Celebratio Mathematica

Martin Scharlemann

A conversation with Martin Scharlemann

by Evelyn Lamb

Martin Scharlemann at the University of California, Berkeley, in 1999.

Marty Schar­le­mann was born in 1948. He re­ceived his un­der­gradu­ate de­gree from Prin­ceton Uni­versity in 1969 and his PhD from the Uni­versity of Cali­for­nia, Berke­ley in 1974 un­der the su­per­vi­sion of Ro­bi­on Kirby. He spent most of his ca­reer at the Uni­versity of Cali­for­nia, Santa Bar­bara, with vis­it­ing po­s­i­tions at sev­er­al oth­er in­sti­tu­tions, in­clud­ing the In­sti­tute for Ad­vanced Study (IAS), In­sti­tut des Hautes Études Sci­en­ti­fiques (IHES), Uni­versity of Texas at Aus­tin, Math­em­at­ic­al Sci­ences Re­search In­sti­tute (MSRI), and the Re­search In­sti­tute for Math­em­at­ic­al Sci­ences in Kyoto, Ja­pan. His re­search has probed sev­er­al areas of low-di­men­sion­al to­po­logy, in­clud­ing Hee­gaard split­tings, knot the­ory, and three-man­i­folds.

The fol­low­ing is a con­densed and ed­ited ver­sion of the au­thor’s con­ver­sa­tion with Schar­le­mann in June 2020, about his col­lab­or­a­tions with Abi­gail Thompson and Mi­chael Freed­man.

Lamb: How did you start work­ing with Abi­gail Thompson?

Schar­le­mann: Rob Kirby and Ray Lick­or­ish or­gan­ized these sum­mer meet­ings of to­po­lo­gists in Cam­bridge, Eng­land. And so on one of these — and I’ve for­got­ten ex­actly the year1  — Bill Menasco, who was also a Kirby stu­dent, brought along a gradu­ate stu­dent, Abby Thompson.2 I star­ted talk­ing to her there about knot the­ory, which we were both in­ter­ested in. She was work­ing on her dis­ser­ta­tion, mostly with Bill Menasco. (Her thes­is ad­visor, Ju­li­us Shaneson, was in­ter­ested in oth­er areas.) At some point, we both ended up at the MSRI in Berke­ley.3 Bill was trans­fer­ring over to SUNY Buf­falo and worked with Abby for half the year. Then I came and he asked me if I would work with her for the next half be­cause we were in­ter­ested in many of the same things. And so I did, and I think she fin­ished her thes­is ba­sic­ally there. I was kind of her ad hoc ad­visor. She was in­ter­ested in some of the well-known and deep prob­lems in knot the­ory, things like prop­erty P and the be­ha­vi­or of knots with cross­ing changes and band sums. At the end of the year at MSRI, she had the prob­lem of wheth­er to go to Buf­falo and work with Bill, or we were hav­ing a spe­cial year in to­po­logy at UC­SB: the to­po­lo­gists were al­lowed to hire all the lec­tures for the de­part­ment that year, with the hope that we would get a good group to­geth­er. And so she, in fact, came to Santa Bar­bara. (I’m sure that the dif­fer­ence between the weath­er in Santa Bar­bara and Buf­falo had noth­ing to do with her de­cision!)

Summer topology meeting in Cambridge (1984). Front row (left to right): Erica Flapan, Dusan Repovs, Abby Thompson, Tim Cochran, Ken Millett, Paulo Ney de Souza. Back row: Bill Menasco, Rae Mitchell, John Calmus, Rob Kirby, Ric Ancel, Mark Feighn, Raymond Lickorish, Marty Scharlemann, Larry Siebenmann, Ric Litherland, Steve Boyer, Bob Edwards, Dennis Barden.
Photo courtesy of Rob Kirby.

Be­cause we were in­ter­ested in the same things, and she had lots of good ideas, we star­ted writ­ing pa­pers to­geth­er. Even­tu­ally, she ended up at UC Dav­is, which is a beau­ti­ful train ride from Santa Bar­bara.

Lamb: Your first pa­per with her is called “Un­knot­ting num­bers, genus, and com­pan­ion tori” [1]. What is that about?

Schar­le­mann: I had be­come very in­ter­ested in something called su­tured man­i­fold the­ory that was de­veloped by Dave Gabai [e5]. I wanted to do it in a slightly dif­fer­ent way, a more com­bin­at­or­i­al way. Dave used fo­li­ations, which didn’t strike me as be­ing at the heart of things, and it wasn’t an area I un­der­stood well. And so I ul­ti­mately was able to do that. Abby had used some the­or­ems in work she did be­fore that [e6], tak­ing some of the out­put of su­tured man­i­fold the­ory to ad­dress prob­lems in knot the­ory. One of the ways that this enters in is in think­ing about cross­ing changes in knots or about band moves. Cross­ing changes are in­tu­it­ive. You just change your cross­ing. Band moves are a little bit more ex­cit­ing. You take a rib­bon from one place in the knot or link and con­nect it to an­oth­er. And then you cut out the middle of the rib­bon.

I think it was in the earli­er pa­per that she real­ized that the the­or­ems that Gabai was cre­at­ing were very rel­ev­ant to ques­tions about band sums and cross­ing changes. And so that first pa­per was an ex­plor­a­tion of that top­ic, how to use su­tured man­i­fold the­or­ems to say things about cross­ings and band sums.

Rob Kirby, Abby Thompson and Marty Scharlemann (left to right) at “Topology in Dimensions 3, 3.5, and 4,” a conference held in their honor at the University of California, Berkeley, June 25–29, 2018.
Photo by Sheila Newbery.

Lamb: I saw that solv­ing the graph planar­ity prob­lem [2] seems to be one of the most im­port­ant prob­lems from your col­lab­or­a­tion. Can you tell me about that?

Schar­le­mann: That came late in our col­lab­or­a­tion, but it was one of the most pleas­ing [pro­jects] be­cause you can ex­plain to any­one what the prob­lem is. You’re giv­en a graph in three-space: ver­tices and arcs con­nect­ing them. And you ask the ques­tion “Can you move it, in three-space, so that it lies on a plane?” There are two as­pects of that. One is the graph prob­lem, which was solved by Kur­atowski [e1]: a graph can­not be put in a plane, even ab­stractly, if and only if it con­tains one of ex­actly two con­fig­ur­a­tions. But bey­ond that, then you ask, “Well, once we know that the graph it­self can be em­bed­ded, can it be moved to the plane in three-space?” And that’s what we were ad­dress­ing.

The simplest ex­ample of that might be just the graph con­sist­ing of a single ver­tex and a single edge, a loop. Of course, that is a planar graph: you just put it in a plane. But it could start out in three-space em­bed­ded as a knot. So then it be­comes the prob­lem: how do you de­cide wheth­er, or can you de­cide wheth­er, a knot in three-space is un­knot­ted, that it can be put in the plane? Well, that was solved by Papakyriako­poulos in the 1950s [e2]. And so our the­or­em could be viewed as an ex­ten­sion of that work, as well. It was a lot of fun. We fairly soon found an idea that seemed very prom­ising via an in­duct­ive step, but it only worked on fairly large graphs. So there was a gen­er­al idea, but then much of the work came in hand­ling the smal­ler graphs as we got closer and closer to the smal­lest in­ter­est­ing one, which would be the Papakyriako­poulos res­ult. Even­tu­ally we got it, and we were very pleased with it.

One of the stor­ies as­so­ci­ated with that is there was ac­tu­ally a bet between two knot ex­perts, Camer­on Gor­don and Jon Si­mon, who pro­posed the prob­lem. It was his con­jec­ture that we proved. They bet \$20 on wheth­er it would ever be solved. Camer­on be­lieved it would nev­er be solved, and Jon bet against that be­cause he was try­ing to en­cour­age people to do it. I think that Jon Si­mon even­tu­ally col­lec­ted his debt, but you’d have to ask him that.

Lamb: So I guess that wasn’t a bet that ac­tu­ally re­mu­ner­ated the people who solved it.

Schar­le­mann: Yeah, where’s our cut?

Lamb: This might be show­ing my naïveté, but look­ing through your pub­lic­a­tions, I see a lot that I would nat­ur­ally think of as be­ing to­po­logy, but I would have thought of graph planar­ity as be­ing more in the graph the­ory/com­bin­at­or­ics world. So how is this con­nec­ted to what we might think of as clas­sic low-di­men­sion­al to­po­logy?

Schar­le­mann: There is a way of look­ing at the struc­ture of three-man­i­folds called Hee­gaard split­tings, in which you start with what’s called a handle­body. A handle­body looks roughly like a thickened-up bou­quet of circles. And it’s an ele­ment­ary the­or­em, prob­ably go­ing back to the 19th cen­tury, that any closed, com­pact three-man­i­fold can be writ­ten as the uni­on of two such things along their bound­ary.

One way of think­ing about a handle­body is by think­ing of a neigh­bor­hood of a graph. Take any fi­nite graph, put it in three-space, and thick­en that up. If you sup­pose the graph lies in a plane and you thick­en it up, and then you ask, “What does the rest of three-space look like?” it doesn’t take long be­fore you dis­cov­er, well, that’s also the neigh­bor­hood of a graph. And so you’ve de­scribed in that way an ac­tu­al Hee­gaard split­ting of the three-sphere. One part is the handle­body ly­ing near the plane, the oth­er is its com­ple­ment. A kind of fun­da­ment­al the­or­em of Hee­gaard split­tings, due to Wald­hausen [e3], is that for the three-sphere, that’s it: any Hee­gaard split­ting of the three-sphere looks like that. So one up­shot of that the­or­em, if you try and think about it in terms of graphs in three-space, is that if you have a graph sit­ting in three-space, and you know that its com­ple­ment also looks like a thickened-up graph, then you have a Hee­gaard split­ting of the three-space. Wald­hausen tells you that you can take this graph sit­ting in three-space, and if its com­ple­ment is right, then you can slide it around, edges over edges and all that sort of thing, un­til fi­nally it be­comes a nice planar graph sit­ting in the plane. What Jon Si­mon asked was, “What if you don’t al­low your­self to slide edges over edges?” If you just leave it as a graph, it’s ob­vi­ously a more dif­fi­cult ques­tion, and so then the con­jec­ture was if you know that every sub­graph can be put in the plane, and you know that the com­ple­ment is right per Wald­hausen’s the­or­em, and every sub­graph of the ori­gin­al graph has the prop­erty that it can be put in­to a plane — then you can put the graph in the plane. So there are all these con­nec­tions to Hee­gaard the­ory, which Abby and I were quite in­ter­ested in at the time. We worked quite a bit on those kinds of is­sues — Hee­gaard the­ory and knot the­ory. Both of those come to­geth­er in that kind of ques­tion.

Lamb: In your col­lab­or­a­tion with Thompson, did you feel like there was an arc, one big thing you were try­ing to push to, or did you feel like you were al­ways find­ing new prob­lems to work on and just kept me­an­der­ing?

Schar­le­mann: I would say it’s more of the lat­ter. We had tech­niques and were think­ing about prob­lems where they might be ap­plic­able. For ex­ample, this graph planar­ity thing. If my memory is cor­rect, at first we star­ted think­ing about ways we might ap­ply su­tured man­i­fold the­ory to that set­ting. In the end, it wasn’t ne­ces­sary. That the­or­em could have been proven without su­tured man­i­fold the­ory, and that’s the way we ended up do­ing it. So it could have been proven dec­ades ago. I don’t think there was any­thing re­mark­ably mod­ern about the tools we ended up us­ing. It was just put­ting them to­geth­er right.

Lamb: How did you work to­geth­er in this col­lab­or­a­tion — what kind of give and take did you have, how did you bounce ideas off of each oth­er?

Schar­le­mann: I think of her as the ar­chi­tect, and I was the plumb­er. She had these ideas; some of them struck me as com­pletely off the wall, and some of them seemed like they could prob­ably work. (There seemed to be no con­nec­tion between my ini­tial im­pres­sion and how things turned out.) But I have to con­fess I get really ex­cited about the tech­nic­al de­tails of ar­gu­ments and try­ing to fig­ure out ex­actly how to nail things down, and she’s more of the idea per­son. So we kind of worked in that way: she would pitch ideas, and I would try and fig­ure out wheth­er there was an ap­proach from, say, su­tured man­i­fold the­ory. Our roles switched back and forth, but I would say pre­dom­in­antly it was like that.

We had a lot of luck. We got a lot of stuff done, and I was de­lighted at how eas­ily it worked. It seemed like most things we star­ted think­ing about went some­where, and some­where in­ter­est­ing.

Lamb: That is lucky!

Schar­le­mann: As I’m sure you know, 99 per­cent of the things you try in math­em­at­ics don’t work, and some­how you don’t get cred­it for all those great ideas that don’t work!

Lamb: Right. It seems very un­fair. Glan­cing through your pub­lic­a­tion list, there’s a mix of single-au­thor pa­pers and things that you’ve writ­ten with col­lab­or­at­ors. How do you feel like the roles of col­lab­or­a­tion and work­ing by your­self have worked in your ca­reer?

Schar­le­mann: I am prob­ably more of a sol­it­ary work­er than is the norm now, though I think when I star­ted my ca­reer it was more of a norm to work on your own. I like to think in­tens­ively in ways that I of­ten have a very hard time ex­plain­ing to people. In par­tic­u­lar, I can’t draw, and this is a real draw­back if you’re a to­po­lo­gist. I can see pic­tures per­fectly in my head, but I can’t draw, and that makes com­mu­nic­a­tion dif­fi­cult. But one of the nice things about col­lab­or­at­ing with Abby was that she could draw. That’s kind of a stu­pid com­ment, but she also had a way of rein­ing me in when I star­ted bor­ing deep­er than was ne­ces­sary for a giv­en prob­lem, kind of bring­ing me back to real­ity and say­ing, “Why don’t we try this in­stead?” It of­ten seemed that I was en­joy­ing think­ing hard about prob­lems in ways that turned out not to be very pro­duct­ive. She was very help­ful in get­ting me back on track. And she had lots of ideas that would nev­er have oc­curred to me, and then I would get ex­cited about ex­actly how to make those ideas con­crete. A lot of that was sol­it­ary work, but we would re­turn and start talk­ing about things, and it of­ten worked out. I def­in­itely get a lot of en­joy­ment in just think­ing about things hard, walk­ing around and think­ing about them. So I prob­ably am not as col­lab­or­at­ive as many.

Lamb: And how did you start work­ing with Mi­chael Freed­man?

Schar­le­mann: I ran in­to him in his first job, which was at Berke­ley. He ar­rived my fi­nal year in gradu­ate school. We were both in­ter­ested in four-man­i­fold stuff. I fol­lowed with great in­terest what he was do­ing, but we didn’t col­lab­or­ate at that time. The pa­pers I’ve writ­ten with him have been from the past five years. But we talked a lot in the in the 1980s. Over the years, I would see him at con­fer­ences, and we’d some­times have con­ver­sa­tions about four-man­i­folds. Freed­man is in­ter­ested in all kinds of stuff, and he re­mem­bers it very well, too. So my over­lap with him be­came nar­row­er and nar­row­er in terms of his broad­en­ing in­terests. I don’t know if we even ever came close to work­ing on a pa­per to­geth­er [earli­er in our ca­reers]. We would bounce ideas off each oth­er, but it nev­er went any­where, though I learned a lot and I hope he learned some.

As to our re­cent col­lab­or­a­tion, he be­came very in­ter­ested in phys­ics and in quantum the­ory, quantum to­po­logy, et cet­era, which I nev­er really warmed to. At some glor­i­ous mo­ment, he ended up in Santa Bar­bara, not as a math­em­atician in the math de­part­ment, but con­nec­ted to the Mi­crosoft quantum com­put­ing lab. I kind of smile at that be­cause it took way, way too long for us to be able to get things set up so that Mike could be as­so­ci­ated with the de­part­ment. But now he’s nom­in­ally a mem­ber of the UC­SB Math­em­at­ics De­part­ment. So we were geo­graph­ic­ally in the same place, and we talked every once in a while about stuff. You know, I have to say, sit­ting in my of­fice work­ing on these prob­lems, I’d feel a little bit like a mole un­der­ground, and then Mike would walk in, and he’d have all these ideas, and it was like the sun sud­denly shin­ing in­to my tun­nel. And it was it was kind of fun.

But there was a ques­tion called the Pow­ell con­jec­ture. Maybe five or six years ago, Mike got very in­ter­ested in this con­jec­ture be­cause he saw it as con­nec­ted to four-man­i­fold the­ory. The ques­tion that was in­volved there was a kind of ana­log to ques­tions about braid the­ory in three-man­i­fold the­ory. He thought it could be quite im­port­ant in four-man­i­fold the­ory, and so we star­ted talk­ing about that. He sent me over a sketch of a way he thought that you might be able to prove it. I was in the pro­cess of send­ing back an email ex­plain­ing to him why I didn’t think that ap­proach could pos­sibly work when I star­ted think­ing, “Wait a minute! Maybe it could.” And so that’s what launched it. Over the years I think we’re get­ting fairly close to something that looks like proof. It’s still un­clear wheth­er it’s go­ing to all work out, but along the way, we’ve proved some nice stuff that gave rise to the pa­pers you can see. Oth­er stuff is in the 99 per­cent that didn’t work, and so there’s kind of no re­cord of that. But it’s an on­go­ing pro­ject.

Mike Freedman speaks at the conference “Topology in Dimensions 3, 3.5, and 4” at UC Berkeley in 2018.
Photo by Kara Stamback.

There’s an in­ter­est­ing back­ground story to the Pow­ell con­jec­ture. Jerome Pow­ell — not the chair of the Fed­er­al Re­serve, but someone with the same name — was a stu­dent of Joan Birman in the early 1980s and proved a nice the­or­em for his thes­is [e4]. I men­tioned Wald­hausen’s the­or­em about how Hee­gaard split­tings of \( S^3 \) look, but you can ask a sort of second-or­der mov­ing pic­ture ques­tion. Giv­en that you have a Hee­gaard split­ting of \( S^3 \), as Wald­hausen de­scribed, sup­pose you just sort of throw it up in space, and then it comes back down and lands in the same place, but maybe it’s moved around or changed dur­ing that pro­cess. And the ques­tion is, are there simple moves that gen­er­ate all pos­sib­il­it­ies of how this handle­body could move in space to re­turn ex­actly to it­self? Pow­ell proved that five par­tic­u­lar moves suf­fice. Many years later, I’d been think­ing about it, and one of the tech­niques that had worked very well in a lot of set­tings, and in par­tic­u­lar with my col­lab­or­a­tion with Abby Thompson, was the no­tion of thin po­s­i­tion. She proved a really im­port­ant the­or­em in three-man­i­folds, the re­cog­ni­tion prob­lem for the three-sphere, us­ing thin po­s­i­tion [e7]. And I thought, “Well, it’d be nice to come up with a mod­ern proof of Pow­ell’s the­or­em that would use thin po­s­i­tion.” It would cer­tainly be easi­er. But I couldn’t fig­ure out how to do it. I would get stuck at a cer­tain point, and I fi­nally de­cided to “cheat” and read Pow­ell to see how he handled this situ­ation. (I hadn’t really read his pa­per care­fully; I just knew the res­ult.) And I dis­covered there was ac­tu­ally a gap, that he had missed something. There seemed to be no way to pro­ceed, and no one knew how to fix it. I spent a lot of time try­ing to fix it us­ing thin po­s­i­tion, but I still couldn’t do it.

Years later, Mike got in­ter­ested in the prob­lem for the reas­ons I’ve de­scribed, and re­membered that I had been in­ter­ested in it for a while, so that’s why he con­tac­ted me. He asked what I thought of his idea. And it’s a ter­rif­ic idea, but it just isn’t quite enough. And we haven’t quite been able to work it out.

Lamb: Are all your pa­pers with Freed­man re­lated to Pow­ell’s con­jec­ture?

Schar­le­mann: They’re all re­lated to the train of thought ini­ti­ated there. One pa­per is a dis­cus­sion of Mike’s big idea of how Pow­ell’s con­jec­ture might be solv­able [3]. It was known to be true for genus two Hee­gaard split­tings. (That goes back to the 1930s.) And one of our con­crete res­ults there was that the Pow­ell con­jec­ture is also true in genus three. It’s not a big thing, but it was at least a con­crete res­ult. We haven’t sub­mit­ted that pa­per for pub­lic­a­tion any­where yet. We view it as more of a dis­cus­sion doc­u­ment. The hope is that we’ll fig­ure out some way to have a happy con­clu­sion to it, but at the mo­ment, it’s not a sat­is­fact­ory pa­per.

At some point we worked on the Dehn’s lemma pa­per [4]. It was Mike’s idea that this might be true, and he really pushed it through. When I first met him back in the 1970s, he was an ex­pert on Dehn’s lemma and just knew it in and out. He real­ized there was a some­what stronger state­ment you could make, and that just floored me. It’s such an old and hon­or­able the­or­em. That there’s a way of mak­ing it even stronger after all this time really sur­prised me.

The pa­per that comes next on my vita is called “A strong Haken’s The­or­em” [5], which solved a prob­lem that goes back to the 1960s. The ar­gu­ment, to­geth­er with ideas that came out of Freed­man’s ori­gin­al ideas for the Pow­ell con­jec­ture, were what led then to the second col­lab­or­at­ive pa­per with Mike Freed­man, “Unique­ness in Haken’s The­or­em” [6]. It’s a weird co­in­cid­ence that the up­grade of Dehn’s lemma was ex­actly the tool that was needed to make the Strong Haken The­or­em work.

Lamb: Do you feel like the style of your col­lab­or­a­tion with Freed­man is sim­il­ar to your col­lab­or­a­tion with Thompson?

Schar­le­mann: There are sim­il­ar­it­ies, but some­how [work­ing with Freed­man] was al­ways a bit more form­al than work­ing with Abby. Part of that prob­ably has to do with the fact that I was her ad­visor, so I kind of felt like the seni­or fig­ure, where­as it’s clear when I work with Mike that he knows sev­er­al or­ders of mag­nitude more than I do. But I would say in both cases, the col­lab­or­a­tion is pretty com­fort­able. And I have to say I am — maybe I shouldn’t be, but I am — con­cerned about wast­ing his time be­cause he’s do­ing so much in so many dif­fer­ent fields. I don’t want to call him up and say, “I just want to talk about this.” I feel like I have to have something to say.

Lamb: How do you feel look­ing back on what you’ve man­aged to do in this field? Are there still big prob­lems that mo­tiv­ate you?

Schar­le­mann: Oh, I really like big prob­lems. But I go to­wards them with the at­ti­tude that I don’t ne­ces­sar­ily ex­pect to solve them. I would like to un­der­stand them, and that of­ten gives rise to oth­er the­or­ems that I end up find­ing really in­ter­est­ing and can solve. I’ll men­tion the Schoen­flies con­jec­ture as an ex­ample. It is just a beau­ti­ful prob­lem, and think­ing about it, try­ing to fig­ure out how you might try vari­ous ap­proaches, has been really pro­duct­ive for my think­ing. I also like tools that I un­der­stand that are visu­al. I get in­ter­ested in how I can use them to try and view dif­fer­ent prob­lems. Su­tured man­i­fold the­ory is an ex­ample. When I learned about that, it seemed like it was kind of on the bor­der­line of something I might be able to un­der­stand if I worked really hard. And that worked out. I did fi­nally un­der­stand it enough to use it in ori­gin­al ways. Thin po­s­i­tion is an­oth­er ex­ample. A lot of the stuff that Abby and I did was a com­bin­a­tion of those two tools. I star­ted out on this Pow­ell con­jec­ture with ex­actly that in mind. I didn’t think the ques­tion it­self was par­tic­u­larly in­ter­est­ing, but I en­joyed try­ing to think about how this new tool of thin po­s­i­tion might be ap­plic­able to Pow­ell’s the­or­em. Well, it went in totally dif­fer­ent dir­ec­tion when it turns out that the the­or­em hadn’t been proven, and I’ve spent a lot of time work­ing on that. It’s not a Schoen­flies con­jec­ture level prob­lem, but it’s a very in­ter­est­ing prob­lem. And it has led to nice pa­pers: a strong Haken the­or­em pa­per with Mike — well, both of the pa­pers are on an at­tempt to solve the Pow­ell con­jec­ture, and the Dehn’s lemma came out of that. So even though we haven’t solved the prob­lem yet, I think we’ve done a lot of nice math­em­at­ics.

No one has ever asked me that ques­tion be­fore and I’ve just made that up in the last five minutes, but I think it’s pretty ac­cur­ate. I get ex­cited by big prob­lems that I can un­der­stand, and I get ex­cited about new tools that I can use.

Lamb: I’ve heard oth­er math­em­aticians say they got a lot of mileage out of know­ing how to use one par­tic­u­lar tool or ar­gu­ment. It sounds like that’s sort of been your ex­per­i­ence, to some ex­tent: “I know how to do this, so let’s see if it will work.”

Schar­le­mann: I think part of the thing that’s in­ter­est­ing in math­em­at­ics is how dif­fer­ent people can be, and still be suc­cess­ful. I think there are quite a few math­em­aticians for whom what I’ve de­scribed wouldn’t be a nat­ur­al way of think­ing about things. That’s one of the ex­cit­ing things about work­ing in it for so long, is see­ing such a vari­ety of per­son­al­ity types and ap­proaches to math­em­at­ics, and see­ing that they can all be suc­cess­ful in dif­fer­ent ways.

Evelyn Lamb is a freel­ance math and sci­ence writer based in Salt Lake City, Utah. She holds a Ph.D. in math­em­at­ics (Rice Uni­versity, 2012) and has writ­ten for such na­tion­al pub­lic­a­tions as Sci­entif­ic Amer­ic­an, Quanta Magazine, Slate, Sci­ence News and Nautilus.


[1] M. Schar­le­mann and A. Thompson: “Un­knot­ting num­ber, genus, and com­pan­ion tori,” Math. Ann. 280 : 2 (1988), pp. 191–​205. MR 929535 Zbl 0616.​57003 article

[2] M. Schar­le­mann and A. Thompson: “De­tect­ing un­knot­ted graphs in 3-space,” J. Diff. Geom. 34 : 2 (1991), pp. 539–​560. MR 1131443 Zbl 0751.​05033 article

[3] M. Freed­man and M. Schar­le­mann: Pow­ell moves and the Goer­itz group. Pre­print, 2018. ArXiv 1804.​05909 techreport

[4] M. Freed­man and M. Schar­le­mann: “Dehn’s lemma for im­mersed loops,” Math. Res. Lett. 25 : 6 (2018), pp. 1827–​1836. MR 3934846 article

[5] M. Schar­le­mann: A strong Haken’s The­or­em. Pre­print, 2020. ArXiv 2003.​08523 techreport

[6] M. Freed­man and M. Schar­le­mann: Unique­ness in Haken’s The­or­em. Pre­print, 2020. ArXiv 2004.​07385 techreport