment, there was a unique way to bring this bookcase into the room, with no wayto turn it around or change its orientation in any other way.And once inside, it turned out that the bookcase fit the room in the only possible position,with the shelves facing the wall and the backside facing the room! The only way to fix this wasto carry the case all 14 flights down the stairs and get it out to the street. Then Fuchs, as theleading topologist, was given the task of finding the correct “initial conditions” in order to startthe process anew. He thought about it, gave the instructions how to turn it, they did all thelifting via 14 flights of stairs up – and the bookcase indeed fit in the room perfectly!S. Tabachnikov recalls (see [4]) “At our first meeting DB asked me what I knew in mathe-matics, and I started to name various books I had read (after I mentioned Spanier’s “Alge-braic topology” DB remarked, characteristically, that it was the first time he met someone whohad managed to finish the book). DB asked a few questions (e.g., which simply connected 4-dimensional manifold I knew), and my answers revealed the obvious: my extensive reading wasalmost a total waste of time, I did not really understand the material. This was a devastatingdiscovery for me, and I mumbled something like: “I should probably quit mathematics, I’vealready lost too much time!” This made DB laugh. “You are 20, aren’t you? It is not too lateto start, even if you had never read anything but ‘Murzilka’ before”.F. Malikov remembers: Fuchs was giving a course on linear algebra and once, trying toentertain us, he devoted a lecture to a proof of Abel’s theorem (impossibility of solving equationsof degree five in radicals). While proving it, he constructed a certain permutation. Thenunexpectedly he turned to A. Kanel’, who was the indisputable genius of the class, and asked:“Alesha, what could you say about this permutation?” Kanel’ replied: “This permutation iseven”. Fuchs said: “Correct, but it’s not its main property – it is the trivial permutation”. Figure 2.
In the office.
Yet another memory: Fuchs was giving a talk at USC. At some point, he said that, fora sufficiently small epsilon, such and such held. One of the attendees, who was drowsing, A Russian magazine for preschoolers.
El. Fuchssuddenly woke up and asked how small this epsilon must be. Fuchs replied immediately: “onetenth of an inch”.B. Khesin remembers the following story. Fuchs recalled that in the 70–80s he was at one ofA. Fomenko’s public lectures on “The New Chronology”, which was a “hot” subject at the time.At some point, when comparing time patterns related to two sequences of the Roman emperors,Fomenko said that the probability that such patterns were independent and not copied fromone another was one out of a million. At this moment Fomenko tried to clarify his point: “Inorder to see how negligible this probability of one millionth is, just imagine that you put a kettleon a hot stove, but instead of boiling the water freezes up! This is how rare such events are!”Fuchs said that once he heard that explanation, he thought: “Hmm . . . Moscow’s population is8 million . . . So every morning 8 people in this city put their kettles on the stove and have thewater in the kettle frozen?!”
One of my first memories of my father was waking up in the morning to find a sheet of mathproblems, handwritten by him and often involving me and my sister, in my room. My fatherwould be gone to work by this point, but it was like a little piece of him that he left for me to enjoy.
Figure 3.
A page from the problem book that Fuchs wrote for his daughters.
When I was a little older, after we moved to America, these sheets of problems came illustratedwith a sketch of me doing something from one of the problems (I remember one picture inparticular where I was diving into a swimming pool, something that he had taught me to do,in fact). Not everyone knows that my father is a wonderful artist: our apartment in Moscowhad several of his oil paintings hanging on the walls. He doesn’t paint much anymore, or talkabout it, but I think he painted those paintings at a sad time in his life when the art helpedhim escape, so perhaps that is a good thing.n the 80th Birthday of Dmitry Borisovich Fuchs 5In fact, I do not remember a time in my life when math did not play a big role in ourrelationship. While we still lived in the USSR, I remember that he was often away on whatI imagined to be an exciting adventure off in some other country. He would always come homewith all sorts of treasures from the West that were unattainable at home at the time: colorfulskirts, stuffed animals that looked like the real thing, fancy new pencils.
Figure 4.
The Fuchs family.
While my sister and I and the other kids in our neighborhood lounged in the sun on a mattressleft on a ping-pong table at our dacha (summer home), my father would have frequent mathvisitors, and sometimes he would take me from there into town on his bike so that I could haveice cream while he met with yet another mathematician. Even when my sister and I did notfully appreciate it, our lives were engulfed in math. Wherever my father took us: to the skatingrink, to a lesson of some kind, he always had his yellow pad of paper on which he would writeand draw what back then looked like amusing doodles. During a month-long stay at St. John’sCollege in Cambridge in the mid-nineties, we would find his math toys at our apartment whenmy parents were away and play ridiculous games with them: one time we found a curvy rulerfor tracing curves that one could shape into any curve one wanted and made it into an S-shapeto advertise Safeway, the American grocery store, out of our apartment window. I am sure theBrits were amused.Rooms with floors covered in chalk dust have been places in which I have felt at home froma very young age, and it is all due to him. Indeed, the first time I remember anyone asking mewhat I wanted to be when I grew up, “mathematician” was always on the long list of professionsI planned to have (along with opera singer, ballerina, and flautist . . . those did not pan out).In the US, I acquired most of my mathematical education until I attended university classesfrom my parents. There was a designated chair in which my father would sit by my desk toexplain new interesting math to me, and I was officially home-schooled in math for 2 years byhim: I might add that those two years probably marked the peak of my teenage rebellion, but Ek. Fuchshe patiently stuck it out, and one lesson I have held on to since those times was to listen tosomeone explaining some mathematics to me no matter if I thought I already knew it: chancesare, my father said, I might learn something new. He was right then and is still right now:I never dismiss mathematics as uninteresting and always try to learn something from it.One might think that our close mathematical relationship through my childhood would meana continued close mathematical relationship in adulthood. That has not been true: in collegeI contemplated majoring in a scientific field besides math and leaning into it, out of a worry thatmathematicians might think any of my success was due to my father’s support. I remembertalking about things along these lines to my father then, and he would often tell me, “yourproblem is not your mathematical ability. It’s your insecurity in it.” It took me a few semestersin university to see that he was right, and I found a passion in number theory, which was fairlyfar removed from his mathematics (although he has told me at least 50 times that, while hewould like to say he has never worked in number theory, that would not be entirely true). Sincethen our mathematical relationship has been much less intense, although I do still have a dreamto produce a Fuchs–Fuchs paper sometime.I have to add that, part of my father’s dedication to my education in math certainly cameout of his genuine care for teaching: even when he was teaching students in the university andnot his children, he would go above and beyond the norm, writing his own lecture notes fornearly every class, shaping his own point of view in a way that I think is rare among universityinstructors. But part of it is a reflection of how much he cares for his family. His passion, I think,is split between two things: family and mathematics. For every fond memory that I have of usdoing math together, I have one where math was not at all a factor. Putting up a tent togetherat countless campsites, hiking dozens of trails together, confiding in him both my biggest joysand fears. Most recently, I have seen the joy he gets out of seeing his grandchildren – my twokids – whenever they are around. I feel truly lucky to have him as my father, and wish him themost wonderful of birthdays, and many more to come.
It seems that the Russian way of raising a baby when I happened to be one involved a lot ofwalking, baby in stroller, outdoors. And so we walked, my Dad and I for hours on end. I woulddo what babies do, I suppose, and sleep, while he pushed the stroller, and did what he continuesto do constantly – think and read. He would tell me how when it rained he would put a bookinto a clear plastic bag so he could read it without the book getting wet, and we would walk on,day in and day out.And then night would fall, and he would read again, or rather recite, poetry to me untilI was soundly asleep. My mom has often told me stories of sitting in the kitchen (how vividlyI remember the layout of our 3 room apartment on
Trinadtsataya Parkovaya , “Thirteenth ParkSt” in Moscow!) and knowing exactly how long it would be until he would be finished, sincehe recited the same poems every time, in his slow, peaceful baritone. Even when my sister andI were older, he would often sit with us as we drifted off into sleep, reciting poetry to us.Poetry continues to be a key motif in my memories of growing up and my dad. One summer,if I had to I would guess the summer of 1988, but I can’t be terribly sure, my father and I decidedtogether that he would help me memorize the entire anthology of Boris Pasternak poetry thatfollowed the novel “Doctor Zhivago”. Some days we would work on one poem, sometimesmore than one, some poems stuck almost like honey in my mind, some were very difficult tointernalize. The poems dealt with some very adult themes, and I remember in many cases ourwork to memorize the poems had to include long conversations about what the poem meant, andwhat the imagery was trying to evoke. We made it through that entire anthology that summer,just like we planned. To this day we occasionally allow ourselves a stroll down memory lane,n the 80th Birthday of Dmitry Borisovich Fuchs 7and see if we can dust off this that or the other poem - while they are rusty and sleepy in myhead, they are unbelievably crisp and alive in his. So many times I’ve said “I’m going to relearnthis one!” and he’s immediately ready to help me with any word or line where I stumble.
Figure 5.
On a hike with Katia and Lyalya.
I suppose this reminds me of another motif, if one could call it that, in my life with dad. Thetime he was willing to devote to basically anything we did together. One year he decided thatthe whole family needed a good solid lesson on the history of the Roman Empire. He immersedhimself in historical texts and made copious notes, which he would then use to teach us (thisincluded myself, my sister, my mom, and the cat). These were incredibly cozy evenings, sittingon the big L shaped sectional in the dining room, the three of us (and the cat) huddled togetheron the long side of the L, him sitting on the short end. I must confess that the evenings gotso cozy, that my sister and the cat and I would often doze peacefully, which didn’t go overparticularly well with our parents.He was then and continues to be now, unquenchably thirsty to teach us things – whether itis beautiful mathematics, or the history of Ancient Rome, or poetry.As a teacher now, I know that my desire to help others learn about beautiful things camefrom the true great fortune of having been surrounded by incredible teachers my whole life. Myfirst teacher, without a doubt, was my dad.I can honestly say that my dad’s influence can be found at the very most foundational core ofwho I am today. His sense of humour, his dedication to his family, his love of art and thirst forknowledge, have shaped me in incredibly important ways. Basically, he’s awesome, and I wishhim a happy birthday, and many, many, many more still to come.
I have known Dmitry Fuchs since 1957, when we were both students at the mathematics de-partment of Moscow State University. We became neighbors quite by accident (our parents hadacquired apartments in adjacent houses in Izmailovo) in our graduate student days and soon A. Sossinskybecame close friends. This friendship has remained, despite the fact that, since 1990, we areusually separated by an ocean and a couple of continents.Mitya, as everyone calls him is, above all, a mathematician, and while mathematics dominateshis life, he has many other interests and activities that shape his unique personality: his sense ofhumor, his fascination for the outdoors, his love of poetry, his interest in the popularization andteaching of mathematics (a rare trait for a research mathematician of his caliber). It is mostlyon these not very mathematical aspects of Dmitry Fuchs’ life that I intend to concentrate here,in the form of “little Fuchs stories” drawn from my vast oral repertoire.
Figure 6.
D. Fuchs, A. Sossinsky, and Katia Fuchs.
Sense of humour.
Mitya’s is always low key, often ironic, it permeates his mathematicallectures and his conversation about people. Here is a related little story. We – a group mainlyconsisting of mathematicians – were downhill skying on the Chiget mountain in the Caucasus.Yulia Gippenreiter, the non-mathematician in our group, was a psychologist and an accomplished(professional-level) Alpine skier. She would generously give us tips on skiing techniques; toMitya, speeding fearlessly downhill, legs spread far apart for better balance, she would shout:“Mitya, parallel skis, keep your skis parallel!” Later, over tea in the cozy Ai chalet, Mityaa would grumble, “These stupid people in the humanities – they think that if your skis are morethan a meter apart, they can’t possibly be parallel”, and we would all burst out laughing.
Poetry . We first discussed poetry with Mitya as neighbors, in Izmailovo, and I must admitthat at first – with a youthful snobishness of which I am now ashamed – I looked down on hisknowledge and taste in Russian poetry. The first “poetry reading” where I began to change mymind occurred in the Kolsky peninsula in the late sixties, and I have described it in detail else-where, [4], but it still often comes to mind: the three of us (the third was our friend and colleagueLenya Erdman) on the Lovozero plateau, standing in a snowstorm near our tent, which was beingprogressively buried in snow, waiting for the blizzard to abate and the late Arctic morning tocome, and – yes, you have guessed correctly – reading poetry to each other, sometimes in two oreven three voices. I remember that Mitya got things started by reading a beautiful poem aboutUlysses by Lugovskoi (whom until then I had regarded as a worthless Soviet establishment poet),Lenya recited a lot of funny poems by Sapgir and Zakhoder, I remember reciting Lermontov’s
Demon and, when my Russian repertoire had dried up, Poe’s
Raven , with my friends pitchingin, like a Greek chorus, shouting out the mandatory “Nevermore” at the end of each stanza.A less dramatic, but just as rewarding, reading took place a couple of years ago at the IHESin Bures-sur-Yvette, where we lived in neighboring cottages (not by accident, but by design).My present for Mitya’s 80-th birthday was a selection of poems by Georgy Ivanov, a Russian´emigr´e poet he was not familiar with, and he responded with a short letter with a very preciseliterary analysis of Ivanov’s verse, as good as those ever written by “people in the humanities”.n the 80th Birthday of Dmitry Borisovich Fuchs 9
Figure 7.
Fuchs and Sossinsky follow the donkeys.
The outdoors.
Mitya could satisfy his love of nature only within the limits of the “ironcurtain” surrounding the USSR, camping along its rivers, in its forests and mountains. This hewould do within the framework of what we call sportivny tourism : whitewater trips (shootingrapids in kayaks), mountain hiking (on foot in summer or on skis in winter), hiking-campingtrips in the rich forests of Russian midland. Of the several camping trips that we took together,I will describe the only one during which I took photographs: a high mountain hiking-campingtrip through the Fanskie mountains, from Dushanbe to Samarkand. It began from the northernoutskirts of Dushanbe, from which we began the long climb up with our backpacks loaded on twodonkeys (see photo), until the path became so steep that the poor overloaded donkeys refusedto continue. Progressively getting used to the increasing altitude, we hauled ourselves and ourbackpacks up to the first mountain pass, Mura, at 3500 meters.
Figure 8.
Fuchs rock climbing.
From then on, it was down and then up over higher and higher mountain passes (the highestwas at 4900 meters), crossing ice-cold streams, overcoming steep glaziers and rock climbing.0 A. SossinskyAnother photograph shows the group resting at the start of one of the glaciers, with two localshepherds gratefully sharing our meal. The local population (nomadic shepherds) was verysparse and practically untouched by 20th century civilization, their self-sufficient way of lifeunchanged for many centuries, most of them having never handled money, been in a city, orseen a photo-camera; they looked at us as if we were creatures from another planet.
Figure 9.
A frugal meal at the foot of a glacier.
One of the high moments of our trip (literally and figuratively) was in the morning at ourcamp site some 4000 meters above sea level, when a huge eagle swooped down on us, grabbeda sausage that Mitya was about to cut up into seven equal pieces and flew off with it; I thinkit was then that Mitya philosophically declared: “Well, the sausage was beginning to spoilanyway”. The whole episode took two or three seconds, and so, unfortunately, there is nophotograph of the eagle, and there are no photographs that do justice to the mountain sceneryand the beautiful and then almost totally uninhabited Iskander Kul’ lake (now a tourist resort),on whose shores we rested for a couple of days before finishing our trip in the historic city ofSamarkand.
Figure 10.
A. Sossinsky, I. Girsanov, D. Fuchs, E. Mochigin, G. Turina, Yu. Chapovsky ( left to right ). There is, however, one photograph that is really worth looking at. It very expressively showssix tired climbers, some still breathing heavily, who have just overcome a mountain pass andn the 80th Birthday of Dmitry Borisovich Fuchs 11look forward with optimism to the continuing trip. Yet this is a tragic photo: five years after itwas taken, only two of the six persons on the picture were still in this world: Girsanov, Mochigin,and Chapovsky were buried in an avalanche in the Sayany mountains in 1967 and Galya Turina,Mitya’s wife, drowned two summers later when her kayak overturned in a white water trip in1970 just above the Arctic Circle in the Urals. But Mitya and I are still here and look back onthis trip as one of the most rewarding experiences of our lives.* * *Perhaps I should conclude this text by shortly recalling how our life lines converged anddiverged in the last 60 years. From our graduate years on, our lives first developed in parallel,with Mitya one year ahead of me (although almost two years younger, he was a year aheadin his studies). After completing our graduate studies and defending our PhD’s, we became,one after the other, associate professors at the topology Chair of P.S. Alexandrov (althoughneither of us were his pupils, and we worked in the rapidly developing field of algebraic topologythat Alexandrov no longer understood). We both became regular participants in the famousGelfand seminar, and also worked together in the math olympiad movement on the national andinternational level.I recall an episode that occurred at the Gelfand seminar: during a break, I walked up to theblackboard and tried to explain to Fuchs how to answer a question raised at the talk precedingthe break; this involved the calculation of the first term of a spectral sequence, I explainedthe filtration and, with Mitya’s help, indicated how the calculation would go. I did not noticethat Gefand was standing by and attentively listening; when I had finished and the seminarwas about to resume, Izrael Moiseevich, addressing us both, said that was quite interesting andasked us to write it up and bring the text next week, he would have it published as a joint paperby the three of us. But we didn’t: Mitya convinced me that the question was not very seriousand the proof too simple to be worth publishing.Perhaps a year later, at the end of the seminar, Gelfand took us aside and proposed usresearch positions at his mathematical biology laboratory. In my case, I think it was the episodedescribed above which decided that I was worth hiring; as to Mitya, it was by then clear thathe was a first rate mathematician who would embellish any research center. Mitya accepted.I refused: I was simply afraid of Gelfand.This was a second step in our moving apart, the first one being geographical: I moved toan apartment in the southern part of town, so we were no longer neighbors. The next stepin that direction occurred when I was forced out of Moscow University, and after a year ofunemployment, started working at the popular science magazine
Kvant . During the 13-yearperiod that I worked there, we would meet with Mitya at the
Kvant headquarters more often thanat research seminars. Mitya became a regular contributor to the magazine, and an unexpectedresult of that was his marriage to Ira Klumova, who then worked as a
Kvant math editor. Itwas my wife, Elena Efimova, not I, who played the role of the matchmaker: two side-by-sidetickets to a theater performance given by Lena to Mitya and Ira started things off . . .Another place where we continued to meet was Bella Subotovskaya’s
People’s University (while it lasted), where Mitya, in this case following my footsteps, became an active teacheruntil this “university” was closed down by the KGB. I have described the dramatic story of thatunusual institution in my article [3].One of our last contacts in Moscow before Mitya’s emigration to the US in 1990 was a seriesof talks on knot polynomials that I gave at the Fuchs–Varchenko seminar. This was extremelyimportant for me – Mitya’s very favorable reaction and encouragement helped me realize thatduring my “exile” to
Kvant
I did not lose the ability of doing serious mathematics.Since 1990, our meetings have been few and far between: at Davis, of course, at Oberwolfach,at the IHES, in Moscow at the Gelfand Centenary celebration – are those that first come to mind.2 A. KirillovBirthday wishes? I have sent them to Mitya by e-mail: to be able to think clearly andmathematically till the very end, to derive from life whatever pleasures we are still capable ofappreciating, and, our healths and God willing – a few more meetings on either side of the ocean.
I was asked to write something for the special issue in honor of 80th anniversary of D.B. Fuchs, myold friend, an outstanding mathematician, and simply a very good man. Of course, I immediatelyagreed. But when push came to shove, it turned out that to do it wasn’t that easy. In general,it is not easy to write a good mathematical paper, and to write it better (or at least not worse)than one’s previous works is even harder. Let alone on a short notice . . .Of course, one could simply write about Mitya, how we lived, worked, what we did in ourfree time. But this immediately forces one to remember many other people and all our life inthe USSR, Russia, and America because in all of this Mytya played an important role. Andstill, I shall give it a try.
Figure 11.
Left: as part of Khrushchev’s campaign to open vast tracts of virgin land in the northernKazakhstan and the Altai region of the USSR, the students were sent to work there in summer. Right:a young topologist, D. Fuchs.
Much was said about the Golden Years of Moscow mathematics (1955–1967). We both hadthe privilege to study at MekhMat of MGU (Mathematics Department of the Moscow StateUniversity) that, without exaggeration, was then the leading mathematical center in the world.Among our professors were P.S. Alexandrov, N.V. Efimov, I.M. Gelfand, A.O. Gelfond,B.N. Delone, E.B. Dynkin, A.N. Kolmogorov, A.G. Kurosh, A.A. Markov, D.E. Menshov,I.G. Petrovski, L.S. Pontryagin, M.M. Postnilov, I.R. Shafarevich, B.V. Shabat, G.E. Shilov. Therecitations and research seminars for youngsters were run by then young, but already well known,mathematicians F.A. Berezin, R.L. Dobrusin, R.A. Minlos, A.S. Schwarz, A.G. Vitushkin.The general atmosphere of those years (1955–1967) was splendid, and it has never beenthe same again. What happens at MekhMat today does not stand comparison. I was lucky tobelong to a group of 20–25 students who were united in their admiration for talent and were gen-uinely interested in each other’s work. (To mention only the most active and well remembered:A. Arkhangelskij, V. Arnold, B. Averbuch, N. Brushlinskaya, A. Chernavskij, L. Churaeva,V. Murskij, R. Sofronitskaya, M. Shur, V. Tutubalin, F. Vetukhnovskij, E. Vinberg.)Later our group extended by students from the next year (D. Fuchs, V. Palamodov, A. Sossin-sky, G. Tyrina) and some previous years (V. Alexeev, I. Girsanov, B. Polyak, Ya. Sinai, V. Tikho-mirov).n the 80th Birthday of Dmitry Borisovich Fuchs 13Besides mathematics, the common trend was tourism. Not in the western form of expensivetours in comfortable conditions, but week-end pedestrian hiking on foot near Moscow, or longertravels during student vacations on ski, on kayaks, in the mountains. These travels were partiallysponsored by the University Tourist Club. One can get some outfit (skis, kayaks, ropes) andsome foodstuffs (condensed milk, braised meat) which was rather difficult to find in the post-warRussia.Mitya was a very important member of our expeditions due to his strength, endurance, andgood nature. Once he saved our kayak when we tried to pass in poor visibility through a narrowpassage in a dam. The kayak was almost folded in half, being pressed by the waterfall and onlyMitya’s power legs and spine allow us to pull it out.Another remarkable situation happened during our winter expedition on the Northern Ural,the coldest place in European Russia, where the temperature in winter falls to −
50 C ◦ . By theway, one of our common discovery with Mitya was that, despite the general opinion, the spittle(and other warm excretions) do not freeze immediately in the air even under this temperature.Preparing to this expedition, we decided to make an experiment and take with us a portablestove working on firewood (the gas stoves did not yet exist in tourist practice at that time). Onthe third or fourth day, when the frost became rather severe, we installed the stove in our tentand started the fire. Soon the air inside the tent became rather warm, but the whole space filledup by the dense smoke. Since none of us had experience of heating a tent with a stove, we thoughtthat it was a natural effect and tried to endure the smoke. But after a short time the smoketurned into flame, and when we finally succeeded to put it out, almost half of our tent had gone.The culprit of the fire was the veneer shovel, incautiously leaned against the stove. (Suchshovels are usually used in winter expedition for digging the snow to make a place for the tentand for the campfire.) In the morning there was a council about one main question: what to do?Some people proposed to fix the tent and to go further. Other preferred to break the expeditionand come back home.Since I was the official head of expedition, my vote was decisive. And with great doubtI decided to return. I was very thankful to Mitya and his wife Galya who did not blame me forthis decision, though I suspect they felt differently. To my surprise, the way back was rathereasy and agreeable. First, going down the river and using our trace in snow, we made the wayback in one day (more precisely, in 12 hours). Second, the frost was softened and when we cameto the railroad station, the temperature was −
18 C ◦ , feeling like a summer day.The mathematical achievements of Mitya are numerous and well-known. Last several yearswe tried to work together. Luckily, I draw to this project my former PhD student V. Ovsienkoand his wife S. Morier-Genoud. Thanks to Valya’s administrative talent and modern socialtrends in scientific politics, the four of us formed a “Square” team at the American Institute ofMathematics (Palo Alto – San Jose).We spend at AIM three one-week terms, and wrote the joint article, where we tried to realizemy idea about the application of the orbit method to the group of triangular matrices overa finite field. I’ll explain here the problem in very informal way.The point is that usually the linearized version of a problem is simpler than the non-linearoriginal. But the classification problem for coadjoint orbits of the triangular matrix group N is still open, notwithstanding that it is the linearized version of the classification problem forthe Borel group orbits in the flag manifold F . The solution of the latter problem is well-known:the B-orbits are labeled by the elements of the Weyl group. I conjectured that the coadjointorbits are in some sense the “shadows” of B -orbits. The naive formalization of this conjectureturns out to be wrong, but we have found a very interested new phenomena and are eager tocontinue.I can not speak about Mitya and not to mention his family. His first wife, Galina Tyurina,was an ardent tourist and a downhill skier. She was also an outstanding mathematician and an4 A. Fialowskiessential member of I.R. Shafarevich’s team in algebraic geometry. She died tragically duringa kayaking expedition.The second Mitya’s wife, Irina Klumova, was my student at MechMat and later my co-author in the magazine “Kvant”, where she worked for many years. They have two beautifuldaughters, Katia and Elena, both talented mathematicians. Thanks to Elena, Mitya becamea happy grandfather of two. Figure 12.
D. Fuchs, A. Kirillov, and I. Klumova. Fuchs, the grandfather.
Now we meet not as often as we wish, but I still look forward in the hope on the further jointwork.
When I was a graduate student (an aspirant) in Moscow from 1979 to 1983, I lived in thedormitory of the Moscow State University, MGU. I regularly visited the Gelfand Seminar onMonday evenings. In a sense, it was a big family in the lecture room: students and professorswere talking to each other, made friends there. One of the main people at those seminars wasDmitry Fuchs, who usually sat with Alexandre Kirillov in the first row. Gelfand often pesteredthem with questions.My first year as an aspirant was quite hard. Partially it was related to the fact that I did nothave much background in the topics which were discussed. MGU was the very best school ofmathematics in the entire world at the time, and the research topics there were numerous andat the highest international level. In comparison, the Hungarian school was mainly influencedby Erd¨os and combinatorics in those days. I liked algebra and functional analysis, and startedstudying Lie groups, Lie algebras, and representation theory. I was also learning Lie algebracohomology from Fuchs, and from his famous book which was already available in Russian. Inorder to use this theory for my project in algebras, I needed help.In those days in Moscow professors did not have offices and used to work with their colleaguesat home. Fuchs invited me to his home and I went to visit him and his mother, EkaterinaIvanovna. We had dinner together, wonderful Russian salads and other stuff. I was touched bytheir openness and hospitality. We worked for a few hours, decided to meet the next week, thenthe next, and thus our Wednesday meetings became regular.Of course, we discussed many subjects: the Fuchses were interested in my country, my family,and Dmitry Borisovich explained to me some details about the situation in Russia, talked abouthis brother, his father, his family. I met Ira, his wife, I was there when they had their first babyn the 80th Birthday of Dmitry Borisovich Fuchs 15
Figure 13.
The Fuchs family. Ekaterina Ivanovna.
Katia, and also when their second baby, Lyalya, was born. I also went with them to their dacha.I will forever be grateful to them for making me feel a part of their family, and the Wednesdaymeetings were really important to me. Living alone in Moscow in those days as a foreigner wasnot so easy, but this way I got a family there!I became especially close with Ekaterina Ivanovna, Fuchs’s mother. She was a lovely lady,and there was something warm and reassuringly peaceful about her. When I was depressed(which unfortunately happened from time to time), she was always able to help me get awayfrom feeling down. I was also there when she was dying. She died in her family, in peace.After my thesis defense, I returned home to Budapest, but I remained in regular contact withmy “Russian family”. Fuchs, Ira, and the girls visited me in Budapest, and I was able to takethem to nice playgrounds on the Gell´ert-hegy, and other places. Dmitry Borisovich had manygood friends in Budapest, like K´aroly M´alyusz, Andr´as Kr´amli, P´eter Major, Doma Sz´asz, andothers. They all got acquainted with Dmitry Borisovich in Moscow, and his lovely personalityattracted many-many people. (Unfortunately K´aroly and Andr´as are not with us anymore . . . )
Figure 14.
The Fuchs family.
Soon after I returned home, I began to get invitations to the West. I was able to go toGermany for a Humboldt fellowship and to attend various conferences. I also received aninvitation to the University of Pennsylvania and later I got a permanent job at the Universityof California at Davis. While I was there I learned that Dmitry Borisovich might be able to6 V. Ovsienkocome to the United States with his family. I immediately suggested to our Department Chair toinvite him to join our Faculty. Eventually, it worked out and since then Dmitry Borisovich Fuchs,together with Albert Schwarz, have graced the Davis mathematics community. UCDavis is luckyto have them! Recently Fuchs’s younger daughter, Elena also joined the Faculty, together withher husband, Martin Thanh Luu.I was able to continue working with Fuchs in Davis as well. We lived just a few housesapart, and I often visited the Fuchs family there too. After my return to Hungary, we have notseen each other very often. But I went to Davis for his 70th birthday celebration, and we alsomet every other summer at the Max-Planck Institute in Bonn, where both his family and mineusually spend one month.I feel lucky that I have such a real friend as Dmitry Borisovich. For me he is not justa colleague with whom it is good to work, but a close friend, with whom we share a lot of valuesin life. I wish him all the very best and long and happy years with his family!
What can be added to the beautiful (and sometimes romantic) descriptions of the mathematicallife in Moscow just before the
Perestroika ? Many remarkable memories have been publishedin recent decades by various authors. “Those days are past now and in the past they mustremain . . . ” I can only talk about personal impressions of a young undergraduate student thatI was at that time.The real life at
Mekhmat started in the afternoon, and I spent them all running from onefascinating seminar to another. This is how I met Dmitry Borisovich. I saw him several timesevery week, at Gelfand’s seminar (that, as we know, was the main mathematical event), Arnold’sseminar, meetings of the Moscow Mathematical Society, etc. Fuchs gave several lectures at MathSociety, where he talked on various subjects. I remember very well his lecture about symplectictopology, a new domain that was making its “first steps”. It was not at all (not yet) among thesubjects of his own research, but Fuchs gave a very clear and precise overview. I was not at allsurprised, it was obvious to me that Fuchs knew everything!My first personal meeting with Dmitry Borisovich was related to the seminar of Arnold. I wasa regular member of the seminar, and Arnold invited me to give a talk about the “LagrangeSchwarzian derivative”, a notion introduced in one of my first research papers. That was myfirst talk outside Kirillov’s seminar for younger students, and I was very anxious. Being out oftown, Arnold asked Fuchs to run the seminar, and Dmitry Borisovich took the job seriously: hesuggested to meet and discuss a few days before the seminar. Our meeting became for me oneof the episodes that remain in memory for the rest of the life. Fuchs asked many questions andmade several comments, that later I used many times. Trying to explain my work to him, I gotan impression of understanding it myself. My talk at the seminar went smoothly . . .My research was largely influenced by the famous papers authored by Gelfand–Fuchs andFeigin–Fuchs. With varied success, I tried to connect their cohomology classes to projectivedifferential geometry, and this task occupied me for many years. During these long years I metDmitry Borisovich from time to time at mathematical conferences and, even more often, at theTabachnikovs’ home. But never could I imagine that I would collaborate with him, until such anopportunity suddenly presented itself. This happened 30 years after the events described above.Five years ago, my teacher, Alexandre Alexandrovich Kirillov, who has always been a closefriend of Dmitry Borisovich, suggested to work on an old unsolved problem of classification ofcoadjoint orbits of the nilpotent Lie algebra of (upper) triangular matrices. Kirillov createda “square-team” consisting of four people: Fuchs, Kirillov, Sophie Morier-Genoud, and myself. A series of survey introductory lectures for a large audience. n the 80th Birthday of Dmitry Borisovich Fuchs 17
Figure 15.
Dmitry Borisovich Fuchs.
We worked within the SQuaREs program at AIM, and later, at MFO, Oberwolfach. Let memention that during our work on the project that lasted for three years and a half, the squareeventually became a pentagon.This work with my teachers was yet another moving experience. Fuchs and Kirillov stayed atthe blackboard, intensively discussing the subject and paying almost no attention to us. Kirillovdirected the general attack, Fuchs generated the technique. Sometimes they asked Sophie aboutdetails of the modern combinatorial theories. I was the chronicler, always with a notebook athand, and I recorded each word of the “oracles”.In the afternoon, the discussion usually became more relaxed. “Do you remember when weclimbed Tian Shan Mountain in 1956?” would ask one of them. “Of course, I remember”, wouldreply the other, “but it was in 1957!”Gradually, our subject evolved. Approximating the orbits, we found ourselves on a more solidsoil, studying Schubert varieties and their tangent cones. Little by little, some understandingand some results started to appear.However, writing of our paper [1] was not an easy promenade. The first version contained twotheorems that would solve our problem completely. I was in charge of reconstructing the detailsof the proofs, outlined on the blackboard by the collective efforts, and spent many sleepless nightson that. Finally, I wrote an SOS message to my coauthors, asking for a permission to call thesecond theorem a “conjecture”. When permission was granted, Sophie found a counterexampleto this brand new conjecture. And, finally, the “only if” part of the first theorem also becamea conjecture, but fortunately, with no counterexamples in sight.
References [1] Fuchs D., Kirillov A., Morier-Genoud S., Ovsienko V., On tangent cones of Schubert varieties,
ArnoldMath. J. (2017), arXiv:1606.07846.[2] Khesin B.A., Tabachnikov S.L. (Editors), ARNOLD: Swimming against the tide, Amer. Math. Soc., Provi-dence, RI, 2014.[3] Sossinsky A., In the other direction, in Golden Years of Moscow Mathematics, Hist. Math. , Vol. 6, Amer.Math. Soc., Providence, RI, 1993, 223–243.[4] Sossinsky A., Tabachnikov S., Roger C., Astashkevich A., Feigin B., Schwarz A., Personal notes, in Dif-ferential Topology, Infinite-Dimensional Lie Algebras, and Applications,