Celebratio Mathematica

I first met Yasha when he vis­ited Prin­ceton for a semester and taught an un­der­gradu­ate to­po­logy class. I had en­joyed ideas like com­pact­ness in be­gin­ning ana­lys­is, and was ex­cited to learn more point-set to­po­logy. Yasha an­nounced on the first day that a semester was really more point-set to­po­logy than we needed as un­der­gradu­ates, so we would spend a third of the course on point-set to­po­logy, a third on the fun­da­ment­al group, and a third on dif­fer­en­tial to­po­logy. Already, from just the course out­line, I had learned something use­ful.

Some stu­dents were con­cerned that they had not stud­ied any al­gebra, so were not ready for al­geb­ra­ic to­po­logy. Yasha as­sured them that the fun­da­ment­al group was a fine place to learn the defin­i­tion of a group. That turned out per­haps not to be uni­ver­sally ac­cur­ate, but the idea that not know­ing the defin­i­tions shouldn’t stop you from think­ing about the ob­jects was again a new and use­ful idea. He did an ex­cel­lent job of giv­ing visu­al in­tu­ition for the sub­ject, and I think every­one in the class re­mem­bers his dis­cus­sion of the wind­ing num­ber, in which he seemed to be stir­ring a huge witch’s cauldron. Part­way through the semester, I looked at his re­search in­terests, and saw he was a sym­plect­ic to­po­lo­gist. We had just learned about sim­pli­cial com­plexes, so I, of course, mis­un­der­stood that as sim­pli­cial to­po­logy. I was sur­prised such a dy­nam­ic per­son would be work­ing on a top­ic that felt so ri­gid, and con­cluded that sim­pli­cial com­plexes must be more flex­ible than it ap­peared.

When I was de­cid­ing where to go to gradu­ate school, Yasha was again vis­it­ing Prin­ceton, this time at the In­sti­tute. I talked with him about my vari­ous op­tions, which in­cluded Stan­ford. He as­sured me they were all good op­tions (though it was clear he felt Stan­ford would be the best), but asked me to come back and tell him what I de­cided. Per­haps partly be­cause I couldn’t get my mind around telling him — in per­son — that I had de­cided to go some­where else, I chose Stan­ford. It turned out to be a good choice for me; the warm at­mo­sphere Yasha helped cre­ate was one key reas­on why.

When I ar­rived at Stan­ford, Yasha had just got­ten an FRG grant1 to work on ap­plic­a­tions of \( J \)-holo­morph­ic curves to low-di­men­sion­al to­po­logy. So, he sug­ges­ted I read and give some talks about Oz­sváth and Szabó’s “Holo­morph­ic disks and to­po­lo­gic­al in­vari­ants” pa­pers. The pa­pers ap­ply the con­struc­tion of Lag­rangi­an in­ter­sec­tion Flo­er ho­mo­logy to a pair of sub­man­i­folds that are only totally real, by us­ing a clev­er ar­gu­ment to prove an en­ergy bound. This seemed im­port­ant to me, but when I star­ted to ex­plain it, Yasha cut me off, in­sist­ing that the pair of sub­man­i­folds must be Lag­rangi­an for an ap­pro­pri­ate sym­plect­ic form. I was con­fid­ent that if that were the case, the au­thors would have said so. (Yasha’s view was per­haps based partly on the fact that totally real sub­man­i­folds are not ri­gid enough for an in­vari­ant like Flo­er ho­mo­logy to be defined, typ­ic­ally.) Even­tu­ally, we agreed to dis­agree. A few years later, Per­utz gave an el­eg­ant con­struc­tion of sym­plect­ic forms for which these sub­man­i­folds are, in­deed, Lag­rangi­an. One les­son I took from this was that ex­cel­lent math­em­aticians (in this case, Oz­sváth and Szabó) hav­ing thought about a prob­lem should not de­ter me from think­ing about it, too.

I star­ted meet­ing weekly with Yasha soon after I ar­rived at Stan­ford. On the second or third such oc­ca­sion, I sug­ges­ted we skip our meet­ing, be­cause I felt I had not ac­com­plished much that week. Yasha said we should meet any­way, be­cause once you skip one meet­ing you start skip­ping lots of them; and that if I didn’t have any­thing to say some weeks, he would tell me things. He was ab­so­lutely right on both points. Over the next few years, in ad­di­tion to help­ing me with my work, he told me a lot of in­ter­est­ing ideas. Many of them have since ap­peared in the work of vari­ous au­thors, some more than a dec­ade later.

Also in my first year at Stan­ford, I got quite sick. I did not men­tion it to the fac­ulty, but Yasha no­ticed, and checked with me sev­er­al times: he spe­cific­ally asked if I was get­ting the med­ic­al at­ten­tion I needed, and if there was any­thing he or the de­part­ment could do to help. I don’t re­mem­ber the words he used, but I do re­mem­ber that it felt caring without be­ing in­trus­ive. His con­cern, and know­ing that I could ask for help if I needed it, meant a lot to me at a dif­fi­cult time.

We had a weekly sym­plect­ic geo­metry sem­in­ar on Monday af­ter­noons. If at­tend­ance was low, Yasha would dart around the de­part­ment re­mind­ing gradu­ate stu­dents about it. Once we were gathered and the talk had star­ted, Yasha would al­most al­ways ap­pear to have fallen asleep. At the end of the talk, he would al­ways wake up and ask ques­tions. Typ­ic­ally, they were in­cis­ive ques­tions about ma­ter­i­al from well in­to the part he had slept through. (Once in a while, they would be in­cis­ive ques­tions about a talk ap­par­ently only he had at­ten­ded.) I as­pire to this level of un­der­stand­ing of some math­em­at­ic­al top­ic.

I was quite fond of sem­in­ar din­ners. The sub­sid­ized food was ap­peal­ing, but the main high­light was the fact that some­times Yasha and the speak­ers would start shar­ing stor­ies. Yasha saw that I en­joyed the meals, and would some­times look at me and smile, al­most as if we were shar­ing a joke, when men­tion­ing that there would be a din­ner. It made me feel a little ri­dicu­lous, but also seen.

Since I gradu­ated in 2006, I have not worked with Yasha for any ex­ten­ded peri­od, though I have been for­tu­nate to con­sult with him about re­search and hear his new ideas from time to time. I have also turned to him for pro­fes­sion­al ad­vice sev­er­al times since then. Al­though he is busy with new stu­dents, postdocs, and col­lab­or­at­ors, he has al­ways made time for me, paus­ing to think about the de­tails of the situ­ation and what op­tions I might have. In all of our in­ter­ac­tions, it strikes me how fo­cused Yasha is on talk­ing to me — think­ing about my back­ground, my in­terests, and what would help me. In that sense, maybe his ap­proach to ad­vising is like his ap­proach to math­em­at­ics: look­ing at is­sues from many per­spect­ives and fo­cus­ing on the ones most use­ful in the mo­ment.

Robert Lip­shitz was a PhD stu­dent at Stan­ford from 2002 to 2006, and is cur­rently a fac­ulty mem­ber at the Uni­versity of Ore­gon. His re­search fo­cuses on ap­ply­ing ideas from sym­plect­ic geo­metry to prob­lems in low-di­men­sion­al to­po­logy.