Celebratio Mathematica — Fuchs — Interview
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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.

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.

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.

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.

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.