Celebratio Mathematica

I re­turned to his sphere soon after when I solved a group the­or­et­ic prob­lem he had men­tioned in his talk: Show that a free product of non­trivi­al cyc­lic groups could not be nor­mally gen­er­ated by a prop­er power [e2]. As Camer­on had ob­served, a pos­it­ive solu­tion to the prob­lem would com­plete the proof that man­i­folds ob­tained by non­in­teg­ral sur­gery on a knot in the 3-sphere are ir­re­du­cible [3].

It would be an­oth­er twelve years be­fore we began to work to­geth­er. To de­scribe how that came about, let’s pause to dig a little deep­er in­to the area of ex­cep­tion­al sur­ger­ies.

A knot man­i­fold is a com­pact, con­nec­ted, ori­ent­able, 3-man­i­fold \( M \) whose bound­ary is an in­com­press­ible tor­us. A slope on the bound­ary of a knot man­i­fold \( M \) is a \( \partial M \)-iso­topy class of (nonori­ented) es­sen­tial simple closed curves in \( \partial M \). Slopes cor­res­pond biject­ively with the set of prim­it­ive classes in \( \alpha \in H_1(\partial M) \), con­sidered up to sign, and para­met­rise Dehn fillings \( M(\alpha) \) of \( M \).

A knot man­i­fold is hy­per­bol­ic if its in­teri­or sup­ports a com­plete, fi­nite volume hy­per­bol­ic met­ric. We call a slope on the bound­ary of a hy­per­bol­ic knot man­i­fold \( M \) ex­cep­tion­al if the as­so­ci­ated filling is not hy­per­bol­ic. The set of such slopes is de­noted by \[ \mathcal{E}(M) = \{ \alpha \; | \; M(\alpha) \mbox{ is not hyperbolic}\}. \] By geo­met­risa­tion, a slope is con­tained in \( \mathcal{E}(M) \) if and only if the as­so­ci­ated filled man­i­fold is re­du­cible, tor­oid­al, or an ir­re­du­cible Seifert fibred man­i­fold which con­tains no es­sen­tial tori. The lat­ter fam­ily is con­tained in the set of small Seifert man­i­folds and in­cludes all closed man­i­folds with cyc­lic or fi­nite fun­da­ment­al groups.

Un­der­stand­ing \( \mathcal{E}(M) \) has been of long-stand­ing in­terest, mo­tiv­ated to a great ex­tent by Thur­ston’s hy­per­bol­ic Dehn filling the­or­em, which im­plies that it is al­ways fi­nite. Ba­sic ques­tions are:

• How large can \( \mathcal{E}(M) \) be and how are its ele­ments dis­trib­uted in the sur­gery plane \( H_1(\partial M; \mathbb R) \)?
• What is the to­po­logy of \( M \) when the car­din­al­ity of \( \mathcal{E}(M) \) is re­l­at­ively large?

Res­ults on the first ques­tion are of­ten phrased in terms of the dis­tance between slopes \( \alpha \) and \( \beta \) which, when they are rep­res­en­ted by prim­it­ive ele­ments of \( H_1(\partial M) \), is giv­en by the ab­so­lute value of the al­geb­ra­ic in­ter­sec­tion num­ber: \[ \Delta(\alpha, \beta) = |\alpha \cdot \beta| . \] A pri­ori bounds on the dis­tances between ele­ments of a set of slopes \( \mathcal{S} \) lead to bounds on the car­din­al­ity of \( \mathcal{S} \). For in­stance, a dis­tance bound of 1 im­plies that \( \#\mathcal{S} \leq 3 \).

One of the first ma­jor res­ults in the area was the cyc­lic sur­gery the­or­em, con­tained in the sem­in­al pa­per [2] that Camer­on coau­thored with Marc Cull­er, John Luecke, and Peter Shalen.

This pa­per was im­port­ant as much for its meth­ods as its res­ults. Its first half con­tained Marc and Peter’s beau­ti­ful rep­res­ent­a­tion-the­or­et­ic meth­ods de­signed to deal with the case that \( M \) is hy­per­bol­ic, while its second half con­tained Camer­on and John’s power­ful de­vel­op­ments of the in­ter­sec­tion graph meth­od to deal with the non­hyper­bol­ic case. This is an overly simplist­ic as­sess­ment of the roles each half plays in the pa­per though. For in­stance, there are situ­ations where the graph the­or­et­ic meth­ods of Camer­on and John are needed when \( M \) is hy­per­bol­ic. See Peter Shalen’s art­icle Camer­on and the Cyc­lic Sur­gery The­or­em: a per­son­al ac­count on this site for a fuller dis­cus­sion of the pa­per.

After Xin­gru and I com­pleted our pa­per [e4], Camer­on in­vited me to Aus­tin in Feb­ru­ary 1995. The day I ar­rived, Camer­on and his wife Sue took me out for mar­gar­itas at a ter­raced res­taur­ant (The Oas­is) high over Lake Trav­is, where I rev­elled in the con­trast between the Ca­na­dian winter morn­ing that I had left and the warm, breezy sun­shine of an Aus­tin af­ter­noon.

I had a bril­liant time that week, as I have on each sub­sequent vis­it. There was plenty of math­em­at­ics to dis­cuss, not only with Camer­on, but also with John Luecke and Alan Re­id. And all of it was fun. The math­em­at­ics flowed back and forth with hanging out, listen­ing to mu­sic, and dis­cuss­ing mu­tu­al in­terests, all en­hanced by good food, drink and laughter. Sue was the ever-gra­cious host and friend, as was Alan’s wife Mara dur­ing later vis­its.

In the late 1990s, Xin­gru and I had mapped out an ap­proach to prov­ing Con­jec­ture 3.1 when \( M \) had first Betti num­ber 2 or more. A case arose that we felt should be doable us­ing graph in­ter­sec­tion tech­niques, but as the de­tails were bey­ond us, we ap­proached Camer­on. His im­me­di­ate re­sponse was yes, his meth­ods did ap­ply, and after he worked out the finer points we had a sketch of a com­plete proof. There were some chal­len­ging gaps to over­come though, but in the end, the three of us com­pleted the pa­per [8].

At this point, Camer­on, Xin­gru and I took on the chal­lenge of show­ing that the dis­tance of a re­du­cible filling slope \( \beta \) to a fi­nite filling slope \( \alpha \) on the bound­ary of a hy­per­bol­ic knot man­i­fold \( M \) was 1, which was con­sist­ent with em­pir­ic­al evid­ence. The res­ults of [e12] had re­duced us to the case that \( M(\alpha) \) and \( M(\beta) \) were very spe­cif­ic man­i­folds, while \( M \) ad­mit­ted an es­sen­tial planar sur­face with four bound­ary com­pon­ents which split \( M \) in­to two pieces with very simple to­po­logy.

We had ar­ranged to work on this prob­lem at a con­fer­ence in Beijing (June 2007). On his flight over, Camer­on real­ised that our con­straints on \( M \) im­plied that it ad­mit­ted an in­vol­u­tion with quo­tient the 3-ball and branch set a two strand tangle. The in­vol­u­tion ex­ten­ded over each Dehn filling \( M(\gamma) \) of \( M \) with quo­tient \( S^3 \) and branch set a link \( L_\gamma \subset S^3 \). Know­ing the to­po­logy of \( M(\alpha) \) and \( M(\beta) \) al­lowed us to de­term­ine \( L_\alpha \) and \( L_\beta \), and com­par­ing them led us to the dis­tance bound of 1. Of course there were de­tails to be sor­ted out, but we over­came the fi­nal hurdles and the pa­per ap­peared a few years later [9].

While still in Beijing, Camer­on, Xin­gru and I real­ised that we might be able to com­bine the tech­niques of [e12] with the in­vol­u­tion tech­nique we were dis­cuss­ing to veri­fy the case of Con­jec­ture 3.1 con­cerned with the dis­tance of a small Seifert filling slope to a tor­oid­al filling slope. More pre­cisely, we hoped to prove this:

Our idea was to use [e12] to re­duce Con­jec­ture 3.2 to the case where \( M \) ad­mit­ted an es­sen­tial genus 1 sur­face \( F \) which split it in­to two re­l­at­ively simple to­po­lo­gic­al pieces, ana­log­ous to what was done in [9]. Set­ting \( m \geq 1 \) to be the num­ber of bound­ary com­pon­ents of \( F \), we knew that when \( m \) was 1 or 2 we could of­ten find an in­vol­u­tion on \( M \), which should be use­ful. This ap­proach failed when \( m \geq 3 \), since \( F \) would not be a “mutat­ing” sur­face, but we hoped that the im­mer­sion graph meth­od could be re­fined to pick up the slack. I don’t think we real­ised how much work this pro­ject would in­volve, cer­tainly I didn’t, and in the end we had to throw everything but the kit­chen sink at it.

First we con­sidered \( m \geq 3 \). Though Camer­on’s ex­per­i­ence with in­ter­sec­tion graphs was es­sen­tial in al­low­ing us to wring the most out of im­mer­sion graphs, his con­tri­bu­tion which sticks out most in my mind oc­curred in the last case we had to con­sider. While dis­cuss­ing it dur­ing a vis­it to Aus­tin, we re­duced the prob­lem to un­der­stand­ing the prop­er­ties of a cer­tain group \( G \). At some point, Camer­on went off to his of­fice to think and came back a little while later to say that we could ap­ply some Tiet­ze trans­form­a­tions to con­vert a nat­ur­al present­a­tion of \( G \) to one with a very par­tic­u­lar re­lat­or. The re­mark­able thing was that he had re­membered that Ed­jvet and How­ie had stud­ied groups with that ex­act re­lat­or some twenty years earli­er, and he thought that their con­clu­sions would be rel­ev­ant to our prob­lem. In short or­der we com­bined their con­clu­sions with some work of Jonath­an Hill­man and some of Xin­gru and mine to com­plete the proof when \( m \geq 3 \) and \( \beta \) was a strict bound­ary slope (e.g., \( M(\beta) \) was neither a tor­us bundle nor a tor­us semibundle). See [10].

Next up was \( m = 1 \), where we were able to re­duce to the situ­ation to when \( M \) ad­mits an in­ter­est­ing in­vol­u­tion. It would be in­cor­rect to say that the re­duc­tion was simple; pro­du­cing the de­tails re­quired a lot of work. Nev­er­the­less, we com­pleted the pa­per in a pre­cise and sat­is­fy­ing way. See [12].

The wheels came off the pro­gramme when we turned to the case \( m = 2 \). We re­duced the prob­lem to the situ­ation that \( F \) was a sep­ar­at­ing non­semi­fibre after a fair amount of work. This was what oc­curred for the fig­ure eight knot ex­ter­i­or and as such, we sus­pec­ted it would be the most chal­len­ging part of the pa­per. Nev­er­the­less, by late fall of 2012 we thought that we were done. Camer­on and I had ar­ranged a two week Re­search in Pairs vis­it to Ober­wolfach in March 2013 to go over the fi­nal de­tails. Our mood, which was tri­umphant and self-con­grat­u­lat­ory, in­stant­an­eously dis­in­teg­rated when we found an er­ror in the ar­gu­ment. (It didn’t help that we were the only hu­man be­ings on the MFO cam­pus over that dark wintry week­end.) Our ana­lys­is was para­met­erised by an in­teger \( d \geq 0 \) as­so­ci­ated to the knot man­i­fold \( M \) and we found a gap in the case that \( d = 1 \), the case that oc­curred for the fig­ure eight ex­ter­i­or! We were stumped, and in spite of some fit­ful at­tempts over the next few years, we de­jec­tedly weaned ourselves off the prob­lem. It re­mained, however, the ever-present ele­phant in the room, fre­quently al­luded to with gal­lows hu­mour. Then out of the blue, some five years later, we real­ised that new work of Bruno Mar­telli based on ma­chine cal­cu­la­tion might help out. It did, and once we figured out the to­po­lo­gic­al side of things, Xin­gru im­ple­men­ted the com­puter cal­cu­la­tions which al­lowed us to fill the re­main­ing gap. At over 120 pages, the pa­per ap­peared in the Mem­oirs of the AMS [20].

The one case which re­mains to com­plete the proof of Con­jec­ture 3.2 is when \( m \geq 3 \) and \( \beta \) is not a strict bound­ary slope. This only arises when \( M(\beta) \) is either a tor­us bundle or a tor­us semibundle, so is highly non­gen­er­ic. It should be an “easy” case, so hope­fully we’ll clear it up at some point.

As for Con­jec­ture 3.1, Lack­enby and Mey­er­hoff veri­fied its first claim \( \#\mathcal{E}(M) \leq 10 \) and \( \Delta(M) \leq 8 \) in [e13]. What re­mains to be es­tab­lished is that \( \Delta(\alpha, \beta) \leq 5 \) if \( \alpha \) is a small Seifert filling slope and either

1. \( \beta \) is a re­du­cible filling slope and a non­strict bound­ary slope, or
2. \( \beta \) is a tor­oid­al filling slope and a non­strict bound­ary slope, and \( m \) (as above) is at least 3, or
3. \( \beta \) is a small Seifert filling slope and \( M \) is not the fig­ure eight knot ex­ter­i­or.

Case (c) is widely open, though two in­ter­est­ing spe­cial cases are known. The first is the cyc­lic sur­gery the­or­em, which we men­tioned above (The­or­em 2.1). The second is the fi­nite sur­gery the­or­em, whose state­ment was con­jec­tured by Camer­on in his ICM ad­dress [4]: If \( M \) is a hy­per­bol­ic knot man­i­fold, then there are at most five slopes \( \alpha \) on \( \partial M \) such that \( \pi_1(M(\alpha)) \) is fi­nite. Fur­ther, the dis­tance between any two such slopes is at most 3. (These num­bers are real­ised when \( M \) is the ex­ter­i­or of the \( (-2, 3, 7) \)-pret­zel knot.) Xin­gru and I veri­fied this con­jec­ture in [e7].

Since 2009, our main fo­cus has shif­ted from ex­cep­tion­al sur­ger­ies to the \( L \)-space con­jec­ture.

A co­ori­ented taut fo­li­ation on a closed ori­ent­able 3-man­i­fold \( M \) is a codi­men­sion-1 fo­li­ation \( \mathcal{F} \) on \( M \) with trivi­al nor­mal bundle for which we can find a trans­verse loop to \( \mathcal{F} \) which passes through each of its leafs. Closed, ori­ent­able 3-man­i­folds ad­mit­ting co­ori­ented taut fo­li­ations are known to be prime and have in­fin­ite fun­da­ment­al groups.

A group is left-or­der­able if it is non­trivi­al and ad­mits a total or­der in­vari­ant un­der left mul­ti­plic­a­tion (a left-or­der). Left-or­der­able groups are tor­sion-free, so are in­fin­ite. An ob­vi­ous ex­ample of a left-or­der­able group is \( \mathbb R \) un­der ad­di­tion. A less ob­vi­ous one is \( \mbox{Homeo}_+(\mathbb R) \). In fact, a non­trivi­al count­able group is left-or­der­able if and only if it is iso­morph­ic to a sub­group of \( \mbox{Homeo}_+(\mathbb R) \). A stronger res­ult holds for 3-man­i­fold groups: The fun­da­ment­al group of an ori­ent­able ir­re­du­cible 3-man­i­fold is left-or­der­able if and only if it ad­mits a ho­mo­morph­ism to \( \operatorname{Homeo}_+(\mathbb R) \) with non­trivi­al im­age [e9].

We say that a closed, con­nec­ted, ori­ent­able 3-man­i­fold \( M \) is

• \( LO \) if \( \pi_1(M) \) is a non­trivi­al left-or­der­able group;
• \( CTF \) if \( M \) ad­mits a co­ori­ented taut fo­li­ation;
• \( NLS \) if \( M \) is not an \( L \)-space.

The one im­plic­a­tion known is \( CTF \implies NLS \), which was proved by Oz­sváth and Szabó for \( C^2 \) fo­li­ations in [e8] and later ex­ten­ded to \( C^0 \) fo­li­ations by Kazez–Roberts [e17] and in­de­pend­ently by Bowden [e15].

When the first Betti num­ber of \( M \) is pos­it­ive, \( M \) is \( LO \) [e9], \( CTF \) [e1], and \( NLS \) (by defin­i­tion), so the in­ter­est­ing case arises when \( M \) is a ra­tion­al ho­mo­logy 3-sphere. Here, it is dif­fi­cult to know wheth­er any of the con­di­tions holds or not, though there are many fam­il­ies for which we know the status of at least one of them and this mo­tiv­ates de­vel­op­ing tech­niques to de­term­ine the status of the oth­ers.

An­oth­er prob­lem on our list was to show that 2-fold cyc­lic branched cov­ers \( \Sigma_2(L) \) of non­split al­tern­at­ing links \( L \) were not \( LO \). (They were known to be \( L \)-spaces [e10].) Serendip­ity played a role at this point. I was sched­uled to meet Camer­on at BIRS on a Re­search in Team vis­it whose goal was to make pro­gress on the case \( m \geq 3 \) of Con­jec­ture 3.2. Our first night there I asked him if he knew any in­ter­est­ing present­a­tions for the fun­da­ment­al group of the 2-fold cyc­lic branched cov­er of a link, ex­plain­ing that Liam and I sus­pec­ted that the group was non-left-or­der­able when \( L \) was non­split al­tern­at­ing. He replied that oddly enough, he had at­ten­ded a talk at the Tech­nion a few weeks earli­er in which Mi­chael Polyak de­scribed a link in­vari­ant he had dis­covered, which took the form of a group defined via a present­a­tion de­term­ined by a dia­gram for the link. Camer­on worked out that the group was \( \mathbb Z * \mathbb Z/3 \) for the tre­foil and \( \mathbb Z * \mathbb Z/5 \) for the fig­ure eight, from which he sur­mised that in gen­er­al it should be the free product of \( \mathbb Z \) and the fun­da­ment­al group of the 2-fold cyc­lic branched cov­er of the link, a fact that he read­ily veri­fied.1 Dur­ing our week in Ban­ff, we dis­cussed how this present­a­tion could be used to study the non-left-or­der­ab­il­ity of \( \pi_1(\Sigma_2(L)) \) in our case of in­terest, and after re­lo­cat­ing to Montreal, Camer­on sug­ges­ted that the dual graph \( \Gamma \) to the black re­gions of a check­er­board sur­face might be use­ful. After a little thought, we saw that any left or­der on \( \pi_1(\Sigma_2(L)) \) in­duced a se­mi­ori­ent­a­tion on \( \Gamma \) re­lated to the dia­gram’s cross­ing pat­tern, and once that ob­ser­va­tion was made, the proof fol­lowed simply.

Thus mo­tiv­ated, Camer­on, Liam and I turned to the prob­lem of show­ing that ra­tion­al ho­mo­logy sphere Sol man­i­folds were \( L \)-spaces. Camer­on and I suc­ceeded in show­ing that roughly 75% of them were us­ing the tri­ad con­di­tion (me) and Némethi’s graph ho­mo­logy ap­proach (Camer­on), but frus­trat­ingly, we couldn’t see how to get all of them. Liam ap­proached the prob­lem via the re­cently de­veloped bordered Hee­gaard Flo­er the­ory. He as­sim­il­ated enough of the the­ory to be able to es­tab­lish the res­ult for a fam­ily of Sol man­i­folds that we then used as the base case of an in­duct­ive ar­gu­ment us­ing tri­ads, thus com­plet­ing the proof.

We also veri­fied the equi­val­ence of \( LO, CTF \) and \( NLS \) in oth­er cases, in­clud­ing in­fin­ite fam­il­ies of hy­per­bol­ic man­i­folds. At this point we felt that \( LO \iff NLS \iff CTF \) might just al­ways hold. This would be rather mys­ter­i­ous since the con­di­tions were of a quite dif­fer­ent nature and we knew no strong heur­ist­ic con­nect­ing them. It was known that \( CTF \implies NLS \), and Oz­sváth and Szabó had asked if the con­verse held. As such, and giv­en the evid­ence we’d com­piled, the equi­val­ence of \( NLS \) and \( CTF \) seemed reas­on­able to con­jec­ture. The pa­per that Camer­on, Liam and I wrote [11] was con­cerned with the con­nec­tion between \( LO \) and \( NLS \), and we con­jec­tured the equi­val­ence of these prop­er­ties in it. An­dras Juhász stated the full con­jec­ture in his sur­vey art­icle [e14]. Though there was a lot of scep­ti­cism ini­tially, it has been a pleas­ure watch­ing the evid­ence mount over the last dec­ade.

In May 2016, Camer­on and I got to­geth­er with Michel Boileau at the ICTP Trieste. Michel poin­ted out that the ca­non­ic­al cyc­lic branched cov­ers \( \Sigma_n(L) \) of a qua­si­pos­it­ive link \( L \) bound Stein do­mains, and if one of these do­mains has an in­def­in­ite in­ter­sec­tion pair­ing, the as­so­ci­ated cyc­lic branched cov­er is not an \( L \)-space. Giv­en that, he wondered what con­sequences there were for qua­si­pos­it­ive links \( L \) for which some \( \Sigma_n(L) \) is an \( L \)-space. Quite a few in fact, two longish pa­pers worth [15], [14], which we worked out in what I like to think of as a Tale of Four In­sti­tutes, since they were com­pleted over the next year dur­ing vis­its to the CRM in Montreal, the New­ton In­sti­tute in Cam­bridge, and fi­nally at the Casa Matemática Oax­aca.

In the first of these two pa­pers, we con­jec­tured that a prime fibred strongly qua­si­pos­it­ive link \( L \) has an \( L \)-space ca­non­ic­al cyc­lic branched cov­er if and only if it was what we now call an \( ADE \) link: i.e., a link ob­tained by plumb­ing pos­it­ive Hopf bands ac­cord­ing to one the \( ADE \) Dynkin dia­grams. Re­pla­cing “\( L \)-space” by “non-\( LO \)” or “non-\( CTF \)” yields two sis­ter con­jec­tures, all three of which I began to dis­cuss with Ying and Camer­on dur­ing the year be­fore the pan­dem­ic hit. ( John Bald­win pitched in dur­ing the ini­tial stages.)

Our am­bi­tions evolved once we un­der­stood what the tech­niques we were de­vel­op­ing could do and what star­ted out as one pa­per be­came two, then three, then four [17], [18], [19], [21]. One of my fa­vour­ite res­ults from our first pa­per was that ir­re­du­cible, tor­oid­al, in­teger ho­mo­logy 3-spheres were \( LO \). This was a lovely ap­plic­a­tion of our res­ults on the slope de­tec­tion and glu­ing tech­no­logy ori­gin­at­ing in [e16], though it had noth­ing to do with our ini­tial study. ( Eftekhary had already shown that such man­i­folds were \( NLS \).) The meth­od also de­term­ined an ap­proach for show­ing that they were \( CTF \) and led to a proof of this un­der an ad­ded fib­ring as­sump­tion.

One of the motive forces in the evol­u­tion of our ideas was Camer­on’s habit of in­cess­antly ques­tion­ing what we really had done and what we could do. Here is a typ­ic­al ex­ample.

We knew in the fall of 2019 how to show that the ca­non­ic­al cyc­lic branched cov­ers of hy­per­bol­ic fibred strongly qua­si­pos­it­ive knots were \( LO \), but we wanted to ex­tend the res­ult to links, and as our fo­cus was else­where, the pro­ject lan­guished. We were us­ing Fen­ley’s asymp­tot­ic circle con­struc­tion to build \( \mbox{Homeo}_+(S^1) \) rep­res­ent­a­tions of the fun­da­ment­al groups of these branched cov­ers and needed to show that their Euler classes van­ish. Dur­ing a chance dis­cus­sion with Adam Clay a few years later, I real­ised that some ideas from a pa­per of his with Id­rissa Ba should do what we wanted, which turned out to be the case.

One con­sequence of this was that the 2-fold cyc­lic branched cov­er of a fibred hy­per­bol­ic strongly qua­si­pos­it­ive link \( L \) was \( LO \). Em­pir­ic­al evid­ence then sug­ges­ted that no mat­ter how the ori­ent­a­tion of \( L \) was altered, the ca­non­ic­al cyc­lic branched cov­ers of the res­ult­ing link were \( LO \). Meet­ing after meet­ing after meet­ing Camer­on would ask how we could prove this. At some point, Ying re­spon­ded “Isn’t chan­ging the ori­ent­a­tion of a link com­pon­ent just the flip­ping op­er­a­tion de­scribed by Calegari and Dun­field?”. It wasn’t long be­fore we saw that not only was it so, but there was a much more gen­er­al op­er­a­tion we called re­cal­ib­ra­tion (and in­form­ally stir-fry­ing) which had much more gen­er­al con­sequences. For in­stance, we could show that all of the cyc­lic branched cov­ers of prime, fibred, strongly qua­si­pos­it­ive, hy­per­bol­ic links were \( LO \), even those where the branch­ing in­dex var­ied with the com­pon­ent of \( L \) [18]. Ow­ing to Camer­on’s per­sist­ent curi­os­ity, the \( ADE \) link con­jec­ture ori­gin­at­ing in [15] now be­came:

As of this writ­ing, we’ve veri­fied most of the \( LO \) case of Con­jec­ture 5.1 [17], [18], [21]. though much of the \( NLS \) and \( CTF \) cases re­main open.

Look­ing back over my ca­reer, it is clear that happy col­lab­or­a­tions fol­low the Anna Karen­ina prin­ciple: many factors are needed to main­tain them; the ab­sence of only one can spell doom. My col­lab­or­a­tion with Camer­on has been a happy one. The mech­an­ics have evolved of course, from e-mails in the nought­ies, to weekly Fri­day af­ter­noon phone calls 2010–20. Ying sug­ges­ted that we switch to zoom meet­ings after the cov­id lock­down hit in early 2020, and these meet­ings helped me main­tain a health­i­er, more bal­anced per­spect­ive dur­ing the shut­down. They re­main a high­light of my week.

A key pleas­ure in col­lab­or­at­ing is ar­riv­ing some­where that you couldn’t have found your­self. I have ex­per­i­enced this pleas­ure in all my work with Camer­on (and my oth­er col­lab­or­at­ors). But I’d like to stress that our col­lab­or­a­tion re­flects much more than a tech­nic­al com­pat­ib­il­ity. There is a shared need to un­der­stand, a shared am­bi­tion to work on chal­len­ging prob­lems, a shared en­joy­ment in do­ing the math­em­at­ics. And I will al­ways be grate­ful for the ever-present ca­marader­ie, mani­fes­ted as much in fail­ure as in suc­cess.