return

Celebratio Mathematica

Yakov M. Eliashberg

⇧

In appreciation of Yasha

by Oleg Lazarev

Yasha-isms

One of the first things I no­ticed as a stu­dent of Yasha’s was his play­ful at­ti­tude to­wards math­em­at­ics, which I think re­flects a deep­er sense that com­plic­ated (and ser­i­ous) things ought to be simple at the end of the day. His talks of­ten in­clude pithy “Yasha-isms”, hu­mor­ous state­ments but with math­em­at­ic­al con­tent. When I was learn­ing about Smale’s h-cobor­d­ism the­or­em as a grad stu­dent, I re­mem­ber Yasha say­ing something along the lines of “Smale’s most im­port­ant ob­ser­va­tion was that a square is the product of an in­ter­val and an in­ter­val.” I also re­mem­ber him say­ing, “An im­port­ant fact is that \( f = f + 0 \)” — but un­for­tu­nately I can’t re­mem­ber what \( f \) here refers to. More gen­er­ally, Yasha pro­motes the per­spect­ive that there should be a key idea in a proof that is easy to ex­plain and un­der­stand, em­bed­ded though it might be in the tech­nic­al de­tails re­quired to com­plete the proof. Talk­ing to him about math forces one to find this key idea. He is also not afraid to poke fun at him­self (and oth­ers). Re­cently, I re­mem­ber him be­gin­ning a talk by stat­ing that he had just con­vinced him­self that the main res­ult was false (after go­ing back and forth mul­tiple times). However, he then noted that one could still learn something from the (at­temp­ted) proof, and pro­ceeded to give an il­lu­min­at­ing talk.

In grad school, this light­hearted, play­ful at­mo­sphere per­vaded the whole cadre of sym­plect­ic geo­met­ers around Yasha, which at one point had sev­en grad stu­dents, in­clud­ing the not­able prank­ster Kyler Siegel. One year on April 1st, I re­ceived an email from Yasha say­ing that he did not want to be my ad­visor any­more and that in­stead I ought to go to Sweden and work with an­oth­er pro­fess­or there. Al­though I was ini­tially sur­prised, I real­ized that it was April Fool’s day and played along, say­ing I would be glad to leave. At this point, Yasha replied say­ing that he wasn’t the one that sent the email. After sev­er­al in­creas­ingly ri­dicu­lous emails, we came to the con­clu­sion that an­oth­er stu­dent had spoofed Yasha’s email and had pranked both of us! Yasha was quite pleased by the prank and im­pressed by the abil­ity of the prank­ster, leav­ing me en­vi­ous that I hadn’t de­vised the prank my­self.

As an advisor

I also re­call hav­ing to ask Yasha to be my ad­visor twice. The first time I asked, when I was already do­ing a read­ing course with him and an­oth­er pro­fess­or, he replied that we could keep meet­ing even if he wasn’t my of­fi­cial ad­visor. In hind­sight, I see that this was a kind ges­ture that would grant me more free­dom, but at the time his re­sponse left me a bit con­cerned about the status of our re­la­tion­ship. So I went back and asked him again. At this point, he laughed and said, “Yes of course, we’re already work­ing to­geth­er! What do you want? A hand­shake?” He shook my hand and we began work­ing in earn­est.

There was quite a bit of flex­ib­il­ity work­ing with Yasha, pun in­ten­ded. After read­ing Mil­nor’s h-cobor­d­ism book (which I think Yasha re­quired many of his stu­dents to do), I worked on some ques­tions about aug­ment­a­tion vari­et­ies and Le­gendri­an knot con­tact ho­mo­logy, which I had star­ted read­ing about be­fore work­ing with Yasha. He was happy to listen and provide feed­back. However I even­tu­ally real­ized that this top­ic was something Yasha had thought about (and helped found as a sub­field) many years ago but that he had not kept up with the latest de­vel­op­ments. I asked Yasha if we could switch to a top­ic he was more fa­mil­i­ar with at the mo­ment. He pro­posed that I think about flex­ible Wein­stein do­mains, on which top­ic he had just writ­ten a book with Kai Cieliebak, and that I fo­cus on their con­tact bound­ar­ies. Even­tu­ally I solved the main prob­lem that he pro­posed and was writ­ing up the proof. In the pro­cess, I had to de­scribe a class of con­tact man­i­folds and de­cided to call them “quasi-good” since there was already a class of con­tact man­i­folds called “good” and mine seemed to obey a weak­er prop­erty that suf­ficed for my pur­poses. In a typ­ic­al Yasha-ism, he replied, “Quasi-good is a quasi-good name.” Later, I re­named these con­tact struc­tures “asymp­tot­ic­ally dy­nam­ic­ally con­vex”, a more de­script­ive (and ser­i­ous-sound­ing) name. The epis­ode made me real­ize that choice of ter­min­o­logy should also re­quire ser­i­ous math­em­at­ic­al thought.

Yasha was also happy to in­tro­duce people to each oth­er and pro­mote col­lab­or­a­tions. Some­time in the middle of grad school, Yasha asked me to join his meet­ings with Sheel Gan­atra, a postdoc at the time. We star­ted talk­ing about flex­ible Lag­rangi­ans, a vari­ant of my thes­is work on flex­ible Wein­stein do­mains. After we made some pro­gress and had an idea of the proof, I wanted to work out the de­tails more care­fully. However, a week later, Yasha pro­duced a first draft sketch­ing the proof we had figured out. I was a bit sur­prised (par­tic­u­larly since I thought that Sheel and I, as the ju­ni­or col­lab­or­at­ors, would be re­spons­ible for the bulk of the writ­ing). This first draft had a lot of er­rors and not quite cor­rect defin­i­tions, but it provided a good frame­work and start­ing point for our pa­per. Later we worked out many of the de­tails and ad­ded new res­ults, but this top-down ap­proach of sketch­ing out the over­all pic­ture first proved much more ef­fi­cient and en­light­en­ing. This epis­ode also re­flec­ted Yasha’s ex­tremely en­er­get­ic and op­tim­ist­ic per­son­al­ity. I re­mem­ber him say­ing, “If you want to learn something, you can do it in two months” — which, in typ­ic­al fash­ion, was a mix of out­rageous and in­spir­ing.

Rigidity vs. flexibility

Yasha Eliashberg (left) with Oleg Lazarev after the latter’s thesis defense in 2017.
Some of my math con­ver­sa­tions with Yasha were al­most philo­soph­ic­al, re­flect­ing a view that math is more than just a set of facts and proofs. For ex­ample, a main theme in Yasha’s work is the ri­gid­ity and flex­ib­il­ity di­cho­tomy. In sym­plect­ic geo­metry, ri­gid­ity usu­ally refers to phe­nom­ena that are dis­tinct from clas­sic­al to­po­logy, such as dif­fer­ent sym­plect­ic struc­tures on the same smooth man­i­fold. Usu­ally, de­tect­ing ri­gid­ity in­volves \( J \)-holo­morph­ic curves or sheaves, pack­aged in some al­geb­ra­ic form. The flex­ible side refers to sym­plect­ic phe­nom­ena that are gov­erned by the un­der­ly­ing to­po­lo­gic­al data, such as h-prin­ciples that pro­duce sym­plect­ic struc­tures on man­i­folds that sat­is­fy the ob­vi­ous to­po­lo­gic­al con­di­tions. Yasha con­trib­uted to both the flex­ible and ri­gid sides but his own view of what is flex­ible versus ri­gid changed over time. For ex­ample, he ex­plained that when he first heard about Smale’s h-cobor­d­ism the­or­em, he thought this was an ex­ample of ri­gid­ity in to­po­logy. In­deed, the proof in­volved al­geb­ra­ic struc­tures like ho­mo­logy and tech­niques like modi­fy­ing the dif­fer­en­tial via row re­duc­tion. But Yasha told me he later real­ized that the h-cobor­d­ism the­or­em is an ex­ample of flex­ib­il­ity, and the al­geb­ra­ic tech­niques are a red her­ring. The main point is that if the ob­vi­ous al­geb­ra­ic con­straints are sat­is­fied, then the geo­met­ric res­ult holds (and the proof does in­volve re­du­cing from al­geb­ra­ic to geo­metry via the Whit­ney trick). I found this type of dis­cus­sion about the place of a par­tic­u­lar res­ult in the math­em­at­ic­al uni­verse in­spir­a­tion­al, typ­ic­al of Yasha’s big-pic­ture think­ing.

The au­thor re­ceived his PhD from Stan­ford Uni­versity in 2017 un­der the su­per­vi­sion of Yasha Eli­ash­berg. He is cur­rently an as­sist­ant pro­fess­or at the Uni­versity of Mas­sachu­setts Bo­ston.