#### 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

**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

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.

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 was, of course we did, but not in
1961.
__\( K \)__-functor

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.

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

**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.

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.

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

**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!