Celebratio Mathematica

Dmitry Fuchs

Interview with Dmitry Borisovich Fuchs

by Boris Khesin, Fedor Malikov, Valentin Ovsienko and Sergei Tabachnikov

The fol­low­ing in­ter­view with Dmitry Fuchs (UC Dav­is) was con­duc­ted by Bor­is Khes­in, Fedor Ma­likov, Valentin Ovsi­en­ko, and Sergei Tabach­nikov in June 2020. It took place via Zoom in the midst of the Cov­id pan­dem­ic and was re­cor­ded in Rus­si­an. The tran­scrip­tion was sub­sequently trans­lated in­to Eng­lish and ed­ited for clar­ity.

1. Family history: path to mathematics

Dmitry Borisovich Fuchs.

Ques­tion: Let us be­gin from the very be­gin­ning. Would you please say a few words about your child­hood, your fam­ily, when and how your in­terest in math­em­at­ics arose.

Fuchs: As for my ori­gin, I keep say­ing the same thing: the names of my grand­fath­ers were Ivan and Ab­ram, and this is very telling.

My ma­ter­nal grand­fath­er, Ivan Fe­dorovich Kozlov, was a peas­ant’s son. His bio­graphy is very in­ter­est­ing. His fath­er was rather rich, had his own farm. But he died very early. The cred­it­ors came and the fam­ily was ruined.

What could peas­ant’s chil­dren in a ruined fam­ily do? It is evid­ent. They went to the Kazan Uni­versity. My grand­fath­er was the lead­ing sur­geon-gyneco­lo­gist pro­fess­or in Kazan. And I can boast to be born in a nurs­ery of the hos­pit­al where my grand­fath­er was the chief phys­i­cian.

He missed my birth be­cause my ar­rival time was es­tim­ated in­cor­rectly and he went fish­ing. He was an avid fish­er­man. When he re­turned after fish­ing, I was right there. I have to add that all my grand­fath­er’s broth­ers, whom I am aware of, were part of the in­tel­li­gent­sia.

My pa­ternal grand­fath­er Ab­ram died in the year of my birth. His main pro­fes­sion was a phar­macist, he was also not a poor per­son, he had his own phar­macy. Be­ing already in adult­hood he went to a Uni­versity, re­ceived a med­ic­al de­gree, and in his last years he was a cor­on­er.

About my in­terest in math­em­at­ics, the first thing that I re­mem­ber is the fol­low­ing. We were in evac­u­ation in Tashkent. I just turned four. And also in Tashkent there was Nikolay Vladi­mirovich Efimov with his fam­ily. We were friends.

And then I got a roy­al birth­day present from him. He presen­ted me with the second copy of his manuscript. I did not care what was in it. But what was im­port­ant, one side of each page of the manuscript was covered by who knows what, but the oth­er side was clean. And I had a dream: to write by pen­cil on pa­per all num­bers from 1 to 1000. And I man­aged to do that thanks to that present. His fam­ily soon left Tashkent, but some­time later we met again. This is the first thing that I re­mem­ber in math.

And the second thing which I re­mem­ber had to do with geo­graphy rather than with math. The point is that every child has a fa­vor­ite book. I also had one. My fa­vour­ite book was the Timetable of Long Dis­tance Trains. I knew all the sta­tions by heart.

And I made a geo­graph­ic­al dis­cov­ery of which neither Colum­bus, nor Magel­lan, nor Drake could even dream. I meas­ured the dis­tance between Vla­divos­tok and Lvov, and it turned out to be equal to about 50 kilo­met­ers.

When my par­ents were sur­prised by such a dis­cov­ery, I proved it to them: I opened the Timetable and showed that there was a loc­al train from Vla­divos­tok to a sta­tion Stal­in­skaya. And there was a sim­il­ar loc­al train between Lvov and Stal­in­skaya. I man­aged to add the dis­tances from Vla­divos­tok and from Lvov to these Stal­in­skay­as and came up with the fig­ure like 51 km or so.

So these are my math achieve­ments from my early youth.

As to oth­er in­terests, the main oth­er in­terest that I had, and still have, is po­etry. I did not write poems my­self but know many poems by heart, and that is an­oth­er story.

As to math­em­at­ics, my fath­er was a math­em­atician, but our main con­tacts were on the sub­ject of an­cient his­tory. He taught his­tory to me, and I still re­mem­ber a lot about Ro­man his­tory, as well as about later epochs.

When I was gradu­at­ing from high school, these were the 1950s, my dad was kicked out of his work (ac­tu­ally he had sev­er­al po­s­i­tions, but he was fired even from a part-time job), and his mood was gloomy. He told me that I should not ap­ply to uni­versity.

As a mat­ter of fact, I entered uni­versity in 1955, and may con­firm that there was vir­tu­ally (or, likely, com­pletely) no dis­crim­in­a­tion by one’s eth­ni­city. At least among my class­mates at the uni­versity, many were Jew­ish, and many of them passed the en­trance ex­ams with the so called semi-passing grade.1

And in Septem­ber of 1953 even my dad had already be­came a pro­fess­or in the all-Uni­on Mech­an­ic­al En­gin­eer­ing In­sti­tute by Cor­res­pond­ence, so he was em­ployed again. But he still had that feel­ing that I would not be ad­mit­ted, and that I should ap­ply to the Ped­ago­gic­al In­sti­tute in­stead.

But my moth­er, secretly from my fath­er, pushed me to take my doc­u­ments from the Ped­ago­gic­al In­sti­tute and ap­ply to the Mo­scow State Uni­versity in­stead. Ap­par­ently my dad wanted that as well, since he knew that this was hap­pen­ing and did not try to dis­suade me. I passed the Uni­versity en­trance ex­ams without prob­lem. In­deed, that was a rather good year.

2. The Mekh-Mat in those days

Figure 1. As part of Khrushchev’s campaign to open vast tracts of virgin land in the northern Kazakhstan and the Altai region of the USSR, the students were sent to work there in summer. This photo dates from 1957, and standing next to Fuchs is A. Vinogradov, holding Pontryagin’s book Topological Groups.

My dad gave me vari­ous math­em­at­ic­al books to read. I re­mem­ber Dis­con­tinu­ous Func­tions by Baire, and I liked it a lot. And then I star­ted study­ing at the Mekh-Mat.2

At those days there were three courses offered in the Mekh-Mat fresh­man year: Ana­lys­is, High­er Al­gebra, and Ana­lyt­ic Geo­metry.

Khinchin was our ana­lys­is lec­turer; his lec­tures were ter­rible. I at­ten­ded only at the first one, where dur­ing the first hour he ex­plained that dia­lectic ma­ter­i­al­ism teaches us that everything changes in the world and this is why we con­sider vari­ables. Dur­ing the second hour he ex­plained that dia­lectic ma­ter­i­al­ism also teaches us that vari­able changes are re­lated, and this is why there are func­tions. I did not go any­more to his lec­tures, but was told that the lec­ture room was nearly empty. Ap­par­ently, Khinchin asked the ad­min­is­tra­tion not to check the stu­dent pres­ence on his lec­tures — oth­er­wise, it would be a cata­strophe.

The al­gebra lec­tures were giv­en by Mikhail Mikhail­ovich Post­nikov.

The lec­tures on ana­lyt­ic geo­metry were giv­en by Bor­is Nikolaevich Delaunay. This was a re­mark­able per­son. He was a mas­ter moun­tain­eer. He seemed to be at the ori­gin of Alpine Camps. He also was the or­gan­izer of the first Mo­scow Math­em­at­ic­al Olympi­ad. He was a charm­ing, but some­what strange per­son.

He or­gan­ized a sem­in­ar for stu­dents. But it was cloaked in secrecy: we should not dis­close that we were par­ti­cip­at­ing in it. There were about six fresh­men at his sem­in­ar, and we were dis­cuss­ing con­tinu­ous frac­tions and re­lated top­ics.

Delaunay had a dream. It is well-known that the con­tinu­ous frac­tions of quad­rat­ic ir­ra­tion­als are peri­od­ic. And he wanted to prove that the con­tinu­ous frac­tions of cu­bic ir­ra­tion­als, al­though not peri­od­ic, have bounded in­com­plete frac­tions. And he even told us what the title, in Ger­man, of our fu­ture pa­per on this top­ic would be: “Über ku­bisch Radikale aus zwei”. By the way, later nu­mer­ic­al ex­per­i­ments on the par­tial ra­tios of the cu­bic root of 2 seem to sug­gest a cer­tain growth, so Delaunay’s con­jec­ture ap­par­ently was in­cor­rect.

However we were study­ing there with en­thu­si­asm. And we even pub­lished a pa­per in the Dok­lady of AN SSSR. One should add that, due to some ty­po­graph­ic­al er­ror, three lines went miss­ing in the for­mu­la­tion of the main the­or­em, so it was im­possible to un­der­stand any­thing in this pa­per. However, the pa­per was pub­lished and that was my first pub­lic­a­tion, in 1958 [1].

While there were sev­er­al people at the sem­in­ar, the most act­ive were Sasha Vino­gradov and my­self. Two of us even­tu­ally came to the con­clu­sion that this was not an ap­pro­pri­ate top­ic for us. Al­though Delaunay tried to con­vince us to stay.

Mean­while we were secretly par­ti­cip­at­ing in the sem­in­ar or­gan­ized by Dynkin. After our sopho­more year, Sasha and I de­cided that we were leav­ing Delaunay’s sem­in­ar and would look for an­oth­er ad­visor. Dynkin was an ap­pro­pri­ate op­tion, but some know­ledge­able friends told us that Dynkin was work­ing en­tirely on prob­ab­il­ity the­ory at the time, and if that was not our top­ic, it was bet­ter to avoid him.

Moreover, it was bet­ter to choose an ad­visor among par­ti­cipants of Dynkin’s sem­in­ar, but not Dynkin him­self. After some time we chose Al­bert So­lomonovich Schwarz as an ad­visor. At the time Schwarz was a third-year gradu­ate stu­dent (gradu­ate stu­dents were al­lowed to ad­vise un­der­gradu­ates back then). Schwarz lived in the dorm­it­ory — he was not a Mo­scov­ite and after de­fend­ing his doc­tor­al thes­is he had worked in Vor­onezh for three years. We kept in touch and pub­lished two joint pa­pers. Sasha had less con­tact with Schwarz and later cre­ated his own area of math­em­at­ics.

Now when I teach a to­po­logy course, I tell my stu­dents that 90% of what I know in to­po­logy, and cer­tainly everything I teach them, I learned dur­ing my third year at the Uni­versity. Usu­ally they don’t be­lieve me, but this is close to the truth.

Schwarz told me to read a lot; already my third-year re­search pro­ject was on to­po­logy, on spec­tral se­quences. Ac­tu­ally spec­tral se­quences, which now are among the most fear­ful to­po­lo­gic­al ob­jects, are the very first thing that I learned in to­po­logy.

When we asked Schwarz to be our ad­visor, he told us to take both of the two courses he was go­ing to give that year. He had no idea how ig­nor­ant in to­po­logy we were at the time. The more ad­vanced course was earli­er on the sched­ule than the in­tro­duct­ory one, and the very first lec­ture I at­ten­ded was on spec­tral se­quences. I did not know ele­ment­ary things, but Schwarz ex­plained so well what the spec­tral se­quence was about, that I learned it back then, and even today I do not un­der­stand why it is so scary to many people.

I changed my math­em­at­ic­al dir­ec­tions sev­er­al times, but my first at­tempts in math were in to­po­logy un­der the guid­ance of Schwarz.

3. Topology in Moscow

Q: You men­tioned once that at the time in Mo­scow al­geb­ra­ic to­po­logy was not quite banned, but it was not ex­actly pro­moted either.

DF: Yes, in­deed. But in the 1940s, right after World War II, and cer­tainly be­fore the war, to­po­logy in Mo­scow was at the highest pos­sible level world­wide. Rokh­lin used to tell me that, at the Pontry­agin sem­in­ar in Mo­scow, when a to­po­logy pa­per ar­rived (at the time there were no e-mails or pre­prints), they already knew its res­ults and how to prove them. So noth­ing really in­ter­est­ing came by mail.

Pontry­agin had very re­mark­able res­ults. Post­nikov did the only — but amaz­ing — work on what is now called Post­nikov towers (or Post­nikov sys­tems).

When Rokh­lin came back from POW camps (his bio­graphy is so much more in­ter­est­ing then mine, so I will not elab­or­ate on that at all), first from the Ger­man camp, and then from the So­viet camp, Pontry­agin man­aged to have Rokh­lin ap­poin­ted as his sec­ret­ary: be­ing blind, Pontry­agin was al­lowed to have a private sec­ret­ary, and he took Rokh­lin for a such po­s­i­tion at the Steklov In­sti­tute. The du­ties of Rokh­lin in­cluded typ­ing and edit­ing all Pontry­agin’s pa­pers. Pontry­agin was able to type on a type­writer but made very many er­rors.

One of the stor­ies is as fol­lows. At the time, the main prob­lem in to­po­logy and, some people (in­clud­ing me) thought, in math­em­at­ics in gen­er­al, was com­pu­ta­tions of the ho­mo­topy groups of spheres. Any to­po­logy pa­per was eval­u­ated against that stand­ard: how does it con­trib­ute to the pro­gress in that dir­ec­tion.

Be­fore World War II, Freudenth­al proved the fam­ous sus­pen­sion the­or­em and com­puted \( \pi_{n+1} S^n \). And Pontry­agin did a beau­ti­ful work re­lat­ing framed man­i­folds and the ho­mo­topy groups of spheres. And he “proved” a wrong the­or­em that \( \pi_{n+2} S^n=0 \).

Ac­tu­ally, any ele­ment of the ho­mo­topy group of a sphere is as­so­ci­ated to a framed man­i­fold. It turns out that to the non­trivi­al ele­ment in that ho­mo­topy group one as­so­ci­ates a framed tor­us, not a sphere. Pontry­agin proved that any fram­ing of a sphere gives a trivi­al ele­ment and de­rived from there that \( \pi_{n+2} S^n=0 \).

This manuscript was not pub­lished. Rokh­lin kept in­sist­ing: “Lev Se­men­ovich, why wouldn’t you pub­lish this re­mark­able res­ult?” Fi­nally, Pontry­agin agreed, sat down to write the proof, found the er­ror, and proved a cor­rect res­ult [e1].

At the time, the Arf in­vari­ant was already known, and the ne­ces­sary tech­niques were de­veloped. His pa­per was sub­mit­ted, but as you know, it takes time for a pa­per to get pub­lished. Mean­while, Pontry­agin gave a talk on that work in the “to­po­lo­gic­al circle”, the main to­po­lo­gic­al sem­in­ar led by Aleksandrov.

And, ac­cord­ing to Rokh­lin, after one of Aleksandrov’s talks some­where in Europe on Pontry­agin’s res­ult, George White­head pub­lished a sim­il­ar res­ult. Now it is called the Pontry­agin–White­head the­or­em. In my opin­ion, White­head was an out­stand­ing to­po­lo­gist and proved it in­de­pend­ently. And later there ap­peared a pa­per by Rokh­lin on \( \pi_{n+3} S^n \), and this is quite a re­mark­able story.

At some point al­gebra in­vaded to­po­logy. Well, al­geb­ra­ic to­po­logy was al­ways al­geb­ra­ic to­po­logy, you know, groups, and not only groups, vari­ous kinds of things. This al­gebra was not ter­ribly com­plic­ated. Nev­er­the­less, it was ho­mo­lo­gic­al al­gebra, which in those times was called“ab­stract non­sense”. It was cre­ated as a chapter of to­po­logy, and only later ac­quired sig­ni­fic­ance in its own right.

Figure 2. A jury member of the Mathematical Olympiad.

To­po­lo­gists of the older gen­er­a­tion, Pontry­agin in­cluded, and Rok­lh­lin too, re­fused to ac­cept this in­va­sion of to­po­logy by al­gebra. For them it was an in­va­sion of their beau­ti­ful and bloom­ing land by an oc­cu­pa­tion­al force.

Pontry­agin left to­po­logy. He later wrote in­ter­est­ing pa­pers on max­im­um prin­ciple — but you know all of this! And along with him went sev­er­al of his pu­pils, of whom the most re­mark­able one was Boltyansky, who had worked in to­po­logy; his doc­tor­al dis­ser­ta­tion was in to­po­logy. Nev­er­the­less this en­tire group walked away from to­po­logy.

While de­part­ing, they re­turned a ver­dict that to­po­logy was no more; that in­stead there had ap­peared al­geb­ra­ic to­po­logy, which only stud­ied it­self — and so it did, later, but not at that time — and there­fore one should not work in al­geb­ra­ic to­po­logy.

It was Schwarz who gave lec­tures in to­po­logy, but as a res­ult of a con­flict with Aleksandrov he left Mo­scow. Post­nikov — I liked him very much, he was a re­mark­able man, but he was un­be­liev­ably lazy, and what he did could not be called teach­ing a class, not really, and so a group of stu­dent-to­po­lo­gists, which did ex­ist, was left on its own.

Novikov, of course, was our lead­er. Apart from Novikov, our group in­cluded Borya Aver­bukh, who, in­de­pend­ently of Mil­nor and at about the same time, did the com­pu­ta­tion of or­tho­gon­al cobor­d­isms; there was Ly­ova Ivan­ovskiy who worked his en­tire life on the ho­mo­topy groups of spheres; then Galya Ty­ur­ina, Sasha Vino­gradov, and I, and that was our group — there also was Dima Anosov, who nev­er worked in to­po­logy, but at­ten­ded the to­po­lo­gic­al gath­er­ings — and in­deed we stud­ied to­po­logy all on our own.

Our friends, fel­low Mekh-Mat stu­dents, ad­mon­ished us, “You are not bad guys, but what you’re work­ing on is non­sense.” Of course our re­ac­tion was ab­so­lutely in­ad­equate.

I re­mem­ber that we, too, treated all oth­er branches of math­em­at­ics in the same way. Take ana­lys­is, for ex­ample. I must say that when I gradu­ated from uni­versity, the pin­nacle of my ana­lys­is know­ledge was in­teg­ra­tion by parts. Later I was forced to learn some more ana­lys­is, but not much more. I nev­er taught ana­lys­is, ser­i­ous ana­lys­is, I mean.

4. Topology books that you authored

Q: We also wanted to ask you about the books. There is the Fuchs–Rokh­lin book [5] and there is the Fuchs–Fo­men­ko book [10]. Would you like to tell us something about them?

DF: Yes, of course. There is the Fuchs–Rokh­lin book. It is uni­ver­sally re­cog­nized that this is not a good book. I have nev­er used it for teach­ing.

Rokh­lin and I spent sev­er­al years writ­ing it. Rokh­lin was not simply a per­fec­tion­ist; he was pre­pared to change every word ad in­fin­itum. In ad­di­tion, he had prin­ciples. These prin­ciples are cor­rect, but it is of­ten the case that when a cor­rect prin­ciple is taken to its lo­gic­al con­clu­sion it be­comes non­sensic­al.

For ex­ample, one of those prin­ciples was: if an ob­ject is unique, it can’t be de­noted by a single let­ter, \( X \) or \( Y \). As a res­ult, there ap­peared ab­so­lutely ugly nota­tions. If truth be told, thank God, we did not change the nota­tions for the ho­mo­logy and ho­mo­topy groups, nor for the sine and co­sine func­tions.

The book is hard to read and, among oth­er things — how can I put it — it’s a thick book that con­tains prac­tic­ally noth­ing. Al­though what it does con­tain is well writ­ten, and if you can ig­nore the sheer volume, those hun­dreds of pages, you’ll find some stuff that may be help­ful for teach­ing and oth­er things.

As to the book writ­ten jointly with Fo­men­ko, ori­gin­ally with Gut­en­mach­er and Fo­men­ko, the story is this.

I have al­ways had a sort of “per­verse” in­clin­a­tion to giv­ing lec­tures, teach­ing. It star­ted as a game, it was in­deed a game. I taught a course — I was a gradu­ate stu­dent at that time — in to­po­logy. I did that ac­cord­ing to how I un­der­stood the sub­ject at that time, and all in all it may have been not so bad.

The course was at­ten­ded by 6 or so stu­dents. We met, I talked about this, that, and some people, To­lya Fo­men­ko was one of them, were tak­ing notes, so that in the end we had a col­lec­tion of course notes that could be quite use­ful.

And once I entered the room, and it was filled to ca­pa­city. The room was not just over­crowded, there simply wasn’t enough space for every­body, and it was clear that we had to find an­oth­er room, which we did.

What had happened? Here is what: the Atiyah–Sing­er for­mula had been dis­covered.

It is fair to say that the in­dex of an el­lipt­ic op­er­at­or prob­lem goes back to Gel­fand, and in­deed Gel­fand posed the prob­lem of ex­press­ing the in­dex of an el­lipt­ic op­er­at­or in to­po­lo­gic­al terms. It is in­deed his idea, he had many such in­ter­est­ing ideas.

This prob­lem had ex­is­ted for a while and then, all of a sud­den, there ap­peared the Atiyah–Sing­er pa­per. The fact that the prob­lem had been solved not by those, in Mo­scow and in Len­in­grad, who had spent there en­tire lives work­ing on it, but by some­body else was not all that ter­rible; the works by Mark Ios­i­fo­vich Vishik, Misha Agran­ovich, and oth­ers were cited, and so, well, it does some­times hap­pen that some­body has proved something.

What was worse is this. Here was an equal­ity: the in­dex of an el­lipt­ic op­er­at­or, in the left hand side, then the equal­ity sign, fol­lowed by something that nobody could read.

Long story short, people’s in­terest was aroused: What was writ­ten there? My course was the only to­po­logy course taught at that time at the Uni­versity.

It must be said that it was quite un­suit­able for such elu­cid­a­tion pur­poses, be­cause my un­der­stand­ing was that the most im­port­ant thing in life was the ho­mo­topy groups of spheres, and, be­lieve it or not, I had no idea what an el­lipt­ic op­er­at­or was — to say noth­ing of its in­dex.

In re­la­tion to this, by the way, I can tell you an­oth­er story. In 1961, an all-Rus­sia — and not all-So­viet — Con­gress of Math­em­aticians took place in Len­in­grad. Among the in­vit­ees were Mil­nor and Hirzebruch, such ma­jor fig­ures. Hirzebruch was giv­ing a talk, Borya Ven­kov was trans­lat­ing.

Hirzebruch star­ted in Ger­man but then, un­ex­pec­tedly for him­self, switched to Eng­lish, which was all the same for Borya Ven­kov, who knew a good dozen of lan­guages, and who con­tin­ued as if noth­ing had happened.

At some point Hirzebruch asked the audi­ence, “Who knows what the \( K \)-func­tor is?” Three or four hands were raised, mine not among them. Borya Ven­kov raised his hand, some­body else did too. This is how it was, in 1961, which is strik­ing, giv­en the fact that most ma­jor works on to­po­lo­gic­al \( K \)-the­ory had been already com­pleted. Well, in 1962, all of us knew what the \( K \)-func­tor was, of course we did, but not in 1961.

Any­way, an in­terest in to­po­logy — a some­what un­healthy in­terest, I’d say — was aroused by the Atiyah–Sing­er for­mula and, in par­tic­u­lar, the notes of my course were in de­mand.

At the be­gin­ning, the au­thors were To­lya Fo­men­ko, Vitya Gut­en­mach­er, and I. Later, when the first part was com­pleted, Vitya stopped par­ti­cip­at­ing in the writ­ing. To­lya Fo­men­ko drew the pic­tures.

I am ashamed to say that I nev­er ed­ited the ori­gin­al text, and it is quite ter­rible — there is an er­ror in each line. Oth­er edi­tions ap­peared. To­lya and I worked on the second part; now To­lya drew pic­tures il­lus­trat­ing a spec­tral se­quence and such and, in the end, the ro­taprint (off­set du­plic­at­or) edi­tion ap­peared. Fur­ther­more, it was trans­lated by Károly Mály­us in­to Eng­lish in Hun­gary, and thus an Hun­gari­an edi­tion ap­peared.

At some point, Seryozha Novikov said that the book must be pub­lished by the “Nauka” pub­lish­ing house, that we must pre­pare the text. Thus the book was pub­lished, some time in the ’80s, I don’t re­mem­ber ex­actly when, by “Nauka”.

This book was con­ceived, and writ­ten, as such a course in to­po­logy, that starts at the very be­gin­ning, noth­ing is as­sumed known, and ends with the res­ults that, at the time, seemed to be the pin­nacle of the to­po­lo­gic­al wis­dom: there was the Adams spec­tral se­quence — an ab­so­lutely mar­velous thing, but no longer needed, un­for­tu­nately — and the chapter on \( K \)-the­ory, which even now seems to me to be well writ­ten. This is how this book came to ex­ist.

Q [ST]: I am not sure you re­mem­ber this, but when I asked you in 1975 to be my ad­visor, you ques­tioned me in de­tail about vari­ous things, in par­tic­u­lar, about a sub­ject I was in­ter­ested in. When I answered: “To­po­logy” you told me: “But why?! To­po­logy is dead!” Could you com­ment on this now?

DF: Well, I could have told you that long time ago, but now let me be more care­ful.

I would say that al­geb­ra­ic to­po­logy is very much alive for the reas­on that it is “com­plete”. It can be learned from the be­gin­ning to the end. The last im­port­ant achieve­ment in al­geb­ra­ic to­po­logy was \( K \)-the­ory. After that, of course, ar­rived bor­d­isms and cobor­d­isms, the spec­tral se­quence of Adams–Novikov, etc., but bright achieve­ments ended.

It is true that the Adams con­jec­ture about the \( J \)-func­tor was proved in the 1970s by Beck­er and Got­tlieb [e3]. out­stand­ing work in al­geb­ra­ic to­po­logy was done when al­geb­ra­ic to­po­logy it­self no longer ex­is­ted.

However, to­po­logy was still alive, since dif­fer­en­tial to­po­logy, or to­po­logy of man­i­folds, re­mained act­ive. This the­ory was pop­u­lar in the 1930s and be­came pop­u­lar again in the 1960s.

At that time it was nat­ur­al to as­sume that a man­i­fold is simply con­nec­ted. Ac­cord­ing to Novikov, the role of the fun­da­ment­al group in the the­ory of man­i­folds be­came the main sub­ject of dif­fer­en­tial to­po­logy in the 1980s. After that, the clas­sic­al dif­fer­en­tial to­po­logy more or less ceased to ex­ist.

Yet, low-di­men­sion­al to­po­logy (to­po­logy of man­i­folds of di­men­sions 3 and 4) re­mains very act­ive and will be act­ive for a long time. This sub­ject res­ists any form of al­geb­ra­iz­a­tion! In par­tic­u­lar, man­i­folds of di­men­sion 3 are very pop­u­lar at our math­em­at­ic­al de­part­ment.

To­po­logy with ad­di­tion­al struc­tures, such as sym­plect­ic to­po­logy, or to­po­logy re­lated to Rieman­ni­an geo­metry, is also a very act­ive area. To fin­ish the dis­cus­sion about al­geb­ra­ic to­po­logy, I re­com­mend to every young math­em­atician to learn it, but I can­not re­com­mend to work on it as a re­search­er.

5. Favorite mathematical works: collaborations with I. Gelfand and B. Feigin

Q: Vladi­mir Arnold, when asked to ex­plain some of his res­ults to the gen­er­al audi­ence, men­tioned ap­plic­a­tions of his the­ory to weath­er fore­cast. How would you an­swer the same ques­tion? Can you ex­plain some of your res­ults to a “ran­dom per­son from the street”?

DF: I am not sure about a ran­dom per­son from the street who can have something more in­ter­est­ing to do, but I could say something to high school stu­dents?

As far as I re­mem­ber my own child­hood, I was de­lighted by two for­mu­las. One of them is about the sum of angles of a tri­angle, the oth­er one (that I learnt at the Delaunay sem­in­ar) is about ra­tion­al ap­prox­im­a­tions of a real num­ber.

Giv­en an ir­ra­tion­al num­ber \( \alpha \), the con­ver­gents of the cor­res­pond­ing con­tin­ued frac­tion provide the best (in many senses) ap­prox­im­a­tions of it. If the ra­tion­al giv­en by a con­ver­gent of the con­tin­ued frac­tion of \( \alpha \) equals \( p/q \), then \( \alpha-p/q = 1/(\lambda q^2) \), where \( \lambda \) is some con­stant that meas­ures the “qual­ity” of the ap­prox­im­a­tion.

The amaz­ing thing is that there ex­ists an ex­pli­cit for­mula for this con­stant for the “best ap­prox­im­a­tion” of \( \alpha \), namely for the con­ver­gents of the con­tin­ued frac­tion. The con­ver­gent is the fi­nite con­tin­ued ob­tained by cut­ting the con­tin­ued frac­tion for \( \alpha \) at some place. The “qual­ity in­dex” for this con­ver­gent equals the sum of the con­tin­ued frac­tion ob­tained by re­mov­ing the con­ver­gent part and the con­ver­gent it­self read from the bot­tom to the top.3 The proof is very simple, but the fact it­self is as­ton­ish­ing.

Figure 3. At “Research in Pairs”, Oberwolfach, 2017. Left to right: Sophie Morier-Genoud, Valentin Ovsienko, Alexandre Kirillov, and Dmitry Fuchs.
Oberwolfach Photo Collection, photo ID: 21783

I don’t know if I can tell something equally in­ter­est­ing about my res­ults. With Gel­fand, we cal­cu­lated the co­homo­logy of the Lie al­gebra of vec­tor fields on the circle [2]. The res­ult turns out to be as­ton­ish­ing, too. All that happened many years ago… I for­got to men­tion one more thing. There is a for­mula that has nev­er been un­der­stood. This is the the­or­em of Fei­gin, Ma­likov and my­self about the sin­gu­lar vec­tors of the Verma mod­ules of a Kac–Moody al­gebra [7]. The strange thing is that the gen­er­at­ors enter this for­mula with strange ex­po­nents which are not even real num­bers. The for­mula is very strange and it re­mains mys­ter­i­ous.

Q: Can you tell us about the Gel­fand–Fuchs the­ory?

DF: First of all, let me tell you how we met. I star­ted to par­ti­cip­ate at Gel­fand’s sem­in­ar. In my stu­dent years that was some­what “against the etiquette”! We were not sup­posed to be in­ter­ested in what was dis­cussed at this sem­in­ar. By the way, the first among to­po­lo­gists who came to Gel­fand’s sem­in­ar was A. S. Schwarz.

Gel­fand liked to in­volve people in­to his circle, and he usu­ally asked many in­ter­est­ing ques­tions.

When I entered in­to a per­son­al con­tact with Gel­fand, he asked me a gen­er­al ques­tion about Lie groups: “How is it pos­sible that for to­po­lo­gists a Lie group and its max­im­al com­pact sub­group is al­most the same ob­ject, they have the same ho­mo­logy, clas­si­fy­ing space, etc., where­as their rep­res­ent­a­tion the­or­ies are com­pletely dif­fer­ent?”

We star­ted to think about this and fi­nally in­tro­duced new clas­si­fic­a­tion spaces. It turned out that there ex­ist non­trivi­al fiber bundles even with the struc­tur­al group ho­mo­top­ic to a point. That was our first joint work. Then we star­ted to work on co­homo­logy Lie al­geb­ras, and con­tin­ued for years. Clev­er people, like Osya Bern­stein and Dima Kazh­dan, ex­plained to us that the spec­tral se­quence that we in­ven­ted and were proud of was well known, and was called the Hoch­shild–Serre spec­tral se­quence.4

Gel­fand had the fol­low­ing view­point. Whatever a (co)ho­mo­logy the­ory we con­sider, it is in­ter­est­ing when a huge in­fin­ite-di­men­sion­al com­plex pro­duces fi­nite-di­men­sion­al (co)ho­mo­logy.

It turned out, for in­stance, that the co­homo­logy of the Lie al­gebra of vec­tor fields on the circle is of this type [2]. It is fi­nite-di­men­sion­al (one-di­men­sion­al in fact!). After this first res­ult, we were un­able to stop, es­pe­cially since we re­ceived the pa­per of God­bil­lon and Vey about char­ac­ter­ist­ic classes of fo­li­ations.

This work was ex­ten­ded by the group of people around us. For four, or even five or six years, we worked with Gel­fand without in­ter­rup­tion, I was of­ten at his home. This ended when my stu­dent Borya Fei­gin “grew up” and we switched to rep­res­ent­a­tion the­ory with him.

To sum­mar­ize our work with Gel­fand, many people asked me about the nature of our col­lab­or­a­tion. Some had an im­pres­sion that Gel­fand gen­er­ated ideas, and I fin­ished all the tech­nic­al work. That was not the case! Gel­fand par­ti­cip­ated in all tech­nic­al work, and our col­lab­or­a­tion al­ways was equal and fair. Both of us pro­duced ideas and res­ults…

Q [BK]: We re­call from one of your auto­bi­o­graph­ic­al art­icles what you wrote about your first en­counter with Gel­fand. He asked you in the cor­ridor about your math­em­at­ic­al in­terests, but then (for­tu­nately!) listened in­at­tent­ively. Was that really like that?

DF: Yes! You re­mem­ber cor­rectly. I also re­mem­ber this sen­tence. I was at Gel­fand’s sem­in­ar and he sug­ges­ted to me to dis­cuss after the sem­in­ar. By the way, that was a night­mare, since Gel­fand was badly or­gan­ized, and there was a risk to wait for a long time. Dur­ing the dis­cus­sion, Gel­fand in­deed asked me about my work, and I did not have much to tell him…Then he sug­ges­ted (as he did with every­one!) to work to­geth­er.

Q [ST]: May we ask one more math­em­at­ic­al ques­tion? This is about ex­per­i­ment­al math­em­at­ics. Around 1980, I came to see you with a the­or­em I just had proved. (I was very proud of it, since the res­ult dis­proved one of your con­jec­tures with Fei­gin.) You listened to the proof, but then said that one ab­so­lutely needs to check this with the help of a com­puter. That was a great sur­prise, since a proof was sup­posed to suf­fice…

DF: In the be­gin­ning I was my­self very far from com­puter as­sisted com­pu­ta­tions. We had a com­puter class at the Uni­versity, but I was not in­ter­ested much.

That was un­til the fol­low­ing event. We were on the Che­get Moun­tain (a part of the Cau­cas­us Moun­tains) for down­hill ski­ing, and the weath­er was aw­ful. It was not even pos­sible to go out. Then my friend, To­lya Kush­nik­en­ko, with whom we stayed at the cab­in, sug­ges­ted to teach me com­puter pro­gram­ming (us­ing the Ba­sic code).

As you know, Ba­sic is still the only com­puter lan­guage I know, and I nev­er needed more. Un­til now, I have the same habit that you de­scribe. Be­fore start­ing to work on a prob­lem, I check ex­amples on the com­puter.

Q [VO]: I re­mem­ber Bor­is Fei­gin’s talk at Gel­fand’s Sem­in­ar about your joint proof of Kac’s for­mula for the Vi­ra­s­oro al­gebra [4]. Gel­fand kept ask­ing Fei­gin why Kac had not pub­lish a proof. Fei­gin replied: “Per­haps he did not have one, per­haps it was the res­ult of com­puter com­pu­ta­tion.” Gel­fand simply could not be­lieve it. What is the story here?

DF: It is like this. Vic­tor Kac has two broth­ers, and one of them, Bor­is, who is about four years young­er, was a pro­gram­mer. And in­deed, this for­mula — tech­nic­ally speak­ing, the for­mula for the de­term­in­ant of the Shapovalov quad­rat­ic form in the case of the Vi­ra­s­oro al­gebra — was con­tained in Kac’s note, along with a vague prom­ise that the proof would ap­pear later. And in­deed, Fei­gin and I proved the for­mula, and this proof was not easy at all.

What can I say here? I have two com­ments: first, what Fei­gin said was true, and second, one should not say it like this. By the way, Kac later re­ferred to our pa­per, say­ing that in­deed we proved his con­jec­ture. And this is cer­tainly true.

Q: A hard ques­tion: which of your pa­pers you like best?

DF: It’s in­deed a hard ques­tion: whatever I will say won’t be com­pletely true.

You know, some­times one needs to fill out a form where one must men­tion not all pub­lished pa­pers, but a small se­lec­tion of the most im­port­ant ones. I al­ways in­clude our joint pa­per with Fei­gin and Ma­likov [7], and it is pre­cisely be­cause it is not com­pletely un­der­stood to this day.

I also very much ap­pre­ci­ate — al­though I am not totally sure that it is fully jus­ti­fied — my pa­per on Le­gendri­an knots [9]. One of its achieve­ments is in­dis­put­able: it is the in­tro­duc­tion of the term “Chekan­ov–Eli­ash­berg al­gebra”.

The point is that Yur­iy Chekan­ov — whom I oth­er­wise very much re­spect — did all he could so that Eli­ash­berg’s name would not be at­tached to this con­struc­tion. It is true that he dis­covered it earli­er, but it is also clear that Eli­ash­berg did it in­de­pend­ently (I at­ten­ded his first talk on the sub­ject — it is re­lated to Flo­er ho­mo­logy, etc.)

And Chekan­ov’s pa­per con­tained a res­ult that Yasha did not have: this al­gebra — as al­geb­ras do — may have an aug­ment­a­tion, and the ex­ist­ence of an aug­ment­a­tion makes the ho­mo­logy cal­cu­la­tions much easi­er. I found a ne­ces­sary and suf­fi­cient geo­met­ric con­di­tion for the ex­ist­ence of aug­ment­a­tion.

Curi­ously, the main geo­met­ric con­struc­tion of my pa­per, which I called “a nor­mal rul­ing”, is also con­tained in the pa­per of Chekan­ov and Pushkar [e5] that ap­peared in­de­pend­ently and sim­ul­tan­eously. And al­though the prob­lem that I solved clearly be­longed to Chekan­ov, they did not ap­ply this geo­met­ric con­struc­tion to aug­ment­a­tions.

So I ap­pre­ci­ate this pa­per of mine. And, start­ing with this pa­per, all the pa­pers on this sub­ject used the term “Chekan­ov–Eli­ash­berg al­gebra”. So I have achieved that Yasha’s name is now firmly at­tached to this con­struc­tion.

Q [ST]: You au­thored the pa­per titled “Quil­len­iz­a­tion and bor­d­ism” [3]. I stud­ied it when I was a stu­dent, and I liked it very much. Would you like to com­ment on it?

DF: What can I say? I al­ways men­tion it as an ex­ample of a work that, on the one hand, is very in­ter­est­ing, but on the oth­er, con­tains no res­ults. It doesn’t con­tain new the­or­ems, but of­fers a new un­der­stand­ing of the ex­ist­ing ones.

I am not aware of ser­i­ous con­tinu­ation of this work ex­cept for one pa­per that, to my shame, I do not know in suf­fi­cient de­tail. It is a pa­per by Eli­ash­berg on this very top­ic, and my pa­per seems to be men­tioned in its title. This is, to the best of my know­ledge, the only con­tinu­ation of my work. I am glad that you liked it; in my time, I was quite fas­cin­ated by this sub­ject.

Q [FM]: To con­tin­ue this thread, here is a ques­tion that is not likely to in­terest any­one, ex­cept me: what is my fa­vor­ite math­em­at­ic­al pa­per (nat­ur­ally, not among my pa­pers, but in gen­er­al)?

There are two, and one of them is yours, jointly with Fei­gin, on the in­vari­ant dif­fer­en­tial op­er­at­ors [4]. I re­mem­ber that I was amazed: you start with a ques­tion about de­riv­at­ives that a fresh­man can un­der­stand, then some du­al­ity oc­curs, and fi­nally the an­swer to this in­no­cent ques­tion fol­lows from the rep­res­ent­a­tion the­ory of the Vi­ra­s­oro al­gebra. And you not only prove a the­or­em; as a by-product, you ob­tain a for­mula for the de­term­in­ant of the Shapovalov quad­rat­ic form. What can you say about this work?

DF: Yes, to be hon­est, I like this work as well. Thank you for ask­ing, I ap­pre­ci­ate this pa­per in­deed. But un­like the pa­per coau­thored with you, there are no mys­ter­ies left in that pa­per with Fei­gin.

And more gen­er­ally, the sev­er­al years of my close col­lab­or­a­tion with Borya Fei­gin were awe­some. You know, not every­one can col­lab­or­ate with him. Borya was al­ways full of ideas. We had weekly meet­ings (on Wed­nes­days). He usu­ally ap­peared with something fas­cin­at­ing. I was puzzled and ex­cited, and was ready to say something about it at our the next meet­ing. But it turned out that at the time of the next meet­ing he even did not re­mem­ber what he said. He had something new, equally fas­cin­at­ing, in­stead.

At some mo­ment, I had to say to him that to gradu­ate from the Uni­versity, he needed to sub­mit a thes­is, so he had to con­cen­trate on something and write a pa­per. He frowned, but brought to our next meet­ing a fully com­pleted art­icle en­titled “Char­ac­ter­ist­ic classes of flags of fo­li­ations”. The work was pub­lished in Func­tion­al Ana­lys­is and its Ap­plic­a­tion [e2], and Borya gradu­ated without a prob­lem.

This is his mode of op­er­a­tion at Kyoto. They have the fam­ous Re­search In­sti­tute of Math­em­at­ic­al sci­ence (RIMS) there, with many su­perb math­em­aticians, and Bor­is also vis­ited there reg­u­larly (in part, for per­son­al reas­ons). And, ac­cord­ing to ru­mors, he came to the In­sti­tute, said something, as if in passing, and left. And his col­leagues there spent the rest of the day try­ing to un­der­stand his re­mark. Of course, Borya was al­ways a wel­comed guest at RIMS.

But I nev­er had dif­fi­culties with him: I al­ways un­der­stood what he said, and our col­lab­or­a­tion as per­fect.

I re­mem­ber that once Vladi­mir Za­khar­ov was vis­it­ing Dav­is (it was at the home of Al­bert Schwarz). Za­khar­ov was the dir­ect­or of the Land­au In­sti­tute of The­or­et­ic­al Phys­ics in Cher­no­go­lovka. So, over a glass of wine, I asked him: “How come that Fei­gin is still a ju­ni­or re­search­er at your in­sti­tute?” He made a long face: “Yes, each time I leave for abroad, I tell them to rem­edy the situ­ation, but they do not listen!” I jok­ingly told Borya that he would be pro­moted to Deputy Dir­ect­or — of course, it nev­er happened.

6. “People’s University”

Q: We would like to ask you about the “People Uni­versity”.5

DF: A lot was writ­ten on this sub­ject already. And, by the way, one of you is a former stu­dent of this uni­versity ([FM]).

In­deed, one day, Valera Sen­der­ov and Borya Kanevsky — who I didn’t know then — vis­ited and asked wheth­er I’d like to give lec­tures to a cer­tain group of stu­dents. These were mostly Jew­ish stu­dents who were not ad­mit­ted to Mekh-Mat, but there were non-Jew­ish ones as well. I replied that it would be my pleas­ure.

In this class, the lec­tur­ers were An­drei Zele­v­in­sky, who taught a very re­mark­able ana­lys­is course; I taught lin­ear al­gebra and something else, equally trivi­al. Alyosha Soss­in­sky taught an al­gebra course, later Borya Fei­gin re­placed him. So the three of us taught and, of course, there was Bella Sub­botovskaya6 without whom noth­ing would hap­pen at all.

Once Valera Sen­der­ov vis­ited and col­lec­ted in­form­a­tion from every­one about their failed at­tempts to enter uni­versity. Ac­tu­ally, it was an un­usu­al year, 1980, the year of the Mo­scow Olympic Games. For this reas­on, the en­trance ex­ams at Mo­scow State took place sim­ul­tan­eously with all oth­er in­sti­tu­tions, un­like a month earli­er, as al­ways. As the res­ult, many stu­dents de­cided not to take their chances at Mekh-Mat, and per­haps they were right to make this choice. And so among our stu­dents the per­cent­age of those who were dis­crim­in­ated against at the en­trance ex­ams wasn’t that high.

Q: Did you real­ize that you were play­ing with fire?

DF: We were all some­what care­less then. Yes, we real­ized that it was dan­ger­ous, but didn’t we do oth­er un­safe things too!

I re­mem­ber how I came to Pet­rovsky, the Rect­or of Mo­scow State, to com­plain that Sasha Ge­r­on­im­us was not ad­mit­ted to Mekh-Mat. Full of in­dig­na­tion, I went to talk to Pet­rovsky and, sur­pris­ingly, his sec­ret­ary didn’t stop me. And com­plain I did!

In­stead of throw­ing me out of his of­fice, as he prob­ably should have done, Pet­rovsky listened to what I had to say calmly and said: “Let him enter as a part-time stu­dent,7 and I prom­ise you…”. Can you ima­gine, Pet­rovsky him­self prom­ises me, al­most a teen­ager! “I prom­ise you that on the next year he will be a full-time stu­dent.” And so he was. Later Ge­r­on­im­us be­came a gradu­ate stu­dent of Man­in.

Some people used this path: it was much easi­er to get ad­mit­ted as a part-time stu­dent (one had to present a proof that one had a day job, but some­times this “proof” was a fake). After the first semester, some Mekh-Mat stu­dents failed fi­nal ex­ams and were ex­pelled, and slots be­came avail­able for these trans­fers.

Q: When they ar­res­ted Sen­der­ov and Kanevsky, and one stu­dent, did you real­ize that you might be the next one?

DF: It happened like this. A per­son came to talk to me whom I had nev­er saw be­fore (Serezha Lvovsky). He came to my of­fice where I happened to be at the time, and he in­vited me to step out­side to chat.8 When we were out­side, he in­formed me about the ar­rest of Sen­der­ov and Kanevsky. Later An­drei Zele­v­in­sky vis­ited, and we re­hearsed what we should say if be­ing in­ter­rog­ated. But neither he nor I was ever summoned.

In gen­er­al, our class was not af­fected much: ap­par­ently, the KGB didn’t have a list of stu­dents who at­ten­ded. That stu­dent who was ar­res­ted, he was a year ju­ni­or, and his class was taught by Sen­der­ov and Kanevsky. And the list of those stu­dents sat on the desk of the KGB of­ficer who talked to Bella Sub­batovskaya. As a mat­ter of fact, it was the list of stu­dents who paid for the sand­wiches that Bella provided to them.

And in­deed, as the res­ult of these ar­rests, our uni­versity ter­min­ated its activ­it­ies, al­though I heard that some activ­it­ies con­tin­ued, in some form, even later. In any event, our fur­ther plans, bey­ond the first two years, were nev­er real­ized. Well, there was a sem­in­ar later, with the same or­gan­izers and more-or-less the same par­ti­cipants.

7. Mathematical olympiads and Kvant magazine

Q: We would like to ask about two, per­haps re­lated, things: your par­ti­cip­a­tion in math­em­at­ic­al olympi­ads and in Kvant magazine.

DF: When I was a stu­dent, math­em­at­ic­al olympi­ads were ex­tremely pop­u­lar. There were very many par­ti­cipants, and the stu­dents were in­vited to grade their works. It so happened that I had reached the top of the hier­archy. In my case, this was the deputy head of the or­gan­iz­ing com­mit­tee of an olympi­ad — the head was a pro­fess­or, Efre­movich, in my case. And the deputy was the highest po­s­i­tion that a non-pro­fess­or could reach.

In the pre­ced­ing years, I was an ed­it­or of col­lec­tions of pre­par­a­tion prob­lems to olympi­ads. One of those col­lec­tions was en­tirely my cre­ation.

Q [ST]: I re­mem­ber that one for these col­lec­tions con­tained a prob­lem whose for­mu­la­tion star­ted as fol­lows: “In the city of Fuchs, they an­nounced that the cit­izens were en­titled to ob­tain ele­phants for free…” Was it one of those that you ed­ited?

DF: Nev­er heard of it! I can say that we had a tra­di­tion: upon the com­ple­tion of an olympi­ad, we drank Geor­gi­an wine whose num­ber co­in­cided with the num­ber of the olympi­ad.

We are talk­ing about the Mo­scow Math Olympi­ad here. The all-Uni­on one ap­peared later. At the be­gin­ning, we in­vited stu­dents from oth­er cit­ies to par­ti­cip­ate in our olympi­ad. This in­cluded Misha Gro­mov — you prob­ably re­cog­nize the name…

The Deputy Sec­ret­ary of Edu­ca­tion of the Rus­si­an Fed­er­a­tion was Markushev­ich.9 Clearly, the in­form­a­tion about the olympi­ads was avail­able to him. And so, the Rus­si­an Fed­er­a­tion, and later, all-Uni­on Olympi­ad came about. Based on its res­ults, the So­viet team was formed to par­ti­cip­ate in the In­ter­na­tion­al Math­em­at­ic­al Olympi­ad, and — I want to em­phas­ize this — in­de­pend­ently of their eth­nic­al iden­tity, the mem­bers of the team were ad­mit­ted, without en­trance ex­ams, to the uni­versit­ies of their choices. And they usu­ally chose Mekh-Mat. An ex­ample is Sasha Gon­char­ov, who, oth­er­wise, would be nev­er ad­mit­ted.

Q: Can you say a few words about Kvant?

DF: Start­ing with my third un­der­gradu­ate year, I was run­ning a math­em­at­ic­al circle at the uni­versity. I was in­tim­ately re­lated with the olympi­ads and circles, so when Kvant star­ted, it was nat­ur­al for me to be­come a con­trib­ut­or. I nev­er was a mem­ber of its ed­it­or­i­al board, but I was a “seri­al” au­thor in­deed.

8. Outside of mathematics: sport and poetry

Figure 4. This picture was drawn by the artist Sergei Ivanov on the occasion of Fuchs’ 70th birthday.

Q: Would you please tell us about sport in your life. Fig­ure 4 shows you on a bi­cycle.

DF: In­deed, this is an nice draw­ing made by Serezha Ivan­ov. I nev­er was a re­cord hold­er. However, I can tell you two things about that. For many years, twice a year, I went hik­ing: in the winter I was ski­ing and in the sum­mer my main spe­ci­al­ity was kayak­ing. This con­tin­ued for 15 years. I vis­ited many places. And I can boast, since nobody will veri­fy, that I rowed very well and was able to over­take any­one. Par­tic­u­larly if I rowed alone and nobody stopped me from do­ing that.

I liked bi­cyc­ling. I would be ashamed to show my bike to ser­i­ous people. Its brand was “Ukraine”, without gears and oth­er gad­gets. I star­ted rid­ing when I was a school stu­dent. And many years later I had an idea, re­lated to our fam­ily coun­try house in the woods close to Kazan: to get by bike from the door­steps of our Mo­scow apart­ment to the deck of our coun­try house.

I fol­lowed the high­ways and such a trip took 6 days. To re­turn, I took a boat from Kazan to Gorky city, and rode the bike only half-way from Gorky to Mo­scow. I did it six times or so. The dis­tance from Mo­scow to Kazan is ex­actly 800 km. I have to ad­mit that the road was nonuni­form. There were hilly places, where everything was slow­ing down. There were some steep parts where I needed to get off the bike and push it next to me. Later, few­er and few­er such places re­mained: I was able to ride up very steep hills. I had a small tent with me and it was a fun trip in­deed.

Q: If you don’t mind, say a few words about po­etry.

DF: I must say that I learned to love po­etry from my fath­er. He loved it very much, al­though our tastes in po­etry were not just dif­fer­ent, they were op­pos­ite.

Figure 5. Fuchs rock climbing.

You know, there was a time when the best spe­ci­mens of Rus­si­an po­etry were not ex­actly banned, but were not pub­lished either. And at those times many people would type un­pub­lished poems for their use. I had a type­writer, and I did this a lot — I still keep some of the poems that I typed then, in spite of the fact that they are pub­lished by now. And it turned out that, by typ­ing, lit­er­ally through my fin­gers, I mem­or­ized the text very well. Once I had typed a poem, I could re­cite it without mis­takes. As a res­ult, I can boast to know very many poems by heart.

One must say that there are ex­perts who in­sist on know­ing all the poems writ­ten by par­tic­u­lar au­thors. But I am just a con­sumer of po­etry: I know only the great poems, but there are more than enough of them!

Some time ago, Laura Givent­al kept a kind of lit­er­ary salon at her home. People would at­tend, in­vited speak­ers gave present­a­tion on vari­ous top­ics. Once I was in­vited, and I gave a two hour lec­ture on the po­etry of Nikolay Gumilev. Of these two hours, I was re­cit­ing poems for an hour and a half, and for half an hour I presen­ted my un­pro­fes­sion­al com­ments.

There is no deny­ing: po­etry is an im­port­ant part of my life.

Q: Once you pub­lished an art­icle in the Kvant magazine on the pros­ody of Rus­si­an verse [8]. It is an un­usu­al top­ic for Kvant. How did it hap­pen?

DF: Un­like you all, I did not at­tend a spe­cial­ized high school. After the war we lived near Mo­scow, and I at­ten­ded a loc­al school that was loc­ated two kilo­met­ers from my home. Many of my class­mates were orphans: their fath­ers did not come home from the war. Their moth­ers worked long hours try­ing to make ends meet, and they were “edu­cated” by the street. Hoo­ligan­ism was wide spread.

Nev­er­the­less, some les­sons were very in­ter­est­ing. In par­tic­u­lar, our teach­er of Rus­si­an lit­er­at­ure or­gan­ized a club where she taught us many things, in­clud­ing pros­ody. And every­one knew that if a poem was writ­ten in iambs, then all even syl­lables were stressed, and all odd ones were not.

But con­sider any ex­ample, say, start read­ing Eu­gene One­gin — and you will see that this defin­i­tion clearly fails. (If you con­tin­ue read­ing, you may find some con­firm­a­tions later, but mostly it fails.) And so I real­ized that since iambs ex­is­ted and could be eas­ily iden­ti­fied, the defin­i­tion must be dif­fer­ent.

Мой дядя самых честных правил,
Когда не в шутку занемог,
Он уважать себя заставил
И лучше выдумать не мог.

(My uncle, man of firm con­vic­tions…
By fall­ing gravely ill, he’s won
A due re­spect for his af­flic­tions
The only clev­er thing he’s done.)10

Even­tu­ally, I came up with a math­em­at­ic­al defin­i­tion, not of iamb only, but of all po­et­ic feet. It de­pends on two num­bers, \( d \), the num­ber of syl­lables between the stresses, and \( n \), the num­ber of the first syl­lable un­der stress. For iamb and tro­chee, \( d=2 \), and for dac­tyl, am­phi­brach, and anapest, \( d=3 \).

The defin­i­tion is as fol­lows. The syl­lables that be­long to this arith­met­ic pro­gres­sion are called strong. And every word (ig­nor­ing some ex­cep­tions) has a stressed syl­lable. So if a word con­tains a strong syl­lable, then its stressed syl­lable is strong. (By the way, ac­cord­ing to this defin­i­tion, dif­fer­ent feet can co­ex­ist).

In any event, this defin­i­tion, giv­en in my art­icle, caused the ire of some people. You prob­ably don’t know the story of how this art­icle got pub­lished.

The point is that the main ex­pert on po­et­ic­al feet, and po­etry in gen­er­al, was A. N. Kolmogorov, who also happened to be a co-Ed­it­or-in-Chief of Kvant. Clearly, it was im­possible to pub­lish such an art­icle without his ex­pli­cit ap­prov­al.

Figure 6. Participants of the conference at UC Davis in honor of D. Fuchs’ 70th birthday.

Kolmogorov was already very old and frail, and people close to him took turns in as­sist­ing him with every­day needs. One of them was Alexey Soss­in­sky, the head of math­em­at­ics de­part­ment of Kvant. So when Alexei was with Kolmogorov, he men­tioned, in passing, that Fuchs wrote something about pros­ody. “It would be very fit­ting if you, An­drei Nikolaevich, wrote such an art­icle”. Kolmogorov sighed: “Yes, it would be bet­ter if I did, but alas, I can­not any­more”. This one could in­ter­pret as a per­mis­sion for pub­lic­a­tion.

And so my art­icle went to print. But it was also sent for a ref­er­ee re­port. The ref­er­ee was Prok­horov — not the prob­ab­il­ist, but an­oth­er stu­dent of Kolmogorov who spe­cial­ized in po­etry. You know, I had the ex­per­i­ence of re­ceiv­ing neg­at­ive re­ports on my art­icles, but this one was not just neg­at­ive — it was dev­ast­at­ing. It didn’t simply say that the art­icle was bad, it claimed that the au­thor was a scoun­drel.

It was simply im­possible to pub­lish the art­icle with such a re­port. But there was a pe­cu­li­ar prop­erty of the So­viet pub­lish­ing pro­cess: after a cer­tain point, one couldn’t stop pub­lic­a­tion (of course, un­less it con­tained a ser­i­ous polit­ic­al mis­take — which did hap­pen, too). And so, in spite of the re­port, my art­icle got pub­lished.

If Prok­horov had sub­mit­ted his re­port earli­er, the story would have ended dif­fer­ently. I main­tained that he simply did not read the art­icle. It was clear from his re­port that he was quite know­ledge­able about the sub­ject. So I offered to meet and dis­cuss it with him. But Yury So­loviev, who su­per­vised math­em­at­ics in Kvant, was afraid that there would be a fist­fight, and he did not au­thor­ize the meet­ing. It is a pity: maybe we would even have writ­ten something to­geth­er.

9. Leaving for the USA

Figure 7. Fuchs at his UC Davis office.

Q: How and when did you leave for the USA? Was it 1990, the year that we all left?

DF: To start with, up to a cer­tain point, there was simply no pos­sib­il­ity to vis­it a for­eign coun­try (I did vis­it Bud­apest once by a private in­vit­a­tion, but this was not for busi­ness). And then, sud­denly, the win­dow of op­por­tun­ity opened, and every­one star­ted to travel.

And so Si­mon Gindikin told me that I should start by vis­it­ing one of the satel­lite So­cial­ist­ic coun­tries. And so I did — went to a con­fer­ence in Czechoslov­akia, a beau­ti­ful place it was. And after that, in 1989, I could vis­it the USA for a month. I vis­ited sev­er­al places, but my host was at the Uni­versity of Mary­land. It was Gindikin, who vis­ited there a month earli­er, who ar­ranged it for me to be in­vited.

In­ter­est­ingly, the amount of pa­per­work to go abroad didn’t change at all com­pared with the stable So­viet time. One still needed to sub­mit eight cop­ies of an of­fi­cial re­com­mend­a­tion. But pre­vi­ously, there was a stage of this pro­cess where the cases were ter­min­ated and closed. Now this stage was elim­in­ated, and the cases went through without obstacles.

When I was at Mary­land, Yasha Eli­ash­berg called. He said that I could vis­it Stan­ford to give a talk, all ex­penses covered. But I already had booked a re­turn flight to Mo­scow in a couple of days.

Then he asked: “If I in­vite you for a year-long vis­it, will you come?” And I replied: “Maybe yes”. But after I told my wife, Ira, the “maybe” part was elim­in­ated.

And so I was in­vited for a year, and I did not have any plans to stay for more than a year. But the situ­ation in Rus­sia had changed from bad to worse, and the tiny amounts of money that we were able to send to our re­l­at­ives there were akin to moun­tains of gold to them.

And so they asked wheth­er it would be pos­sible to stay in the US for an­oth­er year. It was not really an op­tion to stay for an­oth­er year, the real op­tion was to stay for good. And so we did!


[1] A. M. Vino­gradov, B. De­lone, and D. Fuks: “Ra­tion­al ap­prox­im­a­tions to ir­ra­tion­al num­bers with bounded par­tial quo­tients,” Dokl. Akad. Nauk SSSR (N.S.) 118 : 5 (1958), pp. 862–​865. MR 101227 Zbl 0080.​26502 article

[2] I. M. Gel’fand and D. B. Fuks: “Co­homo­lo­gies of the Lie al­gebra of vec­tor fields on the circle,” Funk­cion­al. Anal. i Priložen. 2 : 4 (1968), pp. 92–​93. An Eng­lish trans­la­tion was pub­lished in Funct. Anal. Ap­pl. 2:4 (1968). MR 245035 article

[3] D. B. Fuks: “Quil­len­iz­a­tion and bor­d­isms,” Funk­cion­al. Anal. i Priložen. 8 : 1 (1974), pp. 36–​42. An Eng­lish trans­la­tion was pub­lished in Funct. Anal. Ap­pl. 8:1 (1974). MR 343301 article

[4]B. L. Feĭ­gin and D. B. Fuks: “In­vari­ant skew-sym­met­ric dif­fer­en­tial op­er­at­ors on the line and Verma mod­ules over the Vi­ra­s­oro al­gebra,” Funct. Anal. Ap­pl. 16 : 2 (1982), pp. 114–​126. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Funkts. Anal. Prilozh. 16:2 (1982). MR 659165 Zbl 0505.​58031 article

[5] D. B. Fuks and V. A. Rokh­lin: Be­gin­ner’s course in to­po­logy: Geo­met­ric chapters. Uni­versitext. Spring­er (Ber­lin), 1984. Eng­lish trans­la­tion of 1977 Rus­si­an ori­gin­al. MR 759162 Zbl 0562.​54003 book

[6] D. B. Fuks: Co­homo­logy of in­fin­ite-di­men­sion­al Lie al­geb­ras. Con­tem­por­ary So­viet math­em­at­ics. Con­sult­ants Bur­eau (New York), 1986. Eng­lish trans­la­tion of 1984 Rus­si­an ori­gin­al. MR 874337 Zbl 0667.​17005 book

[7] F. G. Ma­likov, B. L. Fei­gin, and D. B. Fuks: “Sin­gu­lar vec­tors in Verma mod­ules over Kac–Moody al­geb­ras,” Funkt­sion­al. Anal. i Prilozhen. 20 : 2 (1986), pp. 25–​37. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Funct. Anal. Ap­pl. 20:2 (1986). MR 847136 article

[8] D. Fuchs: “On po­et­ic feet,” Kvant 2 (1988), pp. 17–​24. In Rus­si­an. article

[9] D. Fuchs: “Chekan­ov–Eli­ash­berg in­vari­ant of Le­gendri­an knots: Ex­ist­ence of aug­ment­a­tions,” J. Geom. Phys. 47 : 1 (July 2003), pp. 43–​65. MR 1985483 Zbl 1028.​57005 article

[10] A. Fo­men­ko and D. Fuchs: Ho­mo­top­ic­al to­po­logy, 2nd trans­lated edition. Gradu­ate Texts in Math­em­at­ics 273. Spring­er (Cham), 2016. First two chapters writ­ten in col­lab­or­a­tion with Vic­tor Gut­en­mach­er. Second, ex­pan­ded, edi­tion of the 1986 Eng­lish trans­la­tion of the 1968 Rus­si­an ori­gin­al. MR 3497000 Zbl 1346.​55001 book