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Celebratio Mathematica

Yakov M. Eliashberg

Interview with Yakov Eliashberg

by Allyn Jackson

The growth of sym­plect­ic and con­tact to­po­logy stands as one of the most sig­ni­fic­ant de­vel­op­ments in math­em­at­ics in the past few dec­ades. As one of the founders of the sub­ject, Yakov Eli­ash­berg has made pro­found con­tri­bu­tions that re­veal its beauty and cent­ral­ity and have stim­u­lated much sub­sequent re­search. His in­flu­ence has been amp­li­fied by his open­ness and en­thu­si­asm, to­geth­er with a well honed sense of where fer­tile ques­tions lie. He has more than 30 coau­thors and has ad­vised more than 40 doc­tor­al stu­dents.

Born in 1946 in Len­in­grad (now Saint Peters­burg) in­to a Jew­ish fam­ily, Eli­ash­berg earned his doc­tor­ate at Len­in­grad Uni­versity un­der the dir­ec­tion of Vladi­mir Rokh­lin in 1972. At that time the pen­du­lum of So­viet an­ti­semit­ism was on the up­swing, nar­row­ing Eli­ash­berg’s job op­tions to one in­sti­tu­tion, Syk­tyvkar State Uni­versity, about 1000 miles north­east of Len­in­grad, in the re­mote Ko­mi Re­pub­lic. After eight years there, he ap­plied to emig­rate from the So­viet Uni­on and was re­fused. His op­tions nar­rowed fur­ther, and he spent eight years as a com­puter pro­gram­mer in Len­in­grad.

He and his fam­ily fi­nally re­ceived per­mis­sion to emig­rate in late 1987. He be­came a pro­fess­or at Stan­ford Uni­versity in 1989 and in 2004 was ap­poin­ted to his cur­rent po­s­i­tion as the Her­ald L. and Car­oline L. Ritch Pro­fess­or of Math­em­at­ics.

Eli­ash­berg’s hon­ors in­clude the Young Math­em­atician Prize of the Len­in­grad Math­em­at­ic­al So­ci­ety (1973), the Os­wald Veblen Prize of the Amer­ic­an Math­em­at­ic­al So­ci­ety (2001), the Heinz Hopf Prize (2013), the Cra­foord Prize of the Roy­al Swedish Academy of Sci­ences (2016), the Wolf Prize (2020), and the BBVA Found­a­tion Fron­ti­ers of Know­ledge Award in Ba­sic Sci­ences (2023). He was elec­ted to the U.S. Na­tion­al Academy of Sci­ences in 2002 and to the Amer­ic­an Academy of Arts and Sci­ences in 2012.

What fol­lows is an ed­ited ver­sion of an in­ter­view with Eli­ash­berg that took place over three ses­sions in late 2023 and early 2024.

An after-the-war child

Yakov Eliashberg’s mother, Amalya Yakovlevna Eliashberg (1908–1992)…

You were born in 1946, the year after World War II ended. Who were your par­ents, and what was their fate dur­ing the war?

As you said, I am an after-the-war child. I had two broth­ers. Both of them re­cently died; they were much older than me. One was 16 and a half years older, the oth­er 13 and a half years older. My moth­er was trained as a pi­an­ist in Len­in­grad, in the con­ser­vat­ory. She wanted to have a ca­reer as a pi­an­ist, but she got mar­ried to my fath­er, who was a chem­ic­al en­gin­eer. He worked in the pulp and pa­per in­dustry. It was a really big en­ter­prise. He was one of the main en­gin­eers, a deputy dir­ect­or re­spons­ible for the tech­no­lo­gic­al pro­cess. This was of course not in Len­in­grad but in many faraway places. So my moth­er had to fol­low him. At first she was car­ry­ing the pi­ano with her, but that be­came quite dif­fi­cult! Then my broth­ers were born, and then the war star­ted. My fath­er was for quite a while in oc­cu­pied Len­in­grad dur­ing the war, dur­ing the block­ade. Then he was sent to be the dir­ect­or of a pulp and pa­per plant in the Ur­al moun­tains area.

And your moth­er went with him?

…and his father, Matvei Gerasimovich Eliashberg (1905–1968).

My moth­er went with him. Then they re­turned to Len­in­grad, and I was born. Of course I don’t re­mem­ber much about my first years. My fath­er had a sis­ter who lived with my grand­moth­er. My aunt also worked in in­dustry, ac­tu­ally far north, in Kare­lia, and I was sent there every sum­mer.

Was that nice to go there in the sum­mer?

Oh, it was al­ways nice. I was five, six years old. I don’t re­mem­ber much, only gen­er­al im­pres­sions. There was a forest, and they taught me to col­lect mush­rooms. Since that time I love mush­room col­lect­ing very much.

How did your moth­er feel about giv­ing up her ca­reer?

What to do? She gave it up. She had a second ca­reer, after fin­ish­ing at a for­eign lan­guage in­sti­tute. She taught Eng­lish at a col­lege. But she didn’t teach me, so I didn’t know any Eng­lish! I went to school, just a loc­al school in the area.

Yakov Eliashberg is second from left. The others from left to right are his wife Ada, brother Gerasim (Sima), and Gerasim’s wife Lida.

By that time, your broth­ers were grown.

My broth­ers were grown. My old­est broth­er, Ger­asim, or Sima as we called him, be­came a very well known phys­i­cist. He moved to the place called Cher­no­go­lovka, near Mo­scow, where sev­er­al phys­ics in­sti­tutes were star­ted. One is the Land­au In­sti­tute for The­or­et­ic­al Phys­ics. He was among the first three people who worked at the Land­au In­sti­tute. He re­mained there and nev­er left. He died two years ago. He was a mem­ber of the Rus­si­an Academy, so he was well es­tab­lished, a well known phys­i­cist and one of the founders of the the­ory of high tem­per­at­ure su­per­con­duct­iv­ity. If you google “Eli­ash­berg”, you get the Eli­ash­berg Equa­tion, which is not mine but his!

My second broth­er, Vic­tor, was in my view an ex­tremely bril­liant per­son, su­per­i­or to me and prob­ably to my older broth­er as well. He was a very tal­en­ted per­son who could do any­thing. But in his life, it happened that he al­ways had to do something oth­er than what he wanted to do. It was very dif­fi­cult.

When my old­est broth­er was fin­ish­ing uni­versity and Vic­tor was go­ing to study there, it was an ex­tremely bad time. It was the late years of Stal­in’s life, and there was the so-called “Doc­tor’s Case.” This was a man­u­fac­tured cam­paign claim­ing that Jew­ish doc­tors were plot­ting to pois­on and kill Rus­sia’s most im­port­ant people. The cam­paign went on for sev­er­al years. Be­cause of this, my old­est broth­er Sima, al­though he gradu­ated from Len­in­grad Uni­versity and did something quite re­mark­able in his dis­ser­ta­tion, was not able to get any kind of work as a sci­ent­ist. He had to go to work in a chem­ic­al fact­ory.

Yakov Eliashberg (right) with his brother Victor in 1966.

My second broth­er Vic­tor was at that mo­ment go­ing to go to uni­versity. At that time in Rus­sia there was a law that if you gradu­ate from school with a medal, then you can auto­mat­ic­ally be ac­cep­ted at any uni­versity you want. Both of my broth­ers gradu­ated with medals. My old­est broth­er gradu­ated be­fore the time of the “Doc­tor’s Case”, so there was no prob­lem, he could go to uni­versity. But my second broth­er Vic­tor gradu­ated in that bad peri­od, so the uni­versity did not take him and he went in­stead to an elec­tric­al en­gin­eer­ing in­sti­tute in Len­in­grad. He be­came a really high-level en­gin­eer and worked on design­ing con­trol sys­tems. But he al­ways was in­ter­ested in oth­er things.

He ac­tu­ally — well, I will stop be­cause oth­er­wise I will speak a whole hour about his ideas! I was really ex­cited about what he tried to do. Un­for­tu­nately he was not able to do it be­cause of his dif­fi­cult life, but I am still dream­ing that maybe at some time I can con­tin­ue what he was do­ing.

What area was this in? Math?

No, his main in­terest was in the work­ing of the hu­man brain. He had a fant­ast­ic, ab­so­lutely un­con­ven­tion­al the­ory about how it works.

Violin or mathematics?

Your broth­ers were so much older that in some sense you grew up as an only child.

In some sense yes, I was an only child. My moth­er had this dream of real­iz­ing her mu­sic­al as­pir­a­tion in her chil­dren. So she tried with my second broth­er to teach him, but then the war star­ted and it was not pos­sible. I went to a mu­sic­al school, learn­ing vi­ol­in. And I was pretty ser­i­ously play­ing vi­ol­in un­til I was maybe 14 years old. If you would have asked me be­fore that time what I plan to do in my life, I would say I would like to play vi­ol­in. I was not a fant­ast­ic vi­ol­in­ist, but I think I played quite well. So I had this in­spir­a­tion.

Did you think at that time that maybe you would be in an or­ches­tra?

Of course when you study vi­ol­in you think that you will be a so­loist! Who would dream to be in an or­ches­tra?

True! But you thought you might make a ca­reer of the vi­ol­in.

Of course, at this time, when I was 12, 13, I didn’t think in terms of a ca­reer.

How were your par­ents when you were in school? Were they strict? Did they make you do your home­work? Or did they leave it to you be­cause you liked school and you did well?

No. I did well. Nobody ever in­terfered in my school life!

Did you like math­em­at­ics from the start?

My old­est broth­er Sima moved out of the house and went to Cher­no­go­lovka, so I saw him only a few times a year. But Vic­tor, or Vitya as we called him, was liv­ing at home, and he was my main in­spir­a­tion. He got his PhD in con­trol the­ory, but then he de­cided he wanted to do phys­ics. And he was pretty suc­cess­ful in this. He learned phys­ics es­sen­tially by him­self. I was at the time 13 years old, and as he was learn­ing phys­ics, I was try­ing to do something along with him and was learn­ing math.

Vitya also went to Cher­no­go­lovka, to a phys­ics in­sti­tute there — not the Land­au In­sti­tute, a dif­fer­ent one, in sol­id-state phys­ics. Un­for­tu­nately for him, he was a fant­ast­ic en­gin­eer and was ex­tremely good at any kind of in­stru­ment­a­tion ne­ces­sary for the phys­ics ex­per­i­ments. People no­ticed this, and then that’s what he had to do. And that’s ex­actly what he didn’t want, be­cause he really wanted to do phys­ics re­search. So he got dis­il­lu­sioned. At that time he star­ted think­ing about the brain. Then he got mar­ried and re­turned to Len­in­grad and worked there.

Was that what got you in­to math, the con­tact with your broth­er about phys­ics?

He had some in­flu­ence — but no. At that time in the So­viet Uni­on there was a sys­tem of math­em­at­ics olympi­ads. This was really broad, so that, for in­stance, all schools once a year were sup­posed to do some kind of olympi­ad. Each area of Len­in­grad had to have a loc­al olympi­ad, and the win­ners were sup­posed to go to the city olympi­ad, and so on. It was a well struc­tured sys­tem. I think it star­ted from 6th grade. I went through this sys­tem of olympi­ads, and I did pretty well. I got second prize in my first city olympi­ad, without any real pre­par­a­tion.

There was also a sys­tem of math­em­at­ics circles, like math­em­at­ics clubs. A math­em­atician named Nina Me­fant­ievna Mitro­fan­ova1 looked at the list of olympi­ad win­ners and sent me a per­son­al in­vit­a­tion to come to her math circle. It was at the Palace of Pi­on­eers, which was a com­munity cen­ter for chil­dren. She was run­ning two math circles, one for people two years older than me, and the one I was in. This was a com­pletely life-chan­ging ex­per­i­ence for me. She was a fant­ast­ic teach­er. She then got mar­ried, to a math­em­atician, Yuri Bur­ago.2 So I got to know him also.

What you did in the math circles was dif­fer­ent from the math­em­at­ics you had in school.

Math­em­at­ics in school was in some sense not in­ter­est­ing at all. It was kind of stand­ard math­em­at­ics. But the math circle was a com­pletely dif­fer­ent thing. I was there in 7th and 8th grades. Then for grades 9, 10, and 11 there was an ex­per­i­ment in the schools where every school spe­cial­ized in some par­tic­u­lar sub­ject. I was still play­ing ser­i­ous vi­ol­in, and here I had a split mo­ment. Either I need to go to a pro­fes­sion­al school for vi­ol­in, or go to a school spe­cial­iz­ing in the sci­ences and math­em­at­ics. Some of the same people were par­ti­cip­at­ing in the sci­ence and math­em­at­ics school, like Nina Mitro­fan­ova and Yuri Bur­ago. They were not teach­ing there, but they were very much in­volved in the or­gan­iz­a­tion. And I de­cided at this mo­ment to go in this math­em­at­ic­al dir­ec­tion.

What made you choose math over mu­sic?

It was a tough choice ac­tu­ally. But also, it was a some­what ro­mantic in­terest, be­cause there was a girl I liked who was in the school!

You didn’t think math­em­at­ics was more prac­tic­al than mu­sic?

No, no. Prac­tic­al­ity was nev­er a con­sid­er­a­tion. Ac­tu­ally, by this time, maybe there was a little bit of prac­tic­al­ity. I had heard many stor­ies about the mu­sic world. In or­der to make a ca­reer in mu­sic, it’s not suf­fi­cient to be a bril­liant play­er. There is a lot of luck, and a lot of oth­er things. I heard many stor­ies about how some people suc­ceeded, and oth­ers didn’t, and what they did to suc­ceed. And some­how it wasn’t so pleas­ant.

Was that be­cause of the strange­ness of the So­viet sys­tem?

I think this is prob­ably uni­ver­sal.

Uni­ver­sal in the per­form­ing arts?

In the per­form­ing arts, yes. Maybe in the So­viet Uni­on it was even more so.

In the So­viet Uni­on, there was a de­clared egal­it­ari­an prin­ciple, but in real­ity it was just the op­pos­ite, be­cause there were people with priv­ileges, and people without.

Of course, one thing that was uni­ver­sally un­der­stood was that you do not go in­to any kind of hu­man­it­ies stuff, any­thing so­cial or lit­er­ary, or philo­sophy — any­thing that has the taste of polit­ics. Everything was politi­cized, but math­em­at­ics and phys­ics were the least politi­cized. In this sense, mu­sic was prob­ably far from ideal, but, again, I didn’t think in these terms.

Did you have good math­em­at­ics teach­ers in school?

Yeah, there were some good teach­ers there, but I didn’t in­ter­act with them. I didn’t have any­thing in this school that was in­ter­est­ing math­em­at­ic­ally. I es­sen­tially es­caped from the school. It was a three-year school, but I didn’t want to study for three years. I wanted to fin­ish in two years. There was a group of us who star­ted to go to Len­in­grad Uni­versity, to the math­em­at­ics de­part­ment, un­of­fi­cially, to listen to some lec­tures. We wanted to leave school as early as we could! So we asked to take the ex­am to leave school, but we were not al­lowed. Then I just said, Okay, then I’ll leave. In the last half-year, I en­rolled in a school where you have to come once or twice a week and just pass some tests. It was mainly for older people who wanted to get an edu­ca­tion. They im­me­di­ately un­der­stood that I knew quite a lot, so I was teach­ing math­em­at­ics to oth­er stu­dents. I fin­ished that school and then went to the math de­part­ment at Len­in­grad Uni­versity.

Becoming a university student

Eliashberg in 1969.

Did you have to pass an ex­am to be able to enter the uni­versity?

Yes. I passed the ex­am. The his­tory of an­ti­semit­ism in the So­viet Uni­on went in waves. There was al­ways un­der­ly­ing an­ti­semit­ism, but some­times it rose to the level of the state. I told you about the time when my broth­ers were study­ing. The year when I went to uni­versity, it was no prob­lem for me to get in. The cli­mate changed very rap­idly, over the fol­low­ing three or four years, but at first it was pretty nor­mal. I got in­to the uni­versity, and I very much liked my ex­per­i­ence there and met a lot of fant­ast­ic people. My ad­viser was Rokh­lin,3 who was a very re­mark­able math­em­atician. Maybe you know his story?

A little bit. He had a dif­fi­cult time in the war, is that right?

Yes. He star­ted in uni­versity in the 1930s. After the war star­ted, he went as a vo­lun­teer to the front. He was cap­tured by the Ger­mans. Be­cause he was Jew­ish, that was not good. But he was born in Baku, in Azerbaijan, and he man­aged to con­vince them that he was Azerbaijani, be­cause he could speak flu­ent Azerbaijani. So he was not sent to a con­cen­tra­tion camp but to some kind of mil­it­ary pris­on­ers camp. He man­aged to es­cape, and then he was in Europe in dif­fer­ent places, join­ing some res­ist­ance groups. To­wards the end of the war, he man­aged to get back to the So­viet Uni­on. And of course as soon as he got back, he was im­me­di­ately im­prisoned and sent to a camp, be­cause every­body who had been cap­tured by the Ger­mans was not re­li­able and had to be sent to a camp! Be­fore the war he had already been quite bril­liant math­em­at­ic­ally. He had worked with many math­em­aticians, so he was known. So some of them man­aged to get him out of the camp, and he got back to Mo­scow. But of course he couldn’t find any job. Pontry­agin,4 who was one of the people who had helped him get out of the camp, took him on as his help­er. Pontry­agin was blind, so he had a spe­cial po­s­i­tion of help­er.

So Rokh­lin was work­ing at the Steklov In­sti­tute, and he got his second doc­tor­al de­gree, sim­il­ar to the Ha­bil­it­a­tion in Ger­many. Then he was fired be­cause his qual­i­fic­a­tions did not cor­res­pond to his po­s­i­tion as help­er! He couldn’t find a job in Mo­scow, so he worked at places that were an hour or two away by train. But still he ran a sem­in­ar on er­god­ic the­ory in Mo­scow, which was ex­tremely im­port­ant. People like Sinai,5 Arnold,6 and oth­ers went to this sem­in­ar, and later they said it had been really im­port­ant for them.

At some point, the math­em­atician Al­ex­an­drov7 be­came rect­or of Len­in­grad Uni­versity. He per­son­ally in­vited Rokh­lin to Len­in­grad. So Rokh­lin came maybe three years be­fore I entered. Im­me­di­ately he got fant­ast­ic stu­dents. For in­stance among his first stu­dents was Gro­mov.8

That’s im­press­ive that Rokh­lin went through so much in the war but then had the en­ergy to keep his math­em­at­ics go­ing, even in a dif­fi­cult work situ­ation.

Yes — but maybe if you are re­laxed you are not so mo­tiv­ated. It de­pends on the per­son, right? For some people, it’s like the dif­fi­culty helps them to do bet­ter. But of course, you are per­fectly right, it’s re­mark­able.

So Rokh­lin had a sem­in­ar in Len­in­grad, where I star­ted to par­ti­cip­ate, and a lot of fant­ast­ic people were there. I met Gro­mov there. Gro­mov is three years older than I am, but four years ahead in study­ing be­cause he star­ted early. So I didn’t talk to him at first, but he was already quite fam­ous, so I had heard of him.

My first ser­i­ous in­ter­ac­tion with Gro­mov was the fol­low­ing. We had five years in uni­versity, and after the fourth year, Rokh­lin gave me a prob­lem. I was think­ing about this prob­lem in the sum­mer, and I solved it. Then I star­ted to think of something that was in some sense an op­pos­ite ques­tion, a re­verse ques­tion. I un­der­stood how to do that too, and I was very ex­cited. In Septem­ber, I came to Rokh­lin and star­ted telling him what I had done. And he said, “Well, it seems very in­ter­est­ing. Gro­mov just told me something very sim­il­ar. You should go talk to him.”

That was my first ser­i­ous talk with Gro­mov. From that mo­ment, all my math­em­at­ic­al guid­ance was mostly from Gro­mov. I talked with Rokh­lin of course, but with Gro­mov much more.

What was the prob­lem that Rokh­lin gave you?

Rokh­lin and Gro­mov were at that time writ­ing a sur­vey on iso­met­ric em­bed­dings for the Rus­si­an journ­al called Us­pekhi Matem­aticheskikh Nauk. This journ­al was trans­lated as Rus­si­an Math­em­at­ic­al Sur­veys.

In the the­ory of iso­met­ric em­bed­dings, when you want to con­struct such an em­bed­ding, you usu­ally start with a so-called free map. If you have a curve, there is a tan­gent line, but also there is a plane that ap­prox­im­ates the curve in the best way. This is the plane spanned by the ve­lo­city vec­tor and the ac­cel­er­a­tion vec­tor — the first and second de­riv­at­ives. It’s called the os­cu­lat­ing plane. For a man­i­fold of high­er di­men­sion, for in­stance for a sur­face in high­er-di­men­sion­al space, you can look at the os­cu­lat­ing space, which is the space spanned by all first and second de­riv­at­ive vec­tors. For a 2-di­men­sion­al sur­face, the os­cu­lat­ing space in prin­ciple should have the max­im­al di­men­sion of 5, be­cause there are two vec­tors of the first de­riv­at­ive, and three vec­tors of the second de­riv­at­ive.

A map, or an em­bed­ding, is called free if at every point the os­cu­lat­ing space has max­im­al pos­sible di­men­sion, which is 5 in the case of a sur­face. In the case of a curve with an in­flec­tion point, where the first and second de­riv­at­ives align, they span a line in­stead of a plane. That’s a bad point — the em­bed­ding is not free. The pic­ture is sim­il­ar in high­er di­men­sions.

One ques­tion is, What is the min­im­al di­men­sion of, say, Eu­c­lidean space where you can map a man­i­fold freely? When Gro­mov and Rokh­lin men­tioned this in their sur­vey, they wanted some kind of ob­struc­tion, some kind of neg­at­ive res­ult — a way to say for which man­i­folds it’s not pos­sible to em­bed them in such-and-such di­men­sion.

So I thought about this, and it was ac­tu­ally not a hard ques­tion. I un­der­stood it pretty fast and com­puted some ex­amples. But then I star­ted to think about the op­pos­ite ques­tion. There are al­geb­ra­ic ob­struc­tions, in terms of the to­po­lo­gic­al char­ac­ter­ist­ics of the man­i­fold. Sup­pose those ob­struc­tions van­ish. Ques­tion: Can you find such free em­bed­dings?

At that time, this was a fash­ion­able ques­tion, be­cause there was the Smale–Hirsch im­mer­sion the­ory, and the an­swer they give for ex­ist­ence of im­mer­sions was sim­il­ar. So this was a nat­ur­al gen­er­al­iz­a­tion, from my point of view. And I real­ized how to do this. In mod­ern terms, I proved the h-prin­ciple in this case! Though it was not called the h-prin­ciple at the time.

This was 1968, when Gro­mov wrote his fam­ous pa­per9 from which the the­ory of the h-prin­ciple es­sen­tially de­veloped. He was think­ing about a sim­il­ar ques­tion and had found a way of solv­ing that ques­tion that was very close to my idea. It was es­sen­tially the same pro­ced­ure. He told me, “Okay, great, let’s think about all ques­tions we can solve by this meth­od.” That es­sen­tially star­ted our col­lab­or­a­tion. We wrote maybe four pa­pers at that time to­geth­er. It was a pretty pro­duct­ive time.

Rigid and flexible mathematics

What makes Gro­mov such a power­ful math­em­atician?

It’s ex­tremely dif­fi­cult to say. Who knows what makes great people great? But what I see as quite re­mark­able is the way Gro­mov thinks and works. He al­ways is try­ing to un­cov­er the deep­er mech­an­ism be­hind why something is true. I’ll give you an ex­ample.

Be­fore I star­ted talk­ing ser­i­ously to Gro­mov, Rokh­lin or­gan­ized a sem­in­ar for us young­sters who had just star­ted. He asked a gradu­ate stu­dent to run the sem­in­ar, which really was a fant­ast­ic ex­per­i­ence for me. Not long be­fore this, the Smale–Hirsch im­mer­sion the­ory work had ap­peared. I was asked to give a talk about it in the sem­in­ar. So I read all the pa­pers, I thought I un­der­stood them, and I gave a talk. But sev­er­al years later, when I star­ted to in­ter­act with Misha Gro­mov, we talked about this, and then I real­ized I really didn’t un­der­stand any­thing!

Figure 1. Any function that is increasing near the endpoints of an interval extends to the interval with exactly two critical points.

There is a very simple geo­met­ric mech­an­ism show­ing why im­mer­sion the­ory works. Sup­pose you have a func­tion in­creas­ing near the en­d­points of an in­ter­val, and you ask, is it pos­sible to ex­tend the func­tion over the whole in­ter­val without a crit­ic­al point? If the value of the func­tion at the left en­d­point is smal­ler than at the right en­d­point, then the an­swer is yes. Oth­er­wise, it is im­possible. But as soon as you al­low two crit­ic­al points, the pic­ture be­comes ex­tremely flex­ible (see Fig­ure 1).

You can al­ways ex­tend in­side the in­ter­val with pre­cisely two crit­ic­al points; you don’t need to cre­ate more. This pic­ture is the key thing that makes the whole Smale–Hirsch im­mer­sion the­ory work. Everything is based on this. That’s what Gro­mov ob­served. You could say that his PhD dis­ser­ta­tion is an ex­plan­a­tion of this simple pic­ture with the two crit­ic­al points. He trans­formed it in­to a fant­ast­ic­ally power­ful tool.

Here is an­oth­er ex­ample. Gro­mov was also think­ing about the work of John Nash on \( C^1 \) iso­met­ric em­bed­dings. An iso­met­ric em­bed­ding is a way to bend sur­faces without dis­tort­ing dis­tances. For ex­ample, if you take a sheet of pa­per and bend it, that’s like an iso­met­ric em­bed­ding of a plane in the space. Sup­pose now you have a sphere of ra­di­us 1. Without dis­turb­ing dis­tances, can you squeeze it in­to a very small ball?

There was a dis­cov­ery of Nash that this is in­deed pos­sible, but only if the crump­ling is \( C^1 \) smooth — the second de­riv­at­ive does not ex­ist. If you try to do this in such a way that the second de­riv­at­ive ex­ists, so that, be­sides length you can also meas­ure curvature, then it is easy to prove that this sphere can­not be crumpled in this way. It’s really ri­gid. But if you don’t care about curvature, so that the em­bed­ding is only \( C^1 \) smooth, then you can do that. This is the \( C^1 \) iso­met­ric em­bed­ding the­or­em, which was proved by Nash and then im­proved by Kuiper. Many people knew about this, but Gro­mov really un­der­stood what it means. He trans­formed it in­to a meth­od, now called con­vex in­teg­ra­tion, which really un­winds what Nash was do­ing.

I think what makes Gro­mov dif­fer­ent from most of us is that for him, un­der­stand­ing means something com­pletely dif­fer­ent from what it means to oth­er people. For him it means un­der­stand­ing some ba­sic mech­an­ism be­hind a phe­nomen­on. Sud­denly it turns out that this mech­an­ism is much more power­ful than its use in that par­tic­u­lar phe­nomen­on. Then a whole new area might open up.

So Gro­mov is able to see the simple, ba­sic things go­ing on in the back­ground.

Maybe they are not so simple, but you see, there should not be such a thing as a dif­fi­cult math­em­at­ic­al the­or­em, right? If you have a dif­fi­cult the­or­em, that means that the proof is not prop­erly un­der­stood. It’s not struc­tured cor­rectly. A proof could be long, but we need to un­der­stand well what is go­ing on at each step.

Eliashberg (right) with Mikhael Gromov in Masada, Israel, during the conference “Geometries in Interaction” in honor of Gromov’s 50th birthday in 1994.

You and Gro­mov de­veloped the philo­sophy of “soft” and “hard” math­em­at­ics. Can you tell me about this?

I prefer the terms “flex­ible” and “ri­gid”, be­cause this is not about dif­fi­culty of prob­lems. My view­point on all this changed a lot over the years. Now I see it as noth­ing spe­cial.

What are math­em­aticians do­ing? They are in­vent­ing some worlds, and then they are ex­plor­ing these worlds. For ex­ample, say you want to un­der­stand geo­metry in four di­men­sions. In three di­men­sions, we think we un­der­stand more be­cause we have eyes, we have some ex­per­i­ence with our hands. But not in four di­men­sions. There we have to in­vent tools for work­ing and visu­al­iz­ing. Like the idea of Morse the­ory. You take a func­tion on a 4-di­men­sion­al ob­ject and con­sider its level sets, which are 3-di­men­sion­al, and then you ana­lyze how they bi­furc­ate.

Every math­em­atician lives in some math­em­at­ic­al world, which he tries to ex­plore. You need to un­der­stand what is pos­sible in this world and what is pro­hib­ited. So for in­stance, you have a sur­face and you want to con­struct a vec­tor field on it, and you don’t want this vec­tor field to have zer­os. If you take a 2-di­men­sion­al sphere, you real­ize that on a sphere, this is not pos­sible. If you are a very stub­born per­son, you would try and try and try, but maybe at some mo­ment you should stop and think, and then prove that it is not pos­sible.

Or sup­pose you are try­ing to con­struct a gradi­ent vec­tor field on a tor­us, without zer­os. This is also not pos­sible, while a non­gradi­ent vec­tor field without zer­os does ex­ist! It’s not pos­sible be­cause every func­tion on a closed man­i­fold must have a max­im­um and min­im­um. You real­ize that a max­im­um and min­im­um are ne­ces­sary, but ask, Can we do it so that there are no oth­er crit­ic­al points? In to­po­logy there is something called the Morse in­equal­it­ies, which say no, this is not al­ways pos­sible. It de­pends on the to­po­logy. If this were pos­sible, then our man­i­fold would be homeo­morph­ic to a sphere.

There are al­ways these at­tempts to con­struct things in math­em­at­ics, and then something pro­hib­its cer­tain con­struc­tions. So in some areas, it is ex­tremely dif­fi­cult to con­struct any­thing! Al­most everything crashes! In to­po­logy, it’s a more flex­ible world, there are a lot of in­ter­est­ing con­struc­tions you can do. At the same time, there is this ri­gid­ity, called to­po­lo­gic­al in­vari­ants, which pro­hib­it us from per­form­ing cer­tain tasks. What I call flex­ible math­em­at­ics and ri­gid math­em­at­ics are pre­cisely these two as­pects. In one we are try­ing to take our con­struc­tions as far as pos­sible. Then in an­oth­er we are look­ing for the con­straints on our con­struc­tions and try­ing to un­der­stand what is im­possible.

So this is ri­gid math­em­at­ics and flex­ible math­em­at­ics. An ideal math­em­at­ic­al field is when these two as­pects come in­to as close prox­im­ity as pos­sible. That would mean that we know how to live there. We know what we can do and what we should not try to do.

Strong personalities are dangerous

You got your PhD in 1972, with Rokh­lin as your ad­viser. Yuri Matiy­a­sevich10 was also at Len­in­grad Uni­versity at the time. Did you know him then?

I told you about the math circle of Nina Mitro­fan­ova. This was an ex­tremely strong group of young people, and Matiy­a­sevich was among them. So yes, I know him. He is my friend, we grew up to­geth­er.

He solved Hil­bert’s Tenth Prob­lem in 1970. Was this big news in the math­em­at­ics de­part­ment in Len­in­grad?

Yeah, it was of course really big news.

Sev­er­al oth­er people in Len­in­grad were ex­tremely good math­em­aticians. An im­port­ant fig­ure was the older Fad­deev, not Lud­wig Fad­deev,11 but his fath­er, Dmitry Kon­stantinovich Fad­deev.12 He was an al­geb­ra­ist and taught one of the courses I took. He was a very in­ter­est­ing per­son.

The math­em­at­ics de­part­ment was di­vided in­to “chairs” ac­cord­ing to math­em­at­ic­al areas. So there was a chair in al­gebra, in to­po­logy, and also in math­em­at­ic­al phys­ics. In math­em­at­ic­al phys­ics there were two re­mark­able wo­men math­em­aticians, Olga Ladyzhenskaya13 and Nina Ur­alt­seva.14 Both of them were — and Ur­alt­seva still is — won­der­ful people and great math­em­aticians. An­oth­er was Mikhail So­lomyak,15 who emig­rated later and was work­ing at the Weiz­mann In­sti­tute.

On the ana­lys­is chair were sev­er­al math­em­aticians who were very im­port­ant for me. One of them was Vikt­or Pet­ro­vich Hav­in.16 He was in com­plex ana­lys­is, and a lot of high pro­file people went through his school — not ne­ces­sar­ily his stu­dents, but stu­dents of his stu­dents, in­clud­ing some Fields Medal­ists, like Stas Smirnov.17 An­oth­er very re­mark­able per­son was Anato­ly Ver­shik,18 who re­cently passed away.

But at the time I was there, the uni­versity ad­min­is­tra­tion didn’t like bright people. It was nev­er stated, but my im­pres­sion was they pre­ferred to have more me­diocre people. The strong people are al­ways dan­ger­ous be­cause they have their own opin­ions! It is much more dif­fi­cult to ma­nip­u­late them. So Gro­mov nev­er was able to get any reas­on­able po­s­i­tion in Len­in­grad Uni­versity. He was so ob­vi­ously great that, when he gradu­ated, they had to give him a uni­versity po­s­i­tion. But it was ri­dicu­lous, the low­est paid po­s­i­tion. If you went to a metro sta­tion, and you saw an ad­vert­ise­ment for a clean­er in the metro, then the salary was much big­ger than what was paid to Gro­mov! Really ri­dicu­lous.

Was that also be­cause he is Jew­ish?

No — okay, he is half Jew­ish. I doubt it, but it may have played a role. But I would say the more im­port­ant reas­on was that he was too good to have — too dan­ger­ous to have such a great per­son!

Also, Rokh­lin had to re­tire quite early from the uni­versity. Of­fi­cially there was a five-year con­tract that was auto­mat­ic­ally re­newed, and it was un­der­stood that it would be re­newed in­def­in­itely. There was a pen­sion age, which for men at that time was 60 years. Pro­fess­ors at the uni­versity nor­mally would stay un­til they are dead or close it! But not Rokh­lin. They pushed him out as soon as he was 60.

Was there a lot of com­pet­i­tion between Mo­scow and Len­in­grad as math­em­at­ic­al cen­ters?

There was no com­pet­i­tion, be­cause of course Mo­scow won 100 per­cent. There were ab­so­lutely in­com­par­able re­sources there. In­ter­ac­tion with the West was mostly in Mo­scow. In later years, some people were com­ing to Len­in­grad, but that was ex­cep­tion­al. There was no com­par­is­on between the level of in­tens­ity of math­em­at­ic­al life in Mo­scow and Len­in­grad. Of course, Len­in­grad was still a good place com­pared to most.

A lot of people in the West felt there was a spe­cial spir­it in the math­em­at­ic­al life in the USSR. Do you feel that’s true?

There was an ab­so­lutely re­mark­able num­ber of fant­ast­ic math­em­aticians, mainly in Mo­scow. Of course Gel­fand19 was a mag­net, and so was his sem­in­ar, which he cre­ated. No ques­tion about this. There was a group of really amaz­ing math­em­aticians, like Novikov,20 Arnold, Kir­illov,21 Sinai, Fuchs,22 Man­in23  — a fant­ast­ic group of math­em­aticians, more or less of the same age. And as I said, Rokh­lin played some role in the up­bring­ing of this group, be­cause most of them went through his sem­in­ar. There was an ex­tremely good math­em­atician who was not so great a per­son, Sha­far­ev­ich.24 Then of course Kolmogorov25 and the older gen­er­a­tion played an enorm­ous role. So there was no ques­tion, there was a bril­liant group of math­em­aticians in Mo­scow Uni­versity. Anosov26 should be in there too — of course, this list should go on and on. It is a far from an ex­haust­ive list.

What I didn’t like was that in the So­viet Uni­on, on every level, they were re­cre­at­ing the struc­ture of the whole So­viet state, in mini­ature. There were lead­ers who had “schools”, and you needed to be­long to a school. Some of these schools were great, es­pe­cially in Mo­scow and Len­in­grad. But some were quite me­diocre and self-re­pro­du­cing. Of course, this ex­ists in the West as well. One of the great schools was around Gel­fand and his fam­ous sem­in­ar. The Gel­fand sem­in­ar had enorm­ous in­flu­ence on the de­vel­op­ment of math­em­at­ics not only in the USSR but world­wide. But I also know a very good math­em­atician for whom par­ti­cip­a­tion in Gel­fand sem­in­ar was a hurt­ful ex­per­i­ence that he re­membered for a long time.

Eliashberg (left) with Vladimir Arnold in 1995.

I nev­er par­ti­cip­ated in the Gel­fand sem­in­ar in Mo­scow. Much later, in the mid-1990s, I was vis­it­ing the IHES [In­sti­tut des Hautes Et­udes Sci­en­ti­fiques] at Bures-sur-Yvette. The first per­son I met when I ar­rived there was Gel­fand. He was seem­ingly glad to see me, took me to his of­fice, and asked me to tell him what I was cur­rently do­ing. I began to talk, and he al­most im­me­di­ately fell asleep. I stopped and waited for him to wake up. “You nev­er learned how to prop­erly ex­plain things”, Gel­fand told me when he opened his eyes. He in­vited me to talk at his sem­in­ar at the IHES, which I did. I think the sem­in­ar went quite well, though maybe it was not so pleas­ant for Ofer Gab­ber, whom Gel­fand con­stantly called to the black­board to prove what I claimed to be true.

An­oth­er re­mark­able and very in­flu­en­tial fig­ure in So­viet math­em­at­ics was Arnold. In ad­di­tion to Rokh­lin and Gro­mov, I was very much in­flu­enced by Arnold. He was in Mo­scow, not Len­in­grad, but start­ing at some point I in­ter­ac­ted with him al­most every time I was in Mo­scow. I am not his stu­dent, but I am kind of on the peri­phery of his world.

The e-Machine: Modeling the human brain

You men­tioned your broth­er Vic­tor’s ideas about the hu­man brain. You ac­tu­ally worked with him on this.27 Can you tell me briefly about it?

It’s ex­tremely dif­fi­cult to de­scribe in ten minutes! But let me try. He and I shared the point of view that what the brain is do­ing is some kind of com­pu­ta­tion­al pro­cess, though of a dif­fer­ent kind from a con­ven­tion­al com­puter. Still, we are uni­ver­sal com­puters in a sense. And there is a simple the­or­em say­ing that you can­not be a uni­ver­sal com­puter if you don’t have what is called read-write memory. This means you have a memory stor­age, where you not only can store things, but also can re­use the same memory po­s­i­tion for new in­form­a­tion. Con­ven­tion­al com­puter ar­chi­tec­ture uses a sep­ar­ate pro­cessor, which can per­form some op­er­a­tions. All con­ven­tion­al com­pu­ta­tion­al pro­cesses look like this. We take something from memory, send it to the pro­cessor, per­form an op­er­a­tion, write back to the memory, etc. In or­der to do any­thing ser­i­ous in real time, we need to do these things very fast.

But nobody has ever ob­served any­thing like these kinds of pro­cesses in the brain. Ac­cord­ing to cur­rently ac­cep­ted mod­els of memory, in­form­a­tion in the brain is en­coded in con­nectiv­ity of syn­apses — the way the neur­ons at­tach to each oth­er. This syn­aptic memory can­not go very fast be­cause it’s based on chem­ic­al pro­cesses. And nobody has ever ob­served in the brain something like flows of in­form­a­tion go­ing from memory to pro­cessor and back, like in a com­puter.

But as I said, there is a math­em­at­ic­al the­or­em say­ing that if a sys­tem doesn’t have read-write memory, or something that can sim­u­late read-write memory, then it does not have the power of a uni­ver­sal com­puter. There­fore the ques­tion Vic­tor asked is, What does the brain have in­stead of read-write memory? We can play chess, we can per­form ex­tremely com­plic­ated al­gorithms. But whatever it is that car­ries out those al­gorithms can­not be the con­ven­tion­al kind of ar­chi­tec­ture. So what is it?

He came up with a pro­pos­al, which soun­ded to me ex­tremely prob­able, for how this can be done. Ac­cord­ing to him, com­pu­ta­tions are per­formed in the space of func­tions of the write-only, or “long-term,” memory. Roughly speak­ing, his pro­pos­al is the fol­low­ing; it’s not a mod­el of the brain but a meta­phor for an ele­ment­ary block of which our brain is built.

Sup­pose you have some kind of memory stor­age, like a tape, where you can re­cord in­form­a­tion. You have in­form­a­tion com­ing to your eye or ear or whichever sense. At dis­crete times there is some mul­ti­di­men­sion­al vec­tor of in­form­a­tion com­ing in. Let’s do a na­ive thing and sup­pose that you just re­cord in­com­ing in­form­a­tion suc­cess­ively, like on a tape re­cord­er, so in­form­a­tion that comes at time \( t \) you re­cord at po­s­i­tion \( t \), the first avail­able po­s­i­tion.

Sup­pose that at the same time the in­form­a­tion is be­ing re­cor­ded, there is a very simple mech­an­ism that com­pares how this vec­tor that comes at time \( t \) relates to what was re­cor­ded be­fore. It might com­pute a very simple func­tion, for in­stance, the scal­ar product. You have at every mo­ment this func­tion, so you can identi­fy the stored sig­nal most sim­il­ar to the one that just came in. This works like as­so­ci­at­ive memory, so that there is an ac­cess mech­an­ism by ana­logy, by as­so­ci­ation.

But this would be a very stu­pid ma­chine if it just re­peated something that was already prerecor­ded. That would be ab­so­lutely not in­ter­est­ing. So let’s sup­pose there ex­ists the fol­low­ing ad­di­tion­al mech­an­ism, which in neur­os­cience is called “re­sid­ual ex­cit­a­tion”. Sup­pose the func­tion is ac­tu­ally some kind of phys­ic­al field, for ex­ample a po­ten­tial. It was cre­ated at time \( t \). At time \( t + 1 \), it does not dis­ap­pear but in­stead starts to de­cay. But also at time \( t + 1 \), a new sig­nal comes. So this func­tion de­pends not only on what came at this mo­ment, but also on what came be­fore for a cer­tain peri­od of time, be­cause of the re­sid­ual ex­cit­a­tion. This would, for ex­ample, al­low you not only to re­cog­nize let­ters, in­di­vidu­al sig­nals, but to re­cog­nize tem­por­al se­quences, to re­cog­nize words. Vic­tor called this the “e-ma­chine”, where e stands for ex­cit­a­tion.

Both in­put and out­put are di­vided in­to two kinds of sig­nals, sens­ory and mo­tor. We have in­form­a­tion from the out­side world, from our senses, but we also have in­form­a­tion from in­side ourselves. For in­stance, you tell me something, and I move my hand. Whatever I did to move my hand, a cer­tain sig­nal was sent to my muscles in or­der to do this. That sig­nal is also be­ing re­cor­ded, and it is in­form­a­tion to which I have ac­cess. Fur­ther­more, one can con­sider hier­arch­ic­al sys­tems built of e-ma­chines, where the “sens­ory” in­put for e-ma­chines of a high­er level is com­ing from e-ma­chines on a lower level and their “mo­tor re­ac­tion” con­trol ma­chines on the lower level.

I am de­scrib­ing a com­puter ar­chi­tec­ture that is a uni­ver­sal ar­chi­tec­ture. So in prin­ciple whatever it’s pos­sible to do on a com­puter, it’s pos­sible to do in this ar­chi­tec­ture and vice versa. But cer­tain al­gorithms that are simple and nat­ur­al, or nat­ive, for this type of ar­chi­tec­ture re­quire enorm­ous work to be pro­grammed on reg­u­lar com­puters. They are not nat­ur­al there. This sys­tem is much more like us. It is pro­grammed by ex­per­i­ence.

Com­puters have the prop­erty that they are ex­tremely sens­it­ive to pro­gram­ming. If you write a pro­gram with er­rors, then it just does not work. Even if you cre­ate some kind of tol­er­ance so that it is stable with re­spect to some small er­ror, still, it’s ex­tremely sens­it­ive. You have to be very pre­cise in the in­struc­tions for it to work.

But we hu­mans — ima­gine how much garbage has been put in­to our heads dur­ing our life­time! Des­pite the garbage — which you can call in­doc­trin­a­tion — what we do (or at least what some of us do) is ra­tion­al to a large de­gree. If most of our in­form­a­tion is not garbage, then still we are able to func­tion. The sys­tem I de­scribed has this cap­ab­il­ity.

One job option, in faraway Syktyvkar

Let’s go back to Arnold. Can you tell me about your in­ter­ac­tions with him?

I first spoke at Arnold’s sem­in­ar in Mo­scow in spring 1969 dur­ing my last un­der­gradu­ate year. Later on, I gave sev­er­al talks at this sem­in­ar. Mostly they went quite well, but once it was a dis­aster. It was some­time in the fall of 1984. I came to Mo­scow for a few days and called Arnold. As usu­al, he asked if I want to give a talk at his sem­in­ar. At that time Misha Gro­mov was fin­ish­ing his sem­in­al pa­per where he in­tro­duced the tech­nique of holo­morph­ic curves in sym­plect­ic geo­metry, which brought about a re­volu­tion in this sub­ject. Just be­fore I came to Mo­scow, I re­ceived from Gro­mov a pre­lim­in­ary draft of his pa­per. So I told Arnold that I could talk about Gro­mov’s work. For some reas­on, dur­ing my talk Arnold was ex­tremely ag­gress­ive. After I had talked for 10 minutes, he in­ter­rup­ted me and said that be­fore go­ing on I should say what is the main idea. In my view, in that pa­per there are sev­er­al equally im­port­ant ideas. So I tried to start from dif­fer­ent sides, but each time Arnold was not sat­is­fied and didn’t let me speak be­fore I told him what is the main idea. Fi­nally at some point I said something — which in my view is an im­port­ant but def­in­itely not the main idea! — that sud­denly sat­is­fied him, and he said, “Why did you waste al­most two hours of our time and couldn’t start from this point at the very be­gin­ning?”

In Rus­sia, for a PhD dis­ser­ta­tion, there are two of­fi­cial “op­pon­ents,” as they are called — a ref­er­ee, who is sup­posed to read the dis­ser­ta­tion, and the ad­viser. When you have your PhD de­fense, they are sup­posed to come and say what they think about your work. Arnold was one of the op­pon­ents (the second one was Alexei Chernavsky28 ) for my PhD dis­ser­ta­tion. He came on the night train from Mo­scow to Len­in­grad, and I met him about 5 a.m. at the train sta­tion. I was bring­ing him to my home to talk, and on the way there he told me he did not un­der­stand a lemma in my dis­ser­ta­tion. I had writ­ten that this lemma is ob­vi­ous. He said that first of all, he does not be­lieve that it is ob­vi­ous, and second, he be­lieves that it’s just wrong.

We came to my home, and Arnold was ex­tremely sleepy, so he de­cided to sleep for a couple of hours. While he slept, I was think­ing about how to ex­plain this lemma to him. He woke up, and we talked for an hour, and then he said, “Now I be­lieve this is true, but still I don’t see that you have a proof.” At my dis­ser­ta­tion de­fense he didn’t say any­thing about this lemma. But then about a year later, he sent me a pa­per he had writ­ten, with a note say­ing, “Now we have proof of your ob­vi­ous lemma!” The pa­per had a res­ult that im­plied my lemma. I still think it was ob­vi­ous, but — !

After that, each time I was in Mo­scow, which was not that of­ten, I tried to see him. When I was in Syk­tyvkar,29 he came there to par­ti­cip­ate in a con­fer­ence that I or­gan­ized, so I in­ter­ac­ted with him there.

You went to Syk­tyvkar right after your PhD. Can you tell me how you ended up go­ing there?

At the time I fin­ished my PhD, the an­ti­semit­ic situ­ation was pretty bad again. When I entered the uni­versity, it was not bad, but by the time I fin­ished and was sup­posed to go to gradu­ate school, it was bad again. For in­stance, there are en­trance ex­am­in­a­tions for gradu­ate school. One of the ex­ams was about polit­ic­al things — com­mun­ist party his­tory, or sci­entif­ic com­mun­ism. They tried ex­tremely hard to fail me on that ex­am. I didn’t fail only be­cause there was a rep­res­ent­at­ive of the math­em­at­ics de­part­ment on the ex­am com­mit­tee. Nina Nikolaevna Ur­alt­seva — the pro­fess­or of math­em­at­ic­al phys­ics who I men­tioned be­fore — was on the com­mit­tee, and she didn’t let them fail me. So I got the low­est passing grade, but they passed me.

I was ad­mit­ted to gradu­ate school, but I had to sign a pa­per say­ing that, after my PhD, I agree to go any­where they would send me. The Len­in­grad Branch of the Steklov In­sti­tute really wanted to take me for a job there after my PhD. This was a fant­ast­ic place to be. The dir­ect­or of this in­sti­tute was Grig­orii Ivan­ovich Pet­rashen.30 He was an ex­tremely good per­son and a good math­em­atician. He tried really hard to hire me. But this was a branch of the Steklov In­sti­tute, so he needed to get ap­prov­al of the dir­ect­or of the main in­sti­tute in Mo­scow, who was Vino­gradov.31 And Vino­gradov was a very fam­ous an­ti­semite.

The rule es­tab­lished by Vino­gradov said that the Len­in­grad branch is al­lowed to de­cide on its own to hire any­body they want — ex­cept Jews. Any Jew­ish hire was sup­posed to be ap­proved per­son­ally by Vino­gradov. So Pet­rashen would tell me, “Of course I will get his per­mis­sion, but I need to find the ap­pro­pri­ate mo­ment. Next month I go to Mo­scow, and I will bring this up.” And then he would re­turn and say, “The mo­ment was not ap­pro­pri­ate so I didn’t bring it up.” It got ex­tremely close to the time of my gradu­ation, so there was no time to wait, and then he fi­nally did talk to Vino­gradov. When Pet­rashen re­turned to Len­in­grad, I talked to his sec­ret­ary. It turned out that he had entered a psy­chi­at­ric clin­ic after his dis­cus­sion with Vino­gradov! I very much hope that it was not be­cause of me, but any­way Pet­rashen had had some kind of nervous break­down.

Vino­gradov had said no. Then I didn’t have any­where I could go. At this time, a new uni­versity was be­ing or­gan­ized in Syk­tyvkar. Be­cause it was new, Len­in­grad Uni­versity was as­signed as a “par­ent” or­gan­iz­a­tion, to help them get star­ted. The rect­or of the new uni­versity came to Len­in­grad, and she had a carte blanche to choose any new PhD she wanted. And she chose me, be­cause I was ex­tremely con­veni­ent to hire. Stu­dents were usu­ally de­fend­ing their PhD dis­ser­ta­tions in the fall, and I had already de­fen­ded in May. So I was ready to go im­me­di­ately, be­fore the aca­dem­ic year star­ted.

In Syktyvkar in 1975. Left to right: Victor Zvonilov, Aleksandr Pevny, Eliashberg, and Nikolai Mishachev.

So I went to Syk­tyvkar. It was I would say an in­ter­est­ing ex­per­i­ence. I was there for eight years, and I liked my time in Syk­tyvkar. It was in­ter­est­ing, it was a new place, and they really or­gan­ized this uni­versity from scratch. Mostly they hired good young people from Len­in­grad and Mo­scow. There was a very nice at­mo­sphere at the be­gin­ning. They were able to give me an apart­ment, which was very good, be­cause in Len­in­grad get­ting your own apart­ment was dif­fi­cult. But the great things in Syk­tyvkar be­came not so great be­cause, as usu­al, bur­eau­cracy takes over. The at­mo­sphere be­came not as ex­cit­ing.

How was it math­em­at­ic­ally for you there?

Math­em­at­ic­ally for me it was dif­fi­cult, be­cause I had a lot of teach­ing. I star­ted out teach­ing 18 hours per week, which was quite a lot. Then they made me a chair — there was a de­part­ment of math­em­at­ics and phys­ics to­geth­er, and I was chair of one math­em­at­ic­al part. I didn’t have much time. But it was okay, and I did some good work there.

Emigration: An extremely difficult decision

Were you work­ing on the Arnold Con­jec­ture in Syk­tyvkar?

I was work­ing on the Arnold Con­jec­ture already at the be­gin­ning of my time in Syk­tyvkar. When I was fin­ish­ing my dis­ser­ta­tion, I was still col­lab­or­at­ing with Gro­mov a lot.

By the way, Gro­mov left the USSR in 1974. In fact, at some point there was a chance that, in­stead of go­ing to Stony Brook, he would come to Syk­tyvkar! As I told you, Gro­mov had this ex­tremely badly paid po­s­i­tion in Len­in­grad Uni­versity. A good thing about this po­s­i­tion was that he didn’t have many ob­lig­a­tions, so he could just do math­em­at­ics. But it was not really a reas­on­ably paid job. They made a con­di­tion for his pro­mo­tion that he would spend some time in Syk­tyvkar. After that he could go back to Len­in­grad. My feel­ing was that he was really con­sid­er­ing this. In Len­in­grad he didn’t really have any place to live — he was rent­ing a room with his wife. He asked Len­in­grad Uni­versity if they could give him a place to store his stuff while he worked in Syk­tyvkar. But of course nobody ever gave him any­thing. I don’t know if it was for this reas­on or an­oth­er reas­on, but he de­cided against Syk­tyvkar. But he did vis­it Syk­tyvkar and talked to people there.

That was in 1972. I think at that time, he de­cided he would leave the USSR. So he left Len­in­grad Uni­versity and worked at vari­ous places, and then even­tu­ally emig­rated.

Did you think about emig­rat­ing at that time?

No, I didn’t think about it. In 1975, my broth­er and his fam­ily and our moth­er de­cided to emig­rate. They left in Decem­ber 1975. Even­tu­ally my broth­er found a job in Cali­for­nia, so by 1976 they were in Pa­lo Alto.

What about your fath­er?

My fath­er had already died by then. He died in an ac­ci­dent in 1968.

Why did you not think about also go­ing to the US at that time?

At that mo­ment, I didn’t want to go. You know, that was an ex­tremely dif­fi­cult de­cision. At that time it looked like you would not only go through the hu­mi­li­ation and everything, but you were also cut­ting forever any ties with friends — with any­body.

But then, at some mo­ment people in Syk­tyvkar learned that my re­l­at­ives were abroad, and that be­came not such a great thing.

What happened when they found that out?

For in­stance, my po­s­i­tion as chair be­came very weak. Be­ing a chair, your main role is to pro­tect the de­part­ment from all the stu­pid things com­ing from the ad­min­is­tra­tion. I re­mem­ber that once I said something against what the rect­or was pro­pos­ing, and she said to the oth­ers, “Why do you listen to Eli­ash­berg? He has a villa wait­ing for him in Cali­for­nia!” Vari­ous things like that happened.

Gro­mov had left in 1974, which was a blow, be­cause, be­sides be­ing math­em­at­ic­ally an im­port­ant per­son for me, he was my friend. To­wards 1978, I had to make a de­cision about emig­rat­ing. It was an ex­tremely dif­fi­cult de­cision, es­pe­cially at that time. But as more things ac­cu­mu­late, you feel that it’s not really pos­sible to con­tin­ue any­more.

What did your wife think about go­ing or stay­ing? You were mar­ried at this time, right?

Eliashberg with his two sons around 1982.

Yes, we were mar­ried and had two chil­dren at this time. My wife Ada was sup­port­ive of course, but the one thing that she knew for sure was that she did not want for us to re­main forever in Syk­tyvkar. But clearly there was ab­so­lutely no pos­sib­il­ity for me to re­turn and get a de­cent job in Len­in­grad. If I had not been mar­ried and had been alone, maybe I would still be in Syk­tyvkar. Wo­men are mov­ing pro­gress!

So we de­cided to leave. It was tech­nic­ally al­most im­possible to ap­ply for emig­ra­tion from Syk­tyvkar. So we came back to Len­in­grad with the idea of ap­ply­ing for emig­ra­tion from there. We ap­plied some­time around the end of sum­mer 1979. But then in Decem­ber that year, the USSR in­vaded Afgh­anistan, and the war star­ted. Then it was clear that the situ­ation had be­come such that we would not get per­mis­sion, and in­deed in about a month we got re­fused.

One of the pins handed out during the ICM in Berkeley in 1986.

After that, it was pretty bad, be­cause of course I couldn’t find a job. I worked as a sub­sti­tute teach­er in middle school, which was pretty tough. Then one of my friends helped me to get a job do­ing com­puter pro­gram­ming. He es­sen­tially put his ca­reer on the line to help me. In the place where he worked, he was the main per­son on whom everything de­pended. They couldn’t let him go. And he said if they don’t hire me, he is leav­ing. So they hired me. I worked there un­til I man­aged to get per­mis­sion to emig­rate, in 1987.

That was mainly thanks to my broth­er Vic­tor, who was writ­ing to con­gress­men, to every­body he could think of, in Amer­ica. I was in­vited to speak at the In­ter­na­tion­al Con­gress of Math­em­aticians in 1986 in Berke­ley. Of course the au­thor­it­ies didn’t let me go, but my broth­er and his fam­ily and friends or­gan­ized a pe­ti­tion to free me. I still have the pe­ti­tion, with al­most 1500 sig­na­tures.

The cartoon and letter accompanied a petition calling on the USSR to allow Eliashberg to emigrate. Almost 1500 people signed the petition during the ICM in Berkeley in 1986.

They col­lec­ted sig­na­tures dur­ing the ICM?

Yes. Dur­ing the ICM they put a table in front of Evans Hall32 and col­lec­ted sig­na­tures. They also made pins on which it was writ­ten, “Let Yakov Eli­ash­berg Go!” When I came to the US, I got as a present maybe 10 such pins from vari­ous math­em­aticians!

My broth­er col­lec­ted the sig­na­tures and gave the pe­ti­tion to a con­gress­man, and this helped. The con­gress­man went to Mo­scow and talked to some people there. The So­viet Uni­on still was in­tact then, in 1987. People were not leav­ing. The So­viet Uni­on col­lapsed in 1991. But in 1987, 1988, it was dif­fi­cult to leave.

You couldn’t go to the ICM in Berke­ley, so John Math­er gave your talk.

I asked that Misha Gro­mov give my talk, be­cause of course he knew this stuff in­side and out. A per­son I have mixed feel­ings about is Lud­wig Fad­deev. He died some years ago. He was of course a very good math­em­at­ic­al phys­i­cist, and I knew him for many years. I think he was the chair of the Rus­si­an del­eg­a­tion for the Berke­ley ICM. When I got the in­vit­a­tion to speak, I came to him — he was in Len­in­grad. He said of course there is no chance I can get per­mis­sion to go to the ICM, but he offered to take my lec­ture and give it to someone there to read. I asked him to give it Gro­mov. But Fad­deev didn’t do that, be­cause Gro­mov was seen as a bad per­son, from the Rus­si­an point of view. He was polit­ic­ally in­cor­rect!

Exploring mathematical worlds

Let me ask a dif­fer­ent kind of ques­tion. What goes on in your brain when you do math­em­at­ics? Do you see geo­met­ric pic­tures, or do you en­vi­sion pro­cesses, or do you think about cal­cu­la­tions? What goes on in your brain?

I nev­er ana­lyzed my own brain. In math­em­at­ics, it’s like you are a world ex­plorer. You are ex­plor­ing a world, and maybe this world is re­lated to our real world, but it’s a sub­world. For in­stance I am do­ing sym­plect­ic geo­metry, which has some rules, and I try to un­der­stand what is pos­sible in that world, con­strained by these rules. But it’s ex­tremely dif­fi­cult to con­cen­trate and think lin­early in this world. You go in­to this world, but there are a lot of out­side factors that can dis­turb you, and your thought goes com­pletely to the wrong place. You need some dis­cip­line to re­turn back to that world.

There are dis­trac­tions?

It’s not ne­ces­sar­ily out­side dis­trac­tions. It might even be dis­trac­tion from your own brain. Whenev­er I have made pro­gress in my math­em­at­ic­al think­ing, it was be­cause I was able to think for a suf­fi­ciently long time. I am trav­el­ing in this world, and I need to walk there a suf­fi­ciently long time — at least for a few minutes — without get­ting out in between.

So you need enough time to im­merse your­self in that world.

And you come to the place where you wanted to go, but then sud­denly something hap­pens and your brain moves you com­pletely wrong. Then you need to force your­self, no, come back, con­tin­ue, try to see what’s there.

So it’s see­ing, it’s geo­met­ric? Or are you feel­ing around, it’s something tact­ile?

I can­not say. It’s a little bit geo­met­ric. Some people are able to see much more. It’s dif­fi­cult to say.

I can tell you an in­ter­est­ing story that Misha Gro­mov told me. He used to be a smoker, and sev­er­al times he tried to quit but again star­ted smoking. The main thing that ir­rit­ated him was that, be­fore he stopped smoking, he al­ways had in his brain a black­board, and he was writ­ing on the black­board. But when he stopped smoking, this black­board dis­ap­peared. He didn’t have any­thing to write on! I kind of un­der­stand this. I don’t have any black­board, but there is something in this spir­it.

If you talk to an­oth­er math­em­atician about the things you are think­ing about, do you get the im­pres­sion that that per­son is in the same math­em­at­ic­al world as you were, when you were think­ing about the top­ic? Or does it seem like they are in a dif­fer­ent but maybe sim­il­ar place?

People have dif­fer­ent ways of think­ing. I think that’s the most ex­cit­ing thing, that people col­lab­or­ate, and they have dif­fer­ent ways of think­ing. You can say, “This is clear,” and then some­body else says, “It’s com­pletely un­clear. Why is this true?” You had in your brain a cer­tain point of view, and from this point of view something in­deed was pretty clear. But then when you ap­proach it from a dif­fer­ent point of view, then this maybe is not clear at all.

In the world you are ex­plor­ing, there are ob­jects there, and you can look at them from very dif­fer­ent per­spect­ives. So it’s nat­ur­al that dif­fer­ent people have dif­fer­ent views. Of course, there are people with whom you feel more af­fin­ity in the way they think. But the best col­lab­or­at­ors are those who think dif­fer­ently.

A paper never accepted, never withdrawn

What were you work­ing on math­em­at­ic­ally when you were still in the USSR?

When I got my PhD, I was work­ing with Gro­mov on all this stuff about the h-prin­ciple, the flex­ible math­em­at­ics that I men­tioned be­fore. I wrote sev­er­al pa­pers, with and without Gro­mov. I learned about the Arnold Con­jec­ture from Gro­mov. We both thought it’s un­likely to be true. We thought maybe it’s true in di­men­sion 2 as a spe­cial case, but not in high di­men­sions. I had some kind of ar­gu­ment that al­most proved that in­deed it’s not true. But then I real­ized that I was wrong, and I tried to fix the proof. Of course I couldn’t fix it, and then I star­ted to think that maybe it is true. And I star­ted to think about what is now called sym­plect­ic ri­gid­ity.

Eliashberg (left) with Dimitry Fuchs in Syktyvkar in 1977.

In Syk­tyvkar, I was try­ing to prove the Arnold Con­jec­ture for sur­faces, and I even­tu­ally proved it in 1978. I sent the pa­per to the journ­al Func­tion­al Ana­lys­is and its Ap­plic­a­tions, where Gel­fand was the main ed­it­or. Arnold was a deputy ed­it­or, and I think he es­sen­tially was run­ning the journ­al. From time to time I was com­ing to Mo­scow from Syk­tyvkar and talk­ing to one per­son I knew, Dmitry Fuchs. He is older than me but I con­sider him my friend. Usu­ally when I came to Mo­scow I stayed in his home. He or­gan­ized a small sem­in­ar to go through my proof. After I sent the pa­per to the journ­al, I came to Mo­scow, and Fuchs told me that Askold (Asik) Khovanskii,33 a former stu­dent of Arnold whom I knew very well, wanted to talk to me. So I called Asik, and he said, “Well it’s great that you are here, be­cause I am ref­er­ee­ing your pa­per, and I don’t un­der­stand something.” I ex­plained it to him and, I think, con­vinced him. Then, some time after that, I got an ex­tremely angry let­ter from Arnold, who said that I des­troyed his ref­er­ee — 

Be­cause you spoke to him.

 — be­cause the ref­er­ee spoke to me. There­fore Arnold had to start the ref­er­ee­ing pro­cess from scratch. There is some truth in this. Arnold ex­plained that the goal of the ref­er­ee is not only to check wheth­er the proof is cor­rect or not but also if it could be un­der­stood from what is writ­ten. If I have to ex­plain it to the ref­er­ee, then that means it’s not pos­sible to un­der­stand the proof as writ­ten.

This happened around the time when I ap­plied for emig­ra­tion. Of course this be­came known by people on the ed­it­or­i­al board, and they were ex­tremely afraid that pub­lish­ing a pa­per of such a bad per­son like me could be harm­ful for the journ­al.

It’s stag­ger­ing to think that that happened.

Yeah. Gel­fand felt that his main goal was to pro­tect the journ­al. If someone said that the journ­al had pub­lished a pa­per by a per­son who ap­plied for emig­ra­tion, that would be bad for the journ­al.

My pa­per is still at this journ­al. It was nev­er re­jec­ted be­cause they prob­ably didn’t have a math­em­at­ic­al reas­on for re­jec­tion. But it was not ac­cep­ted either. Some­body — I for­got who — called me from the ed­it­or­i­al board and asked me to with­draw the pa­per, be­cause the ed­it­or­i­al board had made a de­cision to not pub­lish pa­pers of people who may have a pos­sib­il­ity to pub­lish in more widely read­able journ­als! But I re­fused to with­draw it. So that means it’s still there.

It nev­er got pub­lished?

It’s nev­er been pub­lished. In the So­viet Uni­on there was a place called VIN­ITI,34 which was an or­gan­iz­a­tion that pub­lished Refer­at­ivnyi Zhurn­al, the Rus­si­an ana­log of Math Re­views and Zen­t­ral­blatt. You could send a pa­per to VIN­ITI, and it would not be pub­lished but it would be re­viewed and the ab­stract would ap­pear in Refer­at­ivnyi Zhurn­al. Then if some­body was in­ter­ested, they could write to the au­thor and re­quest the text. So I sent the pa­per there. In the So­viet Uni­on this was con­sidered as some kind of form­al pub­lic­a­tion.

Did oth­er people prove the same thing that you had in that pa­per?

Yes, later on. It was between the end of 1978 and 1979 that I sent this pa­per to VIN­ITI. Then many things happened. In 1983, there was a pa­per by Con­ley and Zehnder35 that proved the case of tori. I had proved it for all sur­faces, but they proved it for tori of any di­men­sion, so that is a much stronger res­ult. Gro­mov wrote his fam­ous pa­per36 about holo­morph­ic curves in 1985, which im­plied some of what I had proved. And then Flo­er, also around 1985, wrote a pa­per37 where a proof of the same res­ult ap­peared. But my proof is dif­fer­ent from all oth­er cur­rently known proofs, so I think it still has its value.

Everything goes back to Poincaré

Can you say more about the Arnold Con­jec­ture — what it is, and why you and Gro­mov thought it might not be true?

Everything goes back to Poin­caré. Poin­caré in 1912 wrote an ex­tremely dra­mat­ic pa­per.38 It says, “Well, here is a ques­tion I have been look­ing at for a long, long time, and I have con­sidered many spe­cial cases. I am con­vinced this is true but I wasn’t able to find a gen­er­al proof. But I am already old, so I de­cided to write this in­com­plete pa­per.” Ac­tu­ally he was not old, he was 56, but he was sick, and he died a few months later.

The prob­lem Poin­caré wrote about was a geo­met­ric prob­lem to which he came by ana­lyz­ing peri­od­ic or­bits in the so-called re­stric­ted 3-body prob­lem. This is the case of the 3-body prob­lem when one of the bod­ies is small and the ef­fect of its grav­ity on the mo­tion of two oth­er bod­ies can be ig­nored. An ex­ample is the case of a satel­lite mov­ing in the grav­ity of the earth and moon. Poin­caré re­duced this prob­lem to the ques­tion of find­ing fixed points of an area-pre­serving trans­form­a­tion of the an­nu­lus. Here pre­ser­va­tion of area means that, while a sub­do­main in the an­nu­lus might be mapped onto something very com­plic­ated, the area of its im­age re­mains the same. Poin­caré claimed that if, in ad­di­tion, such a trans­form­a­tion ro­tates the bound­ary circles in op­pos­ite dir­ec­tions, then it must have at least two fixed points.

This be­came known as the “last geo­met­ric the­or­em of Poin­caré”. It was ac­tu­ally proved pretty quickly after he died, by George Dav­id Birk­hoff in 1913.39 Poin­caré ob­vi­ously thought of this as a spe­cial case of something very gen­er­al but while Birk­hoff’s proof was very clev­er, it was very spe­cif­ic to this par­tic­u­lar ex­ample. Some­how, it was a dead end. Not much fol­lowed un­til Arnold, around 1965, re­an­im­ated the ap­proach of Poin­caré.

The ques­tion about area-pre­serving trans­form­a­tions arises if you are think­ing about mech­an­ic­al sys­tems with two de­grees of free­dom. If you have a mech­an­ic­al sys­tem with more de­grees of free­dom, then the cor­res­pond­ing ques­tion is not about area-pre­serving trans­form­a­tions, but about trans­form­a­tions of high­er-di­men­sion­al space that pre­serve what is called the sym­plect­ic form. Poin­caré and Birk­hoff un­der­stood this, but nobody ex­pli­citly for­mu­lated con­jec­tures of that sort be­fore Arnold.

In to­po­logy there is a the­ory of count­ing the num­ber of fixed points of a map or dif­feo­morph­ism, which goes back to Brouwer and Lef­schetz. But un­like the prob­lem of Poin­caré, all of this was gov­erned by ho­mo­logy or re­lated al­geb­ra­ic in­vari­ants of the man­i­fold. Arnold real­ized, and ac­tu­ally proved in some very spe­cial situ­ations, that in fact the bound on the num­ber of fixed points is gov­erned by Morse the­ory, so it is gov­erned by the bound on the num­ber of crit­ic­al points of a func­tion.

For in­stance, sup­pose you take a func­tion on the an­nu­lus that is, say, con­stant on the bound­ary and grows if you go in. This func­tion must have at least two crit­ic­al points: the max­im­um and one oth­er point. You can prove this in a stand­ard way. Then — and ac­tu­ally Poin­caré tried to do this in his pa­per — you can re­late the ques­tion of fixed points of the trans­form­a­tion to a ques­tion about crit­ic­al points of the func­tion. But for a gen­er­al trans­form­a­tion it was not clear how to do this.

Arnold star­ted with one very spe­cif­ic con­jec­ture about the 2-di­men­sion­al case, for area-pre­serving maps. Sup­pose you have an area-pre­serving trans­form­a­tion of the tor­us. Then you ask wheth­er there are some fixed points. For the tor­us, an ad­di­tion­al con­di­tion is needed, sim­il­ar to the con­di­tion on the an­nu­lus that the map ro­tate the bound­ary circles in op­pos­ite dir­ec­tions; oth­er­wise, you could just ro­tate the tor­us, and there would be no fixed points. A con­di­tion that Arnold called “pre­ser­va­tion of cen­ter of mass” avoids such cases. Un­der this con­di­tion, Arnold said there must be three, and in the nonde­gen­er­ate case four, fixed points. This was his first con­jec­ture, just for the tor­us. Of course he un­der­stood that it’s not just about the tor­us and that you have a sim­il­ar bound for sur­faces. He made con­jec­tures about man­i­folds of ar­bit­rary di­men­sion over the course of the next few years, up to the be­gin­ning of the 1970s.

I star­ted to think about this, and Gro­mov also thought about it. It was clear that there is some phe­nomen­on in two di­men­sions, be­cause we already had the Poin­caré the­or­em. So for sur­faces, it was very likely that this would be true. At this time Gro­mov for­mu­lated what I call “the Gro­mov al­tern­at­ive,” which said that either es­sen­tially noth­ing in high di­men­sion of this type is true, or everything is!

Sup­pose you have a dif­feo­morph­ism that pre­serves volume and that can be \( C^0 \) ap­prox­im­ated by a dif­feo­morph­ism pre­serving the sym­plect­ic form. It is not clear wheth­er the dif­feo­morph­ism it­self has to pre­serve the sym­plect­ic form, be­cause pre­ser­va­tion of the sym­plect­ic form is a ques­tion about a de­riv­at­ive. If the ap­prox­im­a­tion is only \( C^0 \) close, then a pri­ori there is no reas­on to think that the lim­it will pre­serve this form. So Gro­mov said one of two things is true. Either it’s true that the lim­it has to pre­serve the sym­plect­ic form, or the com­pletely op­pos­ite state­ment is true, that any volume-pre­serving dif­feo­morph­ism can be \( C^0 \) ap­prox­im­ated by sym­plect­ic, so that if you man­age to ap­prox­im­ate by a sym­plect­ic dif­feo­morph­ism any­thing that is not sym­plect­ic, then es­sen­tially you can ap­prox­im­ate whatever you want. If the lat­ter were true, it would im­me­di­ately fol­low that the Arnold Con­jec­ture is wrong in di­men­sions big­ger than 2.

So in some sense, it was not that we be­lieved or did not be­lieve, but there was this al­tern­at­ive. And there was ab­so­lutely no reas­on to think that A is true or B is true.

And they were starkly dif­fer­ent.

Yes, but one of them would say that noth­ing like the Arnold Con­jec­ture can be true in high di­men­sions, which would be an in­ter­est­ing res­ult, but in some sense sad. That would show that the the­ory is not really in­ter­est­ing. There would be noth­ing there.

It’s ex­tremely hard to solve equa­tions of mech­an­ics, but you would still like to be able to make some kind of qual­it­at­ive state­ments, for in­stance, to say wheth­er there are peri­od­ic or­bits — to say something about the sys­tem, short of solv­ing the equa­tions of mo­tion. Poin­caré thought that merely the fact that the ques­tion is based on a mech­an­ic­al sys­tem, to­geth­er with some geo­met­ric con­straints, means that some qual­it­at­ive co­rol­lary should fol­low. And this is ac­tu­ally real­ized. All of the de­vel­op­ment of sym­plect­ic to­po­logy is the proof of this. The Gro­mov al­tern­at­ive was re­solved in fa­vor of ri­gid­ity.

A thriving field, but basic questions open

Eliashberg (right) with Claude Viterbo in 1988.

When you came to the US you were a little over 40 years old.

It was 1988, I had just turned 41.

Some­how I think of you as young­er than your age, partly be­cause you have a youth­ful way about you, but maybe also be­cause the en­ergy you brought to your work after com­ing to the US was that of someone young­er.

The way I think about it is that at least eight years, the time when I was in Len­in­grad after I be­came a re­fusenik, should be sub­trac­ted from my math­em­at­ic­al life. So it should be coun­ted that I left the USSR not at 41 but at 33! In some sense I star­ted a com­pletely new life. So in­deed at 41, I was like a young postdoc in some sense.

Since that time sym­plect­ic and con­tact geo­metry has be­come a thriv­ing field, and you are one of the cent­ral people in it. How do you see that de­vel­op­ment? Did it sur­prise you?

It didn’t sur­prise me. I knew it should hap­pen. I re­mem­ber giv­ing lec­tures and try­ing to con­vince people that it’s something worth do­ing! But I was sure it would hap­pen. Of course, I didn’t pre­dict how it would de­vel­op, and now it has moved in dif­fer­ent dir­ec­tions. With mir­ror sym­metry and Kont­sevich’s ideas, it moved to­ward al­geb­ra­ic stuff. There are many parts of this sub­ject that I now don’t un­der­stand prop­erly but still hope to un­der­stand.

Des­pite this area hav­ing fant­ast­ic suc­cess and many new res­ults, some of the ba­sic ques­tions that I was think­ing about 40 years ago are still not answered. So it’s only a par­tial suc­cess. Clearly we are wait­ing for a new door to open to go some­where else.

Can you give an ex­ample of a ba­sic res­ult that is still open?

You can come to the sym­plect­ic world from dif­fer­ent sides. You can come as a to­po­lo­gist, or you can come as a per­son from Hamilto­ni­an dy­nam­ics and try to get some kind of dy­nam­ic­al con­sequences out of the sym­plect­ic to­po­logy. Both dir­ec­tions are com­pletely val­id, but let’s talk a little bit as a to­po­lo­gist.

As a to­po­lo­gist, you think of a sym­plect­ic struc­ture as a geo­met­ric struc­ture, and you would like to first un­der­stand which man­i­folds ad­mit a sym­plect­ic struc­ture — which man­i­folds are sym­plect­ic, which man­i­folds are con­tact, and so on.

A sym­plect­ic struc­ture is a closed nonde­gen­er­ate 2-form. If you re­move the con­di­tion that the 2-form is closed, then you just have a nonde­gen­er­ate 2-form, which is of course a ne­ces­sary con­di­tion. A the­or­em of Gro­mov from a long time ago says that, for an open man­i­fold, this con­di­tion is suf­fi­cient; the ex­ist­ence of sym­plect­ic struc­tures is a purely ho­mo­top­ic­al con­di­tion. This is an in­stance of the h-prin­ciple, which is the idea that geo­met­ric prob­lems should be solv­able up to ho­mo­topy, once ob­vi­ous to­po­lo­gic­al ob­struc­tions van­ish.

If you have a closed man­i­fold, then you get one more ne­ces­sary con­di­tion, that this sym­plect­ic form defines a 2-di­men­sion­al co­homo­logy class of the man­i­fold, which has the prop­erty that its powers have to be nonzero. So you get one more al­geb­ra­ic-to­po­lo­gic­al con­straint for the man­i­fold to be sym­plect­ic.

The first na­ive ques­tion is, If a man­i­fold sat­is­fies this con­di­tion, does it ad­mit a sym­plect­ic struc­ture, or not? The an­swer is known only in di­men­sion 4, and there the an­swer is no. A com­bin­a­tion of the work of Gro­mov and Taubes im­plies this. But, if we go any­where bey­ond 4 di­men­sions, the ques­tion is com­pletely open. We have neither a con­struc­tion of a counter­example, nor any tools to show that there is something else to ob­struct the man­i­fold to be sym­plect­ic in di­men­sions great­er than 4. It’s com­pletely open.

Nobody has any feel­ing about which way it might go?

Every­body has feel­ings, but these feel­ings are time-de­pend­ent and per­son­al. I don’t have any idea. I would think that prob­ably there is a soft an­swer, or that there are no oth­er con­straints. But I have no idea how this can be proven. All con­jec­tures are ex­tremely ir­re­spons­ible be­cause there is ab­so­lutely no reas­on to be­lieve one way or the oth­er.

There has been so much build-up of the sub­ject, yet this ba­sic ques­tion is still open.

Yes, that’s something. You can ask an­oth­er ques­tion that is very sim­il­ar. Sup­pose you take the space \( \mathbb{R}^{2n} \). In \( \mathbb{R}^{2n} \), there is a stand­ard sym­plect­ic struc­ture, a ca­non­ic­al, con­stant sym­plect­ic struc­ture. Sup­pose you want to find a sym­plect­ic struc­ture that out­side some big ball is stand­ard but in­side the ball it is not. Can you find such a struc­ture? Gro­mov proved that in di­men­sion 4, this is im­possible. All of them are stand­ard. In di­men­sion great­er than 4, we have no idea.

Pro­gress on both of these prob­lems stops at di­men­sion 4. What is dif­fer­ent about di­men­sion 4 that you can do something in that di­men­sion?

In gen­er­al, in di­men­sion 4, to­po­logy is very dif­fer­ent from oth­er di­men­sions. In the sym­plect­ic world, the main reas­on comes from the fol­low­ing. Most of the ri­gid­ity res­ults are proven us­ing some ver­sion of Gro­mov’s the­ory of holo­morph­ic curves on sym­plect­ic man­i­folds. Sup­pose in di­men­sion 4 you have two sur­faces that in­ter­sect trans­vers­ally in fi­nitely many points. The in­ter­sec­tions can have plus or minus — each in­ter­sec­tion has a sign de­pend­ing on the ori­ent­a­tion. For holo­morph­ic curves, the sign is al­ways plus. On the oth­er hand, the total num­ber of in­ter­sec­tion points count­ing with signs is a to­po­lo­gic­al in­vari­ant. There­fore, some­times in di­men­sion 4 you can guar­an­tee that the two sur­faces have no in­ter­sec­tion, just us­ing some to­po­logy. Noth­ing like this ex­ists in high­er di­men­sions. This is ba­sic­ally the reas­on.

Out of the darkness, into the sun

You left the USSR in late 1987. Where did you go right after you left?

There was a stand­ard route of emig­ra­tion. You were sup­posed to come to Vi­enna, and they pro­cessed your doc­u­ments. Then they sent you to Italy, and you stayed in some sub­urb of Rome, wait­ing for a US visa. In mid-Feb­ru­ary of 1988, we got per­mis­sion to go to Amer­ica.

When you ar­rived in the US, did you have a job right away at Stan­ford?

No, but I had an ab­so­lutely fant­ast­ic thing. After the Berke­ley Con­gress, I was in­vited by Alan Wein­stein for a year pro­gram in sym­plect­ic geo­metry at MSRI.40 I got the in­vit­a­tion in 1986, when I was still in Len­in­grad. At the time, I con­sidered it as some kind of joke, be­cause I wouldn’t dream that I could get per­mis­sion to go. But then I emig­rated, so hav­ing the in­vit­a­tion was great.

The MSRI pro­gram star­ted only in Septem­ber 1988. So An­drew Cas­son sug­ges­ted to me to write an ex­tra pro­pos­al on his NSF [Na­tion­al Sci­ence Found­a­tion] grant. I did this and re­ceived the grant, which sup­por­ted me for three months. It was not much, but it was es­sen­tial. Then, from Septem­ber, I was at MSRI.

I should say that I was not sure I would be cap­able of do­ing math­em­at­ics, to be re­born as a math­em­atician. In Syk­tyvkar, I was able to do math­em­at­ics, but then there were the years when I was a re­fusenik. That was an ex­tremely hard peri­od, when I had to work as a com­puter pro­gram­mer. It was a job that really took my brain com­pletely. I caught my­self at night think­ing about how I could send a file from one place to an­oth­er — I was think­ing about that in­stead of math­em­at­ics. And I was too tired to do any­thing else. I tried to do some math­em­at­ics, and I even proved something, but it was not ser­i­ous. So those eight years were es­sen­tially lost for me for math­em­at­ics. There­fore, when I emig­rated, I was pre­pared to not be able to re­in­state my­self as a math­em­atician. I thought, Okay, maybe I’ll just find a job, do something in­ter­est­ing as a pro­gram­mer.

But then, amaz­ingly — es­pe­cially thanks to this grant of An­drew Cas­son — I did quite a lot of nice work dur­ing that first half-year when I came to the US. I went on a tour of lec­tures, to Prin­ceton, Stony Brook, Uni­versity of Pennsylvania, NYU, Uni­versity of Utah, Uni­versity of Mary­land at Col­lege Park, Cal­tech, USC, Stan­ford, Berke­ley. Un­ex­pec­tedly, I got of­fers from most of these places and de­cided to go to Stan­ford.

That is amaz­ing that you could re­start your math­em­at­ics. Why was that? The en­vir­on­ment in Berke­ley? Or were you fi­nally able just to sit and think?

Maybe be­ing able to sit and think, that’s the main thing. I got time to do math­em­at­ics. But of course the en­vir­on­ment was also im­port­ant. I’ll give you an ex­ample.

I was sit­ting in an of­fice in Berke­ley that I shared with a math­em­atician vis­it­ing from Spain, Jesús Gonzalo. Jesús was also work­ing on con­tact man­i­folds. He told me what he was try­ing to prove, and I saw that it was com­pletely im­possible. That in some sense triggered me to think back about con­tact geo­metry. So in­deed the en­vir­on­ment was im­port­ant also.

Eliashberg (center) with Helmut Hofer (right) and Sasha Givental in 2007.

That was over the sum­mer of 1988. Then I star­ted the pro­gram at MSRI, and this was an ab­so­lutely fant­ast­ic time. I met and talked to so many people. That of course was ex­tremely in­spir­ing. I had met Claude Vi­terbo in Italy, but it was at MSRI that I first met Dusa Mc­Duff, Helmut Hofer, and Flo­er. I met S.-S. Chern, Alan Wein­stein, Rob Kirby, John Mil­nor. Also Curt McMul­len was there — he was a postdoc, and I in­ter­ac­ted with him. I also col­lab­or­ated with Tu­dor Ra­tiu.

While I was in Italy right after leav­ing the So­viet Uni­on I got in touch with many Itali­an math­em­aticians. Stefano Mar­chi­afava was es­pe­cially help­ful. He or­gan­ized my first series of lec­tures and paid me for that 360,000 Itali­an lire, which was about US\$300. I was ex­tremely im­pressed by this huge amount and by the ap­par­ent lack of any bur­eau­cracy. With my cur­rent ex­per­i­ence, I strongly sus­pect that Stefano paid me out of his own pock­et. After I gave these talks, they in­vited me for many oth­ers. I was tour­ing Italy giv­ing talks. I got an in­vit­a­tion to a con­fer­ence in Cala Gan­one, in Sardin­ia, which was in Septem­ber 1988. I gave some lec­tures there and also stayed in Rome for some time, maybe for two weeks. And I re­mem­ber one of my best works, which I like very much, is pre­cisely what I thought about while walk­ing in Rome!41

When I first came to the US, I came to Pa­lo Alto be­cause my broth­er and my moth­er were there. It was mid-Feb­ru­ary, it was fant­ast­ic­ally sunny, there were no clouds, noth­ing. In the So­viet Uni­on and Len­in­grad, it was pretty dark. I al­most phys­ic­ally had this feel­ing that all my pre­vi­ous life was un­der dark clouds, and sud­denly I had the sun.