Celebratio Mathematica

Three of the best years of my life were the years I spent work­ing with my thes­is ad­visor Richard Pal­ais. When I ar­rived at Bran­de­is in 1965, where Pal­ais was teach­ing at the time, I was just mar­ried, had a bach­el­or’s de­gree from the Uni­versity of Michigan, one year of gradu­ate school at NYU, and an NSF gradu­ate fel­low­ship. The Bran­de­is gradu­ate pro­gram was in its first few years of ex­ist­ence and some­what ex­per­i­ment­al. Much to my sur­prise, I passed the pre­lims the first fall, and looked about for an ad­visor.

Dick Pal­ais was cer­tainly the best lec­turer among those I tried out, pos­sibly the best ex­pos­it­or I ever took a course from. He com­bined en­thu­si­asm with lec­tures pre­pared down to the last de­tail. What really at­trac­ted me was that the ma­ter­i­al was brand new. Glob­al ana­lys­is was the cur­rent hot top­ic. Over the pre­ced­ing dec­ades, dif­fer­en­tial to­po­logy and, sep­ar­ately, the ba­sic the­ory of el­lipt­ic dif­fer­en­tial equa­tions had been de­veloped. So the time was ripe for the de­vel­op­ment of in­fin­ite-di­men­sion­al man­i­folds spe­cific­ally de­signed to handle prob­lems in non­lin­ear par­tial dif­fer­en­tial equa­tions. Here are some fea­tures of the land­scape I found so en­ti­cing.

I al­ways start with Morse the­ory, which was cent­ral to the work I did with Pal­ais. Morse the­ory con­nects the crit­ic­al points of a func­tion on a man­i­fold with the to­po­logy of the man­i­fold. I find it still strik­ing that Morse de­veloped it to treat an “in­fin­ite-di­men­sion­al” prob­lem, the prob­lem of ana­lyz­ing the crit­ic­al points (which are curves) for the en­ergy func­tion­al defined on curves on a man­i­fold. This is a cal­cu­lus of vari­ations prob­lem defined on an in­fin­ite-di­men­sion­al man­i­fold of curves, but the tech­niques avail­able re­quired ap­prox­im­at­ing the space of curves by broken geodesics, or a fi­nite set of points con­nec­ted by short geodesics. The use of Morse the­ory as a tool to ana­lyze fi­nite-di­men­sion­al man­i­folds is an af­ter­thought. Bott’s proof of the peri­od­icity the­ory us­ing the Morse the­ory of geodesics as a tool and the beauty of the cal­cu­lus of vari­ations en­ticed math­em­aticians to look at the sub­ject fur­ther. However, the tech­nique of ap­prox­im­a­tion by broken curves has no coun­ter­part in ana­lyz­ing the crit­ic­al points of mul­tiple in­teg­rals. Math­em­aticians did try.

In 1963 Pal­ais and Smale pub­lished an an­nounce­ment in the Bul­let­in [1] that de­scribed how a func­tion on an in­fin­ite-di­men­sion­al man­i­fold could be dir­ectly shown to have crit­ic­al points which cor­res­pond to the to­po­logy of the man­i­fold. As Dick tells the story, he was teach­ing a course in to­po­logy at Bran­de­is, and Steve was teach­ing a course in to­po­logy at Columbia, and they both needed the same con­di­tion to com­plete the ex­ist­ence proof. It was some­what ma­gic­al that in each case the con­di­tion was their third con­di­tion, hence the des­ig­na­tion “con­di­tion C”. As­sum­ing the usu­al con­di­tions (which are very much the same as in fi­nite di­men­sions), it is pos­sible to get the fi­nite-di­men­sion­al proof us­ing the gradi­ent flow to work, with one ex­tra con­di­tion, which was for years af­ter­wards known as “Con­di­tion C of Pal­ais and Smale”. This roughly says that a se­quence of ap­prox­im­ate crit­ic­al points in a bounded re­gion has a sub­sequence that con­verges to a crit­ic­al point in the re­gion. Of course, to ap­ply this to the cal­cu­lus of vari­ations, it is ne­ces­sary to de­vel­op the in­fin­ite-di­men­sion­al man­i­fold the­ory of func­tion spaces; the the­ory of or­din­ary dif­fer­en­tial equa­tions on in­fin­ite-di­men­sion­al man­i­folds; a bit about the geo­metry to con­struct gradi­ent curves; and fi­nally to know a lot about the Euler–Lag­range equa­tions, which are the par­tial dif­fer­en­tial equa­tions de­scrib­ing the crit­ic­al points (func­tions in this in­fin­ite-di­men­sion­al set­ting). Dick wrote an el­eg­ant pa­per re-prov­ing the res­ults for the en­ergy in­teg­ral on curves, which pre­vi­ously had only been done us­ing fi­nite-di­men­sion­al ap­prox­im­a­tions.

In an­oth­er land­mark pa­per [e5] from 1963, Eells and Sampson pub­lished a proof that the en­ergy in­teg­ral on maps from a com­pact man­i­fold in­to a man­i­fold of neg­at­ive curvature takes on a unique min­im­um. Later on we will find out why the dir­ect meth­od of Pal­ais and Smale can­not work in this set­ting. Eells and Sampson also dir­ectly use a gradi­ent flow, but this time by a heat equa­tion rather than by the or­din­ary dif­fer­en­tial equa­tions in in­fin­ite di­men­sions used by Pal­ais and Smale. Their ex­ist­ence proof for the non­lin­ear heat equa­tion was so com­plete that it is still of­ten cited in the cur­rent lit­er­at­ure. While the to­po­lo­gic­al res­ults ob­tained from this the­or­em were not new, the idea was.

The year 1963 was a good one for math­em­at­ics. In that same year, Atiyah and Sing­er pub­lished an an­nounce­ment of their cel­eb­rated the­or­em identi­fy­ing the in­dex of a (lin­ear) el­lipt­ic sys­tem on a man­i­fold with a to­po­lo­gic­al in­vari­ant com­puted from the sym­bol of the op­er­at­or [e2]. This new the­or­em was very gen­er­al, al­beit gen­er­al­ized from some very im­port­ant the­or­ems such as the Riemann–Roch the­or­em and the Hirzebruch sig­na­ture the­or­em. Some of the the­or­em’s most im­port­ant ap­plic­a­tions still lay in the fu­ture. In 1963–64 Dick was a mem­ber at the In­sti­tute for Ad­vanced Study, and the fac­ulty mem­ber Ar­mand Borel sug­ges­ted they run a sem­in­ar on this new, ex­cit­ing de­vel­op­ment. Dick was re­l­at­ively ju­ni­or, and was eager to help. At some point dur­ing the term, Borel dis­ap­peared. Dick or­gan­ized the sem­in­ar alone, and the notes be­came a book [2] con­tain­ing the first pub­lished proof of the the­or­em. Dick sus­pects that Atiyah and Sing­er may have re­sen­ted this.

When I ap­peared on Dick’s door­step in the fall of 1965, he was giv­ing a lec­ture course on the cal­cu­lus of vari­ations. He had taught a course on the dif­fer­en­tial to­po­logy of in­fin­ite-di­men­sion­al man­i­folds the pre­vi­ous year, but I was able to fol­low his el­eg­ant, care­fully pre­pared lec­tures from the very start. I was en­thu­si­ast­ic about his lec­tures and glob­al ana­lys­is. Dick was at first not as en­thu­si­ast­ic, and at some point mumbled something about wo­men need­ing to raise a fam­ily. Since this was the cur­rent think­ing at the time, I simply ig­nored the com­ment. I would clearly nev­er get any­where by fight­ing a war that is still, more than 50 years later, not com­pletely won. However, we did start work to­geth­er, and the top­ic nev­er came up again. In ret­ro­spect, I’m sur­prised by the amount of en­cour­age­ment and sup­port I did re­ceive from my (male) fac­ulty ment­ors, giv­en the land­scape of the time. To go back to the ori­gin­al train of thought, the lec­ture notes from that first course I took from him be­came Found­a­tions of Glob­al Non-Lin­ear Ana­lys­is [3].

Dick was an ex­cel­lent lec­turer, both in form­al classes and in ex­tem­por­an­eous an­swers to ques­tions. I have a vivid memory (after some 60 years) of go­ing in­to his of­fice and ask­ing about the heat equa­tion. He pro­ceeded to give me a lec­ture that covered everything I needed to know about the heat equa­tion for sev­er­al dec­ades. I have fond memor­ies of the many ref­er­ences he had me read. I am not a per­son who likes presents. I feel they come with ex­pect­a­tions of pay­back. But my memory of the ref­er­ences I read is that each book or art­icle was a wrapped present, which I had the pleas­ure of slowly un­wrap­ping with a new dis­cov­ery in every pack­age. Per­haps it helped that no one had ex­pect­a­tions for my fu­ture, and hence I didn’t owe any­body any­thing. At first it was ba­sic ref­er­ences: Mil­nor’s Morse The­ory [e4]; Cal­cu­lus of Vari­ations by Gel­fand and Fom­in [e3]; and Land­au and Lif­shitz’s The Clas­sic­al The­ory of Fields [e1]. At this point Mor­rey’s book [e7] on the cal­cu­lus of vari­ations came out, and I read it. To be hon­est, look­ing at it more than 50 years later, I don’t know how I did it. At the time, I felt like I could play with the big boys, even if they would not let me. But it went on from there. There were two pa­pers by Kohn and Niren­berg [e6], [e8] on pseudo-dif­fer­en­tial op­er­at­ors, Giusti and Mir­anda [e9] on vari­ation­al prob­lems whose solu­tions had sin­gu­lar­it­ies, in­ter­pol­a­tion the­ory from a ref­er­ence I no longer re­mem­ber, and No­eth­er’s the­or­em (which I un­der­stood poorly). Without him­self be­ing an ex­pert in the sub­ject, Dick man­aged to give me a back­ground in ana­lys­is that few gradu­ate schools could of­fer their stu­dents. When I think back on it, I am etern­ally grate­ful, but sad that I did not of­fer my own stu­dents a sim­il­arly rich ex­per­i­ence.

Be­ing a gradu­ate stu­dent at Bran­de­is opened up the best of two worlds. We could at­tend courses and col­loquia at MIT and Har­vard (where tea was not much fun for the few wo­men) and hang out at Bran­de­is with Mike Spivak, who over­saw the com­mon room. Bill Fulton, John Polk­ing, Tom Sher­man and Mike Shub all had postdoc­tor­al po­s­i­tions at Bran­de­is while I was there. Paul Mon­sky and I of­ten met on our com­mute from Cam­bridge to Waltham. I still re­mem­ber that Tony Tromba and I gave Mike Shub a hard time about the clos­ing lemma. Stu­dents were in­cluded in the col­loqui­um parties, and on one mem­or­able oc­ca­sion, I got a ride to the col­loqui­um at Har­vard with the one and only John Nash. My gradu­ate days were a good ex­per­i­ence.

What about the re­search I did for my thes­is? As men­tioned be­fore, Dick was pol­ish­ing up the con­struc­tions for what are now known as vari­ation­al prob­lems that are “sub­crit­ic­al”. For these prob­lems, the geo­metry and to­po­logy of the man­i­folds based on a Banach space $H$ of maps between two fi­nite-di­men­sion­al man­i­folds $M$ and $N$ are very rel­ev­ant. We men­tioned the en­ergy in­teg­ral be­fore. Here the points in the in­fin­ite-di­men­sion­al space are maps between two fixed fi­nite-di­men­sion­al man­i­folds. The func­tion­al is the in­teg­ral of the norm squared of the de­riv­at­ive. The catch is, that in or­der to make a man­i­fold, the func­tion space must lie in the space of con­tinu­ous func­tions. But to make con­di­tion C work prop­erly, the func­tion­al has to con­trol the norm. The en­ergy in­teg­ral forces $H=H_1$, the Hil­bert space of func­tions whose de­riv­at­ives are square in­teg­rable. But the only time $H_1$ is in the con­tinu­ous func­tions is when $\dim M=1$. We are stuck back where we star­ted, with suc­cess at treat­ing ODEs but not PDEs. For this reas­on, Eells and Sampson could not use vari­ation­al meth­ods to un­der­stand the crit­ic­al points of the en­ergy in­teg­ral. My ini­tial pro­ject was to find ex­amples in ar­bit­rary di­men­sion of these sub­crit­ic­al vari­ation­al prob­lems. It is straight­for­ward to show that the $p$-en­ergy, defined as the in­teg­ral of the de­riv­at­ive of a func­tion raised to the $p$-en­ergy, has all the right prop­er­ties for $p>\text{dimension }M$. The Euler–Lag­range equa­tions are highly non­lin­ear, and there is a small dif­fi­culty in that it was not known that solu­tions to these prob­lems are ac­tu­ally smooth func­tions. This small prob­lem oc­cu­pied my thoughts for the next 6–7 years. The lim­its of this very beau­ti­ful the­ory is to sub­crit­ic­al prob­lems, and it is de­light­ful but rare to find in­ter­est­ing ex­amples.

However, the back­ground I re­ceived from Dick was more than suf­fi­cient for me to go onto oth­er prob­lems. When I went off to a postdoc­tor­al po­s­i­tion at Berke­ley, he sug­ges­ted I learn re­lativ­ity. I audited the course for phys­i­cists. I can see his hand in every one of my early pa­pers. These in­cluded adding handles in a Banach man­i­fold as op­posed to Hil­bert man­i­folds [e10]; a pa­per us­ing in­ter­pol­a­tion the­or­em to get reg­u­lar­ity for solu­tions of el­lipt­ic equa­tions with dis­con­tinu­ous in two di­men­sions [e11]; an ad­ded two cents to a long line of pa­pers start­ing with Jac­obi, Morse, Ed­wards, Si­mon and Smale on the Morse in­dex the­or­em [e12]; us­ing the tools from in­fin­ite-di­men­sion­al to­po­logy to study gen­er­ic prop­er­ties of ei­gen­func­tions [e14]; and last but not least a Morse the­ory for geodesics on a Lorentz man­i­fold [e13]. All of this came from the com­bin­a­tion of a sol­id back­ground and the en­cour­age­ment to al­ways look fur­ther.

Dick has done well by his stu­dents. My little sis­ters Chuu-Li­an Terng and Jill Mesirov have had dis­tin­guished ca­reers. He has told me of a con­ver­sa­tion he had with Lip­man Bers (who is fam­ous or no­tori­ous, de­pend­ing on your per­spect­ive, for the num­ber of his wo­men stu­dents). Bers claimed the lar­ger num­ber of wo­men stu­dents, but con­ceded qual­ity to Dick’s. Dick Pal­ais is at present the only math­em­atician who can claim an Abel Prize win­ner and a win­ner of the A. M. Tur­ing Award among his former stu­dents. Leslie Lam­port, who won the Tur­ing award in 2013, star­ted out as an un­der­gradu­ate stu­dent of Dick’s and went on to earn a PhD in to­po­logy with Dick after a hi­atus as a pro­fes­sion­al mu­si­cian. Only later, slowly, did he make the switch through pro­gram­ming to the­or­et­ic­al com­puter sci­ence. Dick mod­estly claims that he learned more from his stu­dents than they learned from him. I doubt this.

At the time Leslie was his stu­dent, Dick had de­veloped an in­terest in pro­gram­ming. He re­calls that he “had a lot of fun”. This slowly grew in­to an in­terest in visu­al­iz­a­tion as he de­veloped soft­ware to il­lus­trate ba­sic geo­metry. The mas­ter pro­gram he wrote for this visu­al­iz­a­tion pro­ject is still avail­able on the web, as is a vir­tu­al mu­seum of geo­met­ric ob­jects. I am par­tic­u­larly im­pressed by this move from ab­stract math­em­at­ics in­to an en­tirely new field. Rather than work­ing over the ideas that his early re­search was based on, he headed off in an en­tirely new dir­ec­tion as a pi­on­eer. I have al­ways kept this as my own goal in re­search. Find something new to work on. Math­em­at­ics of­fers won­der­ful op­por­tun­it­ies for this, which are rarely made use of.