Celebratio Mathematica

Ed­it­ors’ note: This talk by Mar­tin Schar­le­mann was giv­en at a con­fer­ence hon­or­ing three to­po­lo­gists: Rob Kirby, Abby Thompson and Marty Schar­le­mann him­self. The three are on the same branch of the math­em­at­ic­al “fam­ily tree”: Marty is Rob’s fourth Ph.D. stu­dent, and Abby is Marty’s first and hence also Rob’s “grand­child”. The con­fer­ence “To­po­logy in Di­men­sions 3, 3.5 and 4” was held on June 25–29, 2018 at the Uni­versity of Cali­for­nia, Berke­ley, in joint cel­eb­ra­tion of Thompson’s 60th, Schar­le­mann’s 70th and Kirby’s 80th birth­days. We are grate­ful to Pro­fess­or Schar­le­mann for his per­mis­sion to pub­lish his re­marks on that oc­ca­sion.

In the early 1980s Rob Kirby and Ray Lick­or­ish or­gan­ized a math-in­tens­ive series of to­po­lo­gic­al sum­mers at Cam­bridge Uni­versity. Bill Menasco was among the former stu­dents of Rob who came along. Bill mem­or­ably brought along two things to Cam­bridge that changed my life. One was something called a Macin­tosh com­puter — to my amazement, it was able to draw pic­tures, where­as I am totally in­ept. The oth­er thing Bill brought along was a gradu­ate stu­dent named Abby Thompson. She could draw too, but also had new and cre­at­ive math ideas. I knew im­me­di­ately that she was smart (she was po­lite enough to laugh at my jokes) and she was clearly math­em­at­ic­ally in­de­pend­ent, with her own ideas for try­ing to solve some clas­sic­al prob­lems in knot the­ory.

The fol­low­ing year MSRI had a big to­po­logy pro­gram that in­cluded Bill and me. Rob made sure that Abby could come as well. Sadly, Bill had to re­turn to Buf­falo in the middle of the year, so I took over as Abby’s ad hoc ad­viser. At the end of the year, Abby de­cided to fin­ish her PhD thes­is in Santa Bar­bara, where we were to have a spe­cial year in to­po­logy, rather than fol­low Bill to Buf­falo. I’m sure that the weath­er — Santa Bar­bara vs. Buf­falo — had noth­ing to do with it.

The to­po­logy year at UC­SB was ex­cit­ing and fun, with lots of emer­ging young to­po­lo­gists, sev­er­al of whom are at this con­fer­ence. Al­most every day we would gath­er for a big lunch and talk about math. One day, when I went by Abby’s of­fice to get her for lunch, I found that she was on the phone with Rob. She seemed re­mark­ably com­fort­able with the con­ver­sa­tion, and I men­tioned to her af­ter­wards how great that was, since Rob still scared me a bit — after all he was my ad­viser. As I nattered on about this Abby in­ter­rup­ted: “Wait…that wasn’t Rob Kirby, that was my Uncle Rob.”

Abby’s uncle Rob was a friendly and in­ter­est­ing man who was then ar­ran­ging to vis­it Abby for some months in Santa Bar­bara. He ad­ded a lot to the am­bi­ence, ar­ran­ging for Abby to give a cello con­cert, etc. One even­ing at a party at our house, he told me how proud he was of how well Abby was do­ing, es­pe­cially since, ac­cord­ing to fam­ily lore, she had been an “ac­ci­dent”. Wow, I thought, that’s pretty cool, since I had learned some years be­fore that I, too, was an ac­ci­dent. So the next day at lunch, sur­roun­ded by a horde of to­po­lo­gists, I men­tioned to Abby how in­ter­est­ing it was that we had that ori­gin story in com­mon. You can guess the up­shot: up un­til that mo­ment Abby had nev­er heard that bit of fam­ily lore.

She did for­give me for blab­bing at lunch (but maybe won’t for bring­ing it up again here!) and, al­most en­tirely on her own, fin­ished her thes­is. So, at least un­of­fi­cially, she be­came my first suc­cess­ful PhD stu­dent. Even­tu­ally we began a dec­ades long math col­lab­or­a­tion. Mod­esty for­bids me from giv­ing our joint math­em­at­ic­al work the praise it de­serves; mostly she was the ar­chi­tect, think­ing up ima­gin­at­ive ap­proaches to old prob­lems, and I was the plumb­er, work­ing to make sure the tech­nic­al ar­gu­ments held wa­ter.

One piece of her own work really does need singling out: after hear­ing Hyam Ru­bin­stein give a talk on how he be­lieved the 3-sphere could be re­cog­nized, Abby in­ven­ted her own ar­chi­tec­ture for such a proof, a plan that com­bined emer­ging ideas in 3-man­i­fold to­po­logy such as thin po­s­i­tion and al­most nor­mal sur­faces. Fur­ther­more, she was able to nail all the tech­nic­al de­tails. She wrote up her proof in a way that con­vinced every­one, and the 3-sphere re­cog­ni­tion prob­lem was fi­nally put to rest. It is in­ter­est­ing to con­tem­plate that if it wer­en’t for an ac­ci­dent 60 years ago, we might still be un­sure wheth­er we could re­cog­nize the 3-sphere!

Abby’s work now con­tin­ues, cur­rently in 4-man­i­folds, and now in col­lab­or­a­tion with my own thes­is ad­viser Rob Kirby. Rob thus con­tin­ues his re­mark­able re­cord of col­lab­or­a­tion and ment­or­ship as he enters his ninth dec­ade. Yikes! I’ll nev­er live up to that sort of role mod­el, but I’m de­lighted to be here to cel­eb­rate with all of you, in­clud­ing Abby and Rob’s on­go­ing con­tri­bu­tions to math­em­at­ics, with a spe­cial ap­pre­ci­ation of the dif­fer­ence that both of them have made to my life and ca­reer.

Mar­tin Schar­le­mann re­ceived his Ph.D. in math­em­at­ics from UC Berke­ley in 1974. Fol­low­ing a year at the In­sti­tute for Ad­vanced Stud­ies and a year at the Uni­versity of Geor­gia, he moved to UC Santa Bar­bara, where he is now a Dis­tin­guished Pro­fess­or Emer­it­us.