Celebratio Mathematica

Ac­cord­ing to what my par­ents told my sib­lings and me, our fam­ily’s ori­gin goes back to the 1890s when our four grand­par­ents emig­rated to the US. All were “Litvaks”, that is, Ashkenazi Jews who grew up in the shtetls near Vil­ni­us in Lithuania, or, as I told my three chil­dren, they all came from “Fid­dler on the Roof” coun­try. For­tu­nately my fath­er was ad­mit­ted to Har­vard in 1916, about six years be­fore strict “Jew­ish quotas” were im­ple­men­ted.

When I was about four years old we moved from my birth town, Salem, fam­ous for its “witches”, to New­ton. There I went to the Angi­er Ele­ment­ary School and to War­ren Ju­ni­or High, and then we moved again to Brook­line where I went to eighth grade and the first year of high school be­fore trans­fer­ring to Gov­ernor Dum­mer Academy for the fi­nal three years of high school.

When I was about twelve, I be­came in­ter­ested in phys­ics, and some­how came across a book titled The Ein­stein the­ory of re­lativ­ity by Lieber and Lieber.1 It made a really pro­found im­pres­sion on me. I pretty well un­der­stood and was fas­cin­ated by the cent­ral phys­ic­al ideas; these were presen­ted in a clear and ele­ment­ary way. But I also very much wanted to un­der­stand the math de­tails, to which the book had a pretty good but rather in­com­plete in­tro­duc­tion. This star­ted me on a long ad­ven­ture. I re­call start­ing out by read­ing the Brit­an­nica art­icle on “In­fin­ites­im­al Cal­cu­lus”, and after many in­ter­me­di­ate steps, even­tu­ally read­ing and par­tially un­der­stand­ing Ed­ding­ton’s The Math­em­at­ic­al The­ory of Re­lativ­ity.2 In any case a side res­ult was that I be­came far ahead of my class­mates in math­em­at­ic­al soph­ist­ic­a­tion dur­ing my high school years (and got an 800 on my Math Col­lege Board ex­am).

In the late 1940s S.-S. Chern came to the US from China, vis­it­ing first at the In­sti­tute for Ad­vance Study (IAS), and then in 1950 An­dré Weil offered him a po­s­i­tion at the Uni­versity of Chica­go Math De­part­ment, where Chern went the next year, after first spend­ing the fall semester of 1951 as a vis­it­ing pro­fess­or at Har­vard. This turned out to be a very for­tu­nate hap­pen­stance for me: Since no one in the Har­vard Math De­part­ment was an ex­pert in the Rieman­ni­an geo­metry of Lorent­zi­an 4-man­i­folds, which my seni­or thes­is was based on, Mackey asked Chern if he would be will­ing to look at my work. Not only did he read my draft, but he then asked me to come talk with him, and he gave me some valu­able sug­ges­tions. He was ap­par­ently im­pressed enough with me that when George Mackey in­formed him in June of 1956 that I had just re­ceived my PhD and was look­ing for a po­s­i­tion the fol­low­ing aca­dem­ic year, Chern ar­ranged for me to have a two-year po­s­i­tion at the Uni­versity of Chica­go, which then had one of the most out­stand­ing math de­part­ments in the US. Chern was of course a ment­or dur­ing those two years and we be­came even closer friends. In later years, I ended up spend­ing more time with him, as a vis­it­or to the UC Berke­ley de­part­ment, as a mem­ber of MSRI the first year after Chern foun­ded it, and later still with vis­its to his In­sti­tute in Tianjin, to­geth­er with Chuu-Li­an Terng, who had her first postdoc­tor­al po­s­i­tion at Berke­ley, thanks also to Chern. (She, like me, be­came close friends with Chern and his wife, and in 1996 Chern asked Chuu-Li­an and me to write a bio­graphy of him [6].)

Even with Chern and Mackey’s help­ful guid­ance, my seni­or thes­is re­quired very hard work and con­cen­tra­tion dur­ing my col­lege years, and it paid off hand­somely. I was one of the eight stu­dents in my class elec­ted to Phi Beta Kappa in our ju­ni­or year, and I gradu­ated summa cum laude with highest hon­ors in math­em­at­ics. Per­haps even more im­port­ant for my de­vel­op­ment, I was awar­ded the Shel­don Trav­el­ing Fel­low­ship of that year.

I was in­tro­duced to the math­em­at­ics of Bourbaki in my un­der­gradu­ate years at Har­vard. I liked the spe­cial way they presen­ted math­em­at­ics, and their work greatly in­flu­enced my later math­em­at­ic­al study and writ­ing style. So I de­cided to go to Nancy, where the Bourbaki group met and worked, for the great­er part of my fel­low­ship. I had only an un­der­gradu­ate edu­ca­tion, could only audit courses, and I did not have much in­ter­ac­tion with the fac­ulty there. But I did meet and be­come friends with Grothen­dieck, who was fin­ish­ing his PhD. He later re­ceived a Fields Medal and one of the most in­flu­en­tial math­em­aticians of the 20th cen­tury.

While in Nancy, I took sev­er­al short side trips to the Uni­versity of Stras­bourg and across the bor­der in­to Ger­many in or­der to ful­fill the re­quire­ment of my fel­low­ship not to spend too much time in one uni­versity or coun­try.

Dur­ing the time I was in Nancy, I also took sev­er­al train trips to Par­is, ar­riv­ing at Gare d’Or­say. I stayed in the Lat­in Quarter, and loved walk­ing around Par­is. The sta­tion was later trans­formed in­to what be­came my fa­vor­ite mu­seum: Musée d’Or­say — just as Par­is had by then be­come my fa­vor­ite city.

While I was trav­el­ing on my fel­low­ship, I felt that I should do math­em­at­ic­al re­search. The end res­ult was my first pub­lished pa­per, “A defin­i­tion of the ex­ter­i­or de­riv­at­ive in terms of Lie de­riv­at­ives” [1].

Look­ing back at how it happened, it seems al­most ac­ci­dent­al that I be­came Andy Gleason’s first PhD stu­dent — and Dav­id Hil­bert was partly re­spons­ible. As an un­der­gradu­ate at Har­vard I had de­veloped a very close ment­or­ing re­la­tion­ship with George Mackey, then a res­id­ent tu­tor in my dorm, Kirk­land House. Be­sides hav­ing meals with him sev­er­al times each week, I took many of his courses. So, when I re­turned in 1953 as a gradu­ate stu­dent, it was nat­ur­al for me to ask Mackey to be my thes­is ad­visor. When he in­quired what I would like to work on for my thes­is re­search, my first sug­ges­tion turned out to be something he had thought about him­self, and he was able to con­vince me quickly that it was un­suit­ably dif­fi­cult for a thes­is top­ic. A few days later I came back and told him I would like to work on re­for­mu­lat­ing the clas­sic­al Lie the­ory of germs of Lie groups act­ing loc­ally on man­i­folds as a rig­or­ous mod­ern the­ory of full Lie groups act­ing glob­ally. Fine, he said, but then ex­plained that the loc­al ex­pert on such mat­ters was a bril­liant young former Ju­ni­or Fel­low named Andy Gleason who had just joined the Har­vard Math De­part­ment. Only a year be­fore he had played a ma­jor role in solv­ing Hil­bert’s Fifth Prob­lem, which was closely re­lated to what I wanted to work on, so he would be an ideal per­son to dir­ect my re­search. I felt a little un­happy at be­ing cast off like that by Mackey, but of course I knew per­fectly well who Gleason was, and I had to ad­mit that George had a point.

Andy was already fam­ous for be­ing able to think com­plic­ated prob­lems through to a solu­tion in­cred­ibly fast. “Johnny” von Neu­mann had a sim­il­ar repu­ta­tion, and since this was the year that High Noon came out, I re­call jokes about hav­ing a math­em­at­ic­al shoot out: Andy vs. Johnny solv­ing math prob­lems with blaz­ing speed at the OK Cor­ral. In any case, it was with con­sid­er­able trep­id­a­tion that I went to see Andy for the first time. Totally un­ne­ces­sary! In our ses­sions to­geth­er I nev­er felt put down. It is true that oc­ca­sion­ally when I was telling him about some pro­gress I had made since our pre­vi­ous dis­cus­sion, part way through my ex­plan­a­tion Andy would see the crux of what I had done and say something like, “Oh! I see. Very nice! And then…,” and in a mat­ter of minutes he would re­con­struct (of­ten with im­prove­ments) what had taken me hours to fig­ure out. But it nev­er felt like he was act­ing su­per­i­or. On the con­trary, he al­ways made me feel that we were col­leagues, col­lab­or­at­ing to dis­cov­er the way for­ward. It was just that when he saw his way to a solu­tion of one prob­lem, he liked to work quickly through it and then go on to the next prob­lem. Work­ing to­geth­er with such a math­em­at­ic­al power­house put pres­sure on me to per­form at a top level, and it was sure a good way to learn hu­mil­ity! My ap­pren­tice­ship wasn’t over when my thes­is was done. I re­mem­ber that shortly after I had fin­ished, Andy said to me, “You know, some of the ideas in your thes­is are re­lated to some ideas I had a few years back. Let me tell you about them, and per­haps we can write a joint pa­per.” The ideas in that pa­per were in large part his, but on the oth­er hand I did most of the writ­ing, and in the pro­cess of cor­rect­ing my at­tempts, he taught me a lot about how to write a good journ­al art­icle. But it was only years later that I fully ap­pre­ci­ated just how much I had taken away from those years work­ing un­der Andy. I was very for­tu­nate to have many ex­cel­lent stu­dents do their gradu­ate re­search with me over the years, and of­ten as I worked to­geth­er with them I could see my­self be­hav­ing in some way that I had learned to ad­mire from my own ex­per­i­ence work­ing to­geth­er with Andy.

Let me fin­ish with one more an­ec­dote. It con­cerns my fa­vor­ite of all of Andy’s the­or­ems, his el­eg­ant clas­si­fic­a­tion of the meas­ures on the lat­tice of sub­spaces of a Hil­bert space. Andy was writ­ing up his res­ults dur­ing the 1955–56 aca­dem­ic year, as I was writ­ing up my thes­is, and he gave me a draft copy of his pa­per to read. I found the res­ult fas­cin­at­ing, and even con­trib­uted a minor im­prove­ment to the proof, as Andy was kind enough to foot­note in the pub­lished art­icle. When I ar­rived at the Uni­versity of Chica­go for my first po­s­i­tion the next year, Andy’s pa­per was not yet pub­lished, but word of it had got­ten around, and there was a lot of in­terest in hear­ing the de­tails. So when I let on that I was fa­mil­i­ar with the proof, Irving Ka­plansky asked me to give a talk on it in his Ana­lys­is Sem­in­ar. I’ll nev­er for­get walk­ing in­to the room where I was to lec­ture and see­ing Ed Span­i­er, Mar­shall Stone, Saun­ders Mac Lane, An­dré Weil, Ka­plansky and Chern all look­ing up at me. It was pretty in­tim­id­at­ing and I was suit­ably nervous! But the pa­per was so el­eg­ant and clear that it was an ab­so­lute breeze to lec­ture on it, so all went well, and this “in­aug­ur­al lec­ture” helped me get off to a good start in my aca­dem­ic ca­reer.

After re­ceiv­ing my PhD de­gree from Har­vard, I ap­plied for my first aca­dem­ic po­s­i­tion to sev­er­al dis­tin­guished Math­em­at­ics De­part­ments. In par­tic­u­lar, Pro­fess­or Chern of Uni­versity of Chica­go en­cour­aged me to ap­ply there, and I am pretty sure he played an im­port­ant role in get­ting me an of­fer there. That de­part­ment was a power­house then with Shi­ing-Shen Chern, An­dré Weil, Ant­oni Zyg­mund, and Saun­ders Mac Lane as seni­or fac­ulty, and Paul Hal­mos, Irving Segal, and Ed­win Span­i­er as as­sist­ant pro­fess­ors. Steve Smale and Siggy Hel­gas­on were also be­gin­ning in­struct­ors there and we three be­came life-long good friends.

I was sup­posed to be in Chica­go for the fall quarter, but since my first child, Ju­lie, was born in Septem­ber, I re­ceived the De­part­ment’s per­mis­sion to start in the winter quarter. Ju­lie had a long ca­reer as a po­lar gla­ci­olo­gist and made im­port­ant con­tri­bu­tions to cli­mate change re­search, study­ing vol­can­ic fal­lout in ice cores from both Green­land and Ant­arc­tica. She served as the Pro­gram Dir­ect­or of the Ant­arc­tic Gla­ci­ology Pro­gram at the Na­tion­al Sci­ence Found­a­tion (NSF) from 1990 to 2016. Both the Pal­ais Gla­ci­er and Pal­ais Bluff in Ant­arc­tica were named in her hon­or, and she re­ceived the Ex­plorers Club Low­ell Thomas Award in 2007 for her work in cli­mate change re­search. She also re­ceived an hon­or­ary de­gree from the Uni­versity of New Hamp­shire, and the Goldth­wait Po­lar Medal from the Byrd Po­lar and Cli­mate Re­search Cen­ter at The Ohio State Uni­versity in 2019.

The first day there while set­ting up my of­fice, Chair­man Mac Lane came in and said “Good to see you,” and with a big smile and a pat on my back said “We liked the work you have been do­ing so much, we will raise your an­nu­al salary from \$5,000 to \$5,500.”

One piece of re­search I ac­com­plished at Chica­go was the equivari­ant em­bed­ding the­or­em for \( G \)-man­i­folds, a res­ult which was also in­de­pend­ently dis­covered by Mostow at Yale. We each pub­lished our in­di­vidu­al pa­per with no claim to pri­or­ity and re­mained friendly. This the­or­em has be­come known as The Mostow–Pal­ais The­or­em.

After two years at the Uni­versity of Chica­go, I felt honored to re­ceive a two-year mem­ber­ship at the In­sti­tute for Ad­vanced Study (IAS) in Prin­ceton. Among the es­teemed fac­ulty were Ar­mand Borel and Deane Mont­gomery, both lead­ing ex­perts in the the­ory of trans­form­a­tion groups, which was also my re­search fo­cus. They con­duc­ted a sem­in­ar on this sub­ject that ul­ti­mately served as the found­a­tion for a book pub­lished in the An­nals of Math­em­at­ics. As a young mem­ber, I act­ively par­ti­cip­ated in the sem­in­ar and con­trib­uted a chapter to the book [e1].

The IAS dir­ect­or, J. Robert Op­pen­heimer, held private meet­ings with each new mem­ber and oc­ca­sion­ally in­vited a se­lect group of us to his home for din­ner. I fondly re­call Mont­gomery in­vit­ing me to din­ner at Chung-Tao Yang’s house, where I met Yang’s son Deane, who was just a baby at the time. Deane Yang is now a pro­fess­or of math­em­at­ics at New York Uni­versity’s Cour­ant In­sti­tute of Math­em­at­ics.

Dur­ing my time at IAS, my son Robert was born in Prin­ceton, and he has since be­come a math­em­atician at the Uni­versity of Utah, spe­cial­iz­ing in the in­ter­sec­tion of ge­net­ics and math­em­at­ics. He re­ceived pat­ents for meth­ods that have been widely used for new­born screen­ing and in rap­id tests for in­fec­tious dis­eases such as Cov­id, and with me and H. Karch­er for math­em­at­ic­al visu­al­iz­a­tion tools. He is also a very good rock climber (and has climbed El Cap­it­an five times).

Many young IAS mem­bers had of­fices in the fam­ous Elec­tron­ic Com­puter Pro­ject (ECP) build­ing, which housed the von Neu­mann com­puter. Not­able young mem­bers like Mi­chael Atiyah, Fritz Hirzebruch, Steve Smale, and Moe Hirsch were also part of the vi­brant com­munity dur­ing that peri­od. Mem­bers lived in on-cam­pus hous­ing, had reg­u­lar lunch at the din­ing room, and daily tea time (4 p.m.) with fresh baked cook­ies. It was a won­der­ful place to be for math­em­at­ic­al col­lab­or­a­tions and friend­ships.

I was thrilled to se­cure a ten­ure-track po­s­i­tion at Bran­de­is Uni­versity in Waltham, a sub­urb of Bo­ston, where my fam­ily — par­ents, broth­ers, and a sis­ter — were all nearby (as were Har­vard and MIT). Named after Su­preme Court Justice Louis Bran­de­is, the first Jew­ish justice, the Uni­versity was es­tab­lished by the Jew­ish com­munity in 1948. While I may not have been re­li­gious, I cer­tainly felt a strong eth­nic con­nec­tion to my Jew­ish her­it­age.

When I joined the fac­ulty, Arnold Sha­piro served as the Chair, and I was wel­comed by col­leagues such as Maurice Aus­lander and Edgar Brown. Around the same time, Dav­id Buchs­baum, Har­old Lev­ine, Jerry Lev­ine, Ter­uhisa Mat­su­saka and Robert See­ley joined the de­part­ment, fol­lowed later by Paul Mon­sky, Alan May­er, and Dav­id Eis­en­bud in the 60s. Most of my col­leagues were of sim­il­ar age, fos­ter­ing a friendly at­mo­sphere where we of­ten en­joyed lunch to­geth­er at the Fac­ulty Club. It was an en­er­get­ic and vi­brant de­part­ment, and we had a great time de­vel­op­ing both the un­der­gradu­ate and gradu­ate pro­grams. Since sev­er­al of us had con­nec­tions to the Uni­versity of Chica­go, where in­struct­ors were in­vited to de­part­ment meet­ings, we modeled our own de­part­ment after the one in Chica­go.

I had the priv­ilege of be­ing friends with Fritz Hirzebruch, the vis­ion­ary be­hind the Arbeit­sta­gung (an­nu­al math­em­at­ics meet­ing in Bonn). The meet­ing in­tro­duced a unique ap­proach: rather than hav­ing a pre­set agenda, the pro­gram was shaped col­lab­or­at­ively by the par­ti­cipants through open dis­cus­sions. This in­nov­at­ive format en­sured that the talks high­lighted the latest ad­vance­ments in math­em­at­ics, al­low­ing many sig­ni­fic­ant res­ults to be un­veiled for the first time at the con­fer­ence. The pa­pers fea­tured in this con­fer­ence, au­thored by lead­ing math­em­aticians, ex­plored a di­verse ar­ray of top­ics, in­clud­ing al­geb­ra­ic geo­metry, to­po­logy, ana­lys­is, op­er­at­or the­ory, and rep­res­ent­a­tion the­ory. They col­lect­ively show­cased the im­press­ive breadth and depth of pure math­em­at­ics that has con­sist­ently defined the Arbeit­sta­gung. I at­ten­ded this gath­er­ing nearly every year for many years.

In 1962, Atiyah provided a thor­ough ex­plan­a­tion of the proof of the Atiyah–Sing­er In­dex The­or­em at the Arbeit­sta­gung. As I ex­ited the lec­ture hall, Borel re­marked on the suc­cess of our sem­in­ar on trans­form­a­tion group the­ory at IAS in 1959 and pro­posed that we con­duct a sem­in­ar on the Atiyah–Sing­er In­dex The­or­em to­geth­er. I re­ques­ted a leave of ab­sence from Bran­de­is to spend the fall semester at the In­sti­tute for Ad­vanced Study (IAS). However, in the middle of sum­mer, Borel in­formed me that he would need to be in France for the first semester. He offered to as­sist in or­gan­iz­ing and out­lining the sem­in­ar, leav­ing me to man­age it in­de­pend­ently. The sem­in­ar turned out to be a suc­cess, fea­tur­ing en­ga­ging weekly lec­tures. I sub­sequently com­piled the lec­tures in­to a book titled Sem­in­ar on the Atiyah–Sing­er In­dex The­or­em [2].

My young­est child Dav­id was born in April 1963. He ob­tained a PhD de­gree in geo­logy from the Ari­zona State Uni­versity at Tempe, did two years of postdoc­tor­al re­search, then went to work Ar­row­head wa­ter com­pany for a num­ber of years, and now is the Primo Wa­ter Cor­por­a­tion’s man­ager for the west­ern US. While he was a gradu­ate stu­dent, he went to Ant­arc­tica to do re­search while his sis­ter Ju­lie was there, too. It is prob­ably quite un­usu­al to have sib­lings in Ant­arc­tica for re­search at the same time.

My ex-wife El­lie and I were very lucky to have three won­der­ful chil­dren, all had PhD de­grees, good jobs, and happy fam­il­ies.

The launch of Sput­nik by the So­viet Uni­on in 1957 promp­ted the U.S. gov­ern­ment to in­crease fund­ing for sci­entif­ic re­search. Rep­res­ent­at­ives from the Na­tion­al Sci­ence Found­a­tion (NSF) and the De­part­ment of De­fense (DOD) vis­ited our de­part­ment to en­cour­age us to sub­mit grant pro­pos­als. I wrote a short pro­pos­al every three years de­tail­ing my past work and fu­ture plans. For math­em­aticians of my gen­er­a­tion, there was re­l­at­ively little ef­fort needed to se­cure grants, as long as we main­tained a reas­on­able level of re­search out­put.

The Bran­de­is De­part­ment func­tioned smoothly and am­ic­ably over­all, though I did have a couple of dis­agree­ments with my col­leagues. One was re­gard­ing the ten­ure de­cision for Mi­chael Spivak. He was an out­stand­ing teach­er and com­mu­nic­at­or, and I be­lieved that while his re­search may not have been par­tic­u­larly strong, his teach­ing and con­tri­bu­tions to edu­ca­tion­al re­sources made him an ex­cel­lent can­did­ate for ten­ure in an un­der­gradu­ate pro­gram. An­oth­er con­ten­tious ten­ure de­cision in­volved Mike Shub, a re­mark­able re­search­er in dy­nam­ic­al sys­tems. Many of my col­leagues op­posed his ten­ure be­cause he had sup­por­ted stu­dents protest­ing the Vi­et­nam War and helped pre­vent their draft by re­fus­ing to give them fail­ing grades.

In 1995, Bran­de­is Uni­versity offered early re­tire­ment in­cent­ives: fac­ulty could re­tire in 1997 with 90% of their pay and re­duced teach­ing re­spons­ib­il­it­ies. Since I would turn 66 that year in 1997 and wanted to de­vote more time to math­em­at­ic­al visu­al­iz­a­tion pro­grams in col­lab­or­a­tion with Her­mann Karch­er, I de­cided to ac­cept the of­fer. I con­tin­ued to teach one course per year at Bran­de­is from 1997 to 2003, when my wife, Chuu-Li­an Terng, re­ceived an of­fer from the Uni­versity of Cali­for­nia, Irvine (UCI). We moved to Irvine in 2004, where I con­tin­ued to work on my math­em­at­ic­al visu­al­iz­a­tion pro­jects, en­gage in math­em­at­ic­al re­search, and teach one course per year at UCI un­til 2009.

I en­joyed work­ing with gradu­ate stu­dents, of­ten feel­ing I learned as much from them as they did from me. Many of my PhD stu­dents went on to pur­sue aca­dem­ic ca­reers, while oth­ers have found suc­cess in high-tech and fin­an­cial in­dus­tries. It has been a great pleas­ure of my life to main­tain con­nec­tions with many of them, who con­tin­ue to vis­it me from time to time. I take pride in all my gradu­ate stu­dents, and would like to high­light three of them here.

Kar­en Uh­len­beck re­ceived her Ph.D. in 1968 and is now a pro­fess­or emer­it­us of math­em­at­ics at the Uni­versity of Texas at Aus­tin, as well as a dis­tin­guished vis­it­ing pro­fess­or at the In­sti­tute for Ad­vanced Study. In 2019, she was awar­ded the pres­ti­gi­ous Abel Prize for her pi­on­eer­ing con­tri­bu­tions to geo­met­ric par­tial dif­fer­en­tial equa­tions, gauge the­ory, and in­teg­rable sys­tems, which have had a pro­found im­pact on ana­lys­is, geo­metry, and math­em­at­ic­al phys­ics. Not­ably, she is the first and, thus far, only wo­man to re­ceive this hon­or since the prize’s in­cep­tion in 2003. Demon­strat­ing her com­mit­ment to ad­van­cing the field, she gen­er­ously donated half of her prize money to or­gan­iz­a­tions that pro­mote great­er en­gage­ment by wo­men in re­search math­em­at­ics.

Leslie Lam­port earned his PhD in math­em­at­ics in 1972. He is renowned for his ground­break­ing work in dis­trib­uted sys­tems and is the ori­gin­al de­veloper of the \( \mathrm{\LaTeX} \) doc­u­ment pre­par­a­tion sys­tem, for which he au­thored the first manu­al. In re­cog­ni­tion of his sig­ni­fic­ant con­tri­bu­tions, Lam­port was awar­ded the 2013 Tur­ing Award for es­tab­lish­ing clear and co­her­ent frame­works with­in the of­ten chaot­ic realm of dis­trib­uted com­put­ing sys­tems, where mul­tiple autonom­ous com­puters com­mu­nic­ate via mes­sage passing. His de­vel­op­ment of im­port­ant al­gorithms and form­al mod­el­ing and veri­fic­a­tion pro­to­cols has greatly en­hanced the cor­rect­ness, per­form­ance, and re­li­ab­il­ity of com­puter sys­tems.

Bing-Le Wu is a part­ner at Cap­ula In­vest­ment Man­age­ment LLP, one of Europe’s largest hedge funds, with ap­prox­im­ately \$30 bil­lion in as­sets un­der man­age­ment as of 2024. Since 2020, Bing-Le has served as a Trust­ee at Bran­de­is Uni­versity and gen­er­ously es­tab­lished the en­dowed Richard Pal­ais Gradu­ate Fel­low­ship for the Bran­de­is Math­em­at­ics De­part­ment, demon­strat­ing his com­mit­ment to sup­port­ing fu­ture gen­er­a­tions of math­em­aticians.

I feel grate­ful for the op­por­tun­ity to ment­or such tal­en­ted stu­dents and to wit­ness their re­mark­able achieve­ments in their re­spect­ive fields.

Someone told me after listen­ing to my lec­ture on “Morse The­ory on Hil­bert Man­i­folds” at the Har­vard to­po­logy sem­in­ar that he just heard a talk very sim­il­ar to mine by Steve Smale at Columbia. I wrote to Smale and we real­ized that we had very sim­il­ar res­ults. In fact, we each give three of the same con­di­tions (a), (b), (c) for a func­tion on Hil­bert man­i­folds that our Morse the­ory would work for, and the third one is the es­sen­tial one and later was known as Con­di­tion C or the Pal­ais–Smale Con­di­tion C. We de­cided to pub­lish a joint an­nounce­ment in the Bul­let­in of the AMS and to pub­lish our in­di­vidu­al pa­pers sep­ar­ately. Com­mu­nic­a­tion was so dif­fer­ent in the old days; we re­lied on reg­u­lar mails and pri­or­ity fights were un­com­mon.

The ori­gin­al proof of Schwar­zchild’s solu­tion of the Ein­stein equa­tions was very long, but an al­tern­ate proof by Her­mann Weyl was short and el­eg­ant: He var­ied the Hil­bert–Ein­stein func­tion­al on the set of the met­rics in­vari­ant un­der SO(3). Since such met­rics only de­pend on the ra­di­us, the Euler–Lag­range equa­tion be­comes an or­din­ary dif­fer­en­tial equa­tion, which al­lowed him to write down the ex­pli­cit solu­tion. In oth­er words, he stated that if a func­tion \( f \) is in­vari­ant un­der a group, then to find a crit­ic­al point of \( f \) that is fixed by \( G \), we only need to find crit­ic­al points of the re­stric­tion of \( f \) to the set of fixed points of \( G \). I found that this was not al­ways true, but it is true if the group or the iso­tropy sub­group of the ac­tion is com­pact, which is the case for a lot of ap­plic­a­tions in geo­metry and phys­ics. I wrote up the pa­per [3] and sub­mit­ted it to the ed­it­or (Sydney Cole­man, a phys­i­cist at Har­vard) of the Com­mu­nic­a­tions of Math­em­at­ic­al Phys­ics. The pa­per was ac­cep­ted right away and Cole­man told me that he was sur­prised this prin­ciple may not al­ways work. This has be­come one of my most quoted pa­pers.

I went to Taiwan with my wife Chuu-Li­an Terng in the sum­mer of 1981 and gave a sum­mer course on math­em­at­ic­al phys­ics at Na­tion­al Tsing-Hua Uni­versity in Taiwan. I wrote up my lec­ture notes and it was pub­lished there [4].

Around 1982, I was work­ing on po­lar ac­tions, that is, iso­met­ric ac­tions on a Rieman­ni­an man­i­fold that ad­mit a sec­tion (a closed sub­man­i­fold that meets all or­bits and meets them or­tho­gon­ally). A typ­ic­al ex­ample is the ac­tion of SO\( (n) \) on the space of \( n\times n \) sym­met­ric real matrices by con­jug­a­tion, and the set of di­ag­on­al matrices is a sec­tion. Chuu-Li­an was read­ing a pa­per by Carter and West that stud­ied codi­men­sion 2 iso­pa­ra­met­ric sub­man­i­folds of spheres, and tried to find the the­ory for any codi­men­sion. She real­ized that prin­cip­al or­bits of po­lar ac­tions on spheres are the ho­mo­gen­eous iso­pa­ra­met­ric sub­man­i­folds in spheres. So we worked out a struc­ture the­ory of po­lar ac­tions and Chuu-Li­an de­veloped the the­ory of iso­pa­ra­met­ric sub­man­i­folds in space forms. Gud­laugur Thorbergs­son and Enrst Heintze joined us to de­vel­op a the­ory of hy­per­pol­ar ac­tions in the 1990s.

Shi­ing-Shen Chern in­vited Chuu-Li­an and me to give two sum­mer courses at Nankai In­sti­tute in Tianjin, China, in 1987. We each gave a one-month “mini course”. Chern was try­ing to help build math­em­at­ic­al re­search in China after the Cul­tur­al Re­volu­tion by in­vit­ing west­ern math­em­aticians to go to China for con­fer­ences or to give mini courses. Nankai would get stu­dents from many dif­fer­ent uni­versit­ies to at­tend these courses. Chuu-Li­an gave a course on iso­pa­ra­met­ric the­ory and I gave a course on Morse the­ory. We spent one semester to write up the lec­tures we gave there as the book Crit­ic­al Point The­ory and Sub­man­i­fold Geo­metry [5].

I liked the res­ults in the pa­per “Pois­son ac­tions and scat­ter­ing the­ory for in­teg­rable sys­tems” by Chuu-Li­an Terng and Kar­en Uh­len­beck [e2]. In this pa­per, they ex­plain the com­mut­ing Hamilto­ni­an flows, the scat­ter­ing and the in­verse scat­ter­ing the­ory of soliton equa­tions in terms of group ac­tions. When we vis­ited the Max Planck In­sti­tute for Math­em­at­ics in Bonn as mem­bers in 1997, I was in­vited to give a “mini course” at the Uni­versity of Bonn. Since I liked the joint pa­per by Chuu-Li­an and Kar­en, I worked very hard to give an in­tro­duc­tion to the his­tory of soliton the­ory and present their res­ults on sym­met­ries of solitons. This ef­fort res­ul­ted in the pa­per “The sym­met­ries of solitons” [7]. Chuu-Li­an and Kar­en com­men­ted that it was such a nice present­a­tion that people might read my pa­per in­stead of theirs.

In the later years of my ca­reer, I de­veloped a strong in­terest in math­em­at­ic­al visu­al­iz­a­tion, and cre­ated a pro­gram called 3D-XplorMath. This is a tool for aid­ing in the visu­al­iz­a­tion of a wide vari­ety of math­em­at­ic­al ob­jects and pro­cesses. Based on what I learned from my ex­per­i­ence in writ­ing this pro­gram, I wrote an es­say called “The visu­al­iz­a­tion of math­em­at­ics: To­wards a math­em­at­ic­al ex­plor­at­or­i­um” [8]. Her­mann Karch­er worked jointly with me on de­vel­op­ing 3D-XplorMath (3d-xplormath.org) and the Vir­tu­al Math Mu­seum (vir­tu­al­math­mu­seum.org). Sci­ence magazine ran a yearly visu­al­iz­a­tion chal­lenge (co­sponsored by the NSF) which graph­ic artist Luc Bern­ard and I jointly won in 2006 (the first year of this chal­lenge). Our win­ning visu­al­iz­a­tion was dis­played on the cov­er of an is­sue of Sci­ence (see Figure 1).