Celebratio Mathematica

James Simons

James Harris Simons

by Rob Kirby

James Har­ris Si­mons was born on April 25, 1938, in New­ton, Mas­sachu­setts. His fath­er was in sales, ini­tially selling films for 20th Cen­tury Fox to theat­ers, then later en­ter­ing the shoe busi­ness with his fath­er-in-law. His moth­er went to art school and was, ac­cord­ing to Jim, “a pretty good paint­er”,1 though as a stay-at-home mom, she did not ex­hib­it com­mer­cially.

Jim at­ten­ded New­ton High School, and at the age of 17 entered MIT, gradu­at­ing in three years in 1958. Hear­ing that Chern was go­ing to Berke­ley, he de­cided to go there, too, to be able to work with him. Chern delayed his ar­rival in Berke­ley by a year, however, and Jim ended up writ­ing his PhD thes­is [1] un­der Bert Kostant, fin­ish­ing in 1961 (but gradu­at­ing form­ally in 1962). Jim tells the story of be­ing in a sem­in­ar dur­ing his second year, and see­ing a dis­tin­guished Chinese man walk in­to the sem­in­ar room; he asked who that was, and was startled to hear it was Chern. Jim had thought Chern was an ab­bre­vi­ated form of an east­ern European name, such as “Chernavski”.

Jim star­ted teach­ing at MIT in the fall of 1961 as a Moore in­struct­or. His stay there las­ted a year, at which point he resigned to go in­to busi­ness with friends who launched a floor tile man­u­fac­tur­ing con­cern in Colom­bia, South Amer­ica. This was the first of three oc­ca­sions when Jim de­cided to “leave” math­em­at­ics, though this ini­tial de­par­ture was short-lived. As­sem­bling the man­u­fac­tur­ing equip­ment in Colom­bia turned out to be a lengthy pro­cess, and in the mean­time Jim took a sum­mer job cal­cu­lat­ing and pro­gram­ming Bessel func­tions for a radar com­pany. It didn’t take long for him to real­ize that he dis­liked the work and wished to re­turn to aca­demia.

Jim men­tioned his change of heart to Raoul Bott at Har­vard, who im­me­di­ately offered to in­clude him on his up­com­ing NSF con­tract for the aca­dem­ic year 1962–63. Not long after, he was offered as­sist­ant pro­fess­or­ships by Har­vard and MIT both. De­cid­ing between the two was a struggle, but Jim had grown up in Bo­ston and was swayed by Har­vard’s stature.

Jim spent two years at Har­vard, but was rest­less and de­cided to leave aca­demia a second time. In 1964 he went to work at the In­sti­tute for De­fense Ana­lyses (IDA) in Prin­ceton and be­came a code crack­er for four years. The IDA was flex­ible, al­low­ing him to spend half his time work­ing on their pro­jects, and half on his own math re­search.

The fo­cus of Jim’s math­em­at­ics at this time was min­im­al vari­et­ies. The Plat­eau prob­lem asks if a codi­men­sion-2 man­i­fold \( M^{n-2} \) in \( \mathbb{R}^n \) can bound a smooth \( (n-1) \)-di­men­sion­al man­i­fold of min­im­al area. The an­swer was known for \( n=3 \) (Jesse Douglas re­ceived the Fields Medal in 1936 for his solu­tion) and \( n=4 \) [e1]. Pre­vi­ous at­tempts were made us­ing ana­lys­is and meas­ure the­ory, but Jim was work­ing from a geo­met­er’s per­spect­ive, and think­ing about what it meant to have mean curvature zero. It was known that any sin­gu­lar­it­ies in a min­im­al sur­face are cones over min­im­al sur­faces in the \( (n-1) \)-sphere; thus the sin­gu­lar­it­ies are 0-di­men­sion­al. Jim found a meth­od to de­form a cone sin­gu­lar­ity to a smooth man­i­fold of smal­ler area, which worked for \( n \leq 7 \), thereby solv­ing Plat­eau’s prob­lem in those di­men­sions.

The solu­tion also gave a proof (for \( n\leq 8 \)) of the gen­er­al­ized Bern­stein’s con­jec­ture which stated that if \( f:\mathbb{R}^{n-1} \to\mathbb{R} \) is a \( C^{\infty} \) func­tion whose graph is a min­im­al sur­face in \( \mathbb{R}^n \), then \( f \) is lin­ear.

But the meth­od didn’t hold for the case of \( n=8 \). Jim could show that the cone on \( \mathbb{S}^3 \times\mathbb{S}^3 \) (em­bed­ded smoothly in \( \mathbb{S}^7 \)) in the 8-ball was loc­ally stable and loc­ally min­im­al. But he did not know how to show it was glob­ally min­im­al. He was non­ethe­less pleased with this body of res­ults, wrote them up, and cir­cu­lated the pa­per, which later ap­peared in the An­nals of Math­em­at­ics [2].

E. Bom­bieri, E. de Giorgi and E. Giusti [e2] then showed that the cone was in­deed a glob­al min­im­um, and this settled Plat­eau’s prob­lem and the Bern­stein con­jec­ture neg­at­ively in high­er di­men­sions. Years later, Bom­bieri told Jim that after hear­ing word of Si­mons’ res­ults, the three Itali­ans had worked for two days straight without sleep to ob­tain their res­ult.

Shortly after pub­lic­a­tion of the min­im­al sur­faces work, Jim was fired from the IDA be­cause of his out­spoken op­pos­i­tion to the Vi­et­nam War. In Septem­ber 1968, he ac­cep­ted an of­fer from the State Uni­versity of New York at Stony Brook to be­come chair of the math de­part­ment. He re­cog­nized this as an op­por­tun­ity to build the de­part­ment: Her­shel Far­kas, Ir­win Kra and Tony Phil­lips had joined Stony Brook the same year as Jim, so they had a nuc­le­us of young math­em­aticians, and right off Jim was able to hire James Ax, Jeff Chee­ger, Ron Douglas, De­tlef Gro­moll, and Ro­ger Howe (who later slipped away to Yale).

While at Stony Brook, Jim de­cided he wanted to em­bark on something new, hav­ing fin­ished his work on the Bern­stein con­jec­ture. So began his study of char­ac­ter­ist­ic classes, and his at­tempt to find a com­bin­at­or­i­al for­mula for the first Pontry­agin class \( p_1 \) of a 4-man­i­fold. Only later did he dis­cov­er that many math­em­aticians had so tried — and failed.

As Chern and Si­mons stated ([3], page 48), “The hope was that by in­teg­rat­ing the char­ac­ter­ist­ic curvature form […] sim­plex by sim­plex, and re­pla­cing the in­teg­ral over each in­teri­or by an­oth­er on the bound­ary, one could eval­u­ate these bound­ary in­teg­rals, add up over the tri­an­gu­la­tion, and have the geo­metry wash out, leav­ing the sought after com­bin­at­or­i­al for­mula. This pro­cess got stuck by the emer­gence of a bound­ary term which did not yield to a simple com­bin­at­or­i­al ana­lys­is”, and which “seemed in­ter­est­ing in its own right”.

This bound­ary term, a 3-form on the frame bundle over the link of a sim­plex, the 3-sphere, could also be con­sidered for any ori­ent­able 3-man­i­fold. Jim saw that the 3-form was closed, thus rep­res­ent­ing a real co­homo­logy class on the frame bundle. A cross-sec­tion of the frame bundle would pull the co­homo­logy class down to the base 3-man­i­fold, and dif­fer­ent cross-sec­tions would change the an­swer by an in­teger, so Jim now had an in­ter­est­ing co­homo­logy class over \( \mathbb{R}/\mathbb{Z} \) on the 3-man­i­fold, which, eval­u­at­ing on the 3-man­i­fold, gives an in­vari­ant in \( \mathbb{R}/\mathbb{Z} \).

But all this de­pends on a met­ric, so Jim wondered how the in­vari­ant would change un­der a con­form­al change in the met­ric — though, in fact, it turned out not to change. Then came an epi­phany dur­ing a flight to Min­nesota with Jeff Chee­ger. Jim wanted to cal­cu­late what hap­pens if the 3-man­i­fold is con­form­ally em­bed­ded in \( \mathbb{R}^4 \), and he dis­covered to his dis­ap­point­ment that the in­vari­ant was zero. But Jeff thought it was a “fant­ast­ic” res­ult, for then the in­vari­ant gives an ob­struc­tion to a con­form­al em­bed­ding of a con­form­al 3-man­i­fold in­to \( \mathbb{R}^4 \).

That’s when Jim real­ized he was onto something with his bundle of res­ults. Chern got very ex­cited when Jim showed it to him, and that was the be­gin­ning of their col­lab­or­a­tion. As Jim ex­plained, “I nev­er pub­lished the 3-man­i­fold res­ults be­cause it was sub­sumed in this big­ger bundle.”

The for­mula for the 3-form on the frame bundle is the now fam­ous \[ \operatorname{Tr}\mkern2mu[A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A], \] whose in­teg­ral gives the ac­tion in Chern–Si­mons the­ory, a 3-di­men­sion­al to­po­lo­gic­al quantum field the­ory dis­covered by Ed Wit­ten. Jim re­calls, “The phys­i­cists love that term. I re­mem­ber telling Yang at a cer­tain point that I’ve got a bunch of num­bers, but he didn’t want to pur­sue it, and I didn’t know any phys­ics. It wasn’t un­til six or sev­en years later that Wit­ten picked it up. So it was a fluke; I didn’t set out to change phys­ics but you nev­er know what you’re go­ing to get when you set out with a good prob­lem.”

Jim was honored in 1975 by the eighth Veblen Prize in Geo­metry for the above­men­tioned math­em­at­ics. The oth­er Veblen Prize that year went to Bill Thur­ston.

After his col­lab­or­a­tion with Chern, Jim worked with Jeff Chee­ger on what turned in­to dif­fer­en­tial char­ac­ters [4], and this led to fur­ther work (men­tioned in [e3]) which was quite frus­trat­ing to Jim and partly ex­plains his de­cision to leave math for the third time.

The dif­fer­en­tial char­ac­ters were sec­ond­ary char­ac­ter­ist­ic classes of \( G \)-bundles which could be de­scribed more geo­met­ric­ally in the spe­cial case of \( \mathit{SO}(n) \)-bundles over a sim­pli­cial com­plex, where \( \mathit{SO}(n) \) is con­sidered as a dis­crete group. Di­vid­ing out by \( \mathit{SO}(n-1) \) to get an \( (n-1) \)-sphere fiber, and tak­ing a sec­tion over the bound­ary of an \( (n-1) \)-sim­plex, one gets (gen­er­ic­ally) an \( (n-2) \)-sphere de­term­ined by the \( n \) ver­tices and the geodes­ic faces they de­scribe. Nor­mal­iz­ing the met­ric so that the volume of the sphere is one, the volume of the \( (n-1) \)-sim­plex is an in­ter­est­ing real num­ber mod \( \mathbb{Z} \).

As­sum­ing the di­hed­ral angles are ra­tion­al, Jim and Jeff ex­pec­ted the volumes to be ir­ra­tion­al, des­pite some atyp­ic­al cases where the volume is ra­tion­al. They were un­able to find any such cases, however, and this prob­lem re­mains open today.

Jim spent 1975–76 at the Uni­versity of Geneva do­ing math re­search, but also some trad­ing on the stock mar­ket. When he re­turned in 1976–77, he real­ized he was good at in­vest­ing, and that he liked it, so he de­cided to go in­to it full time. This de­cision led to the birth of Renais­sance Tech­no­lo­gies.

Mean­while, Jim’s oth­er en­tre­pren­eur­i­al ven­tures flour­ished. The jazz club Mt. Au­burn 74 was a re­sound­ing suc­cess. His friend Steve Kuhn who was at Har­vard with him played there of­ten and has since be­come a fam­ous jazz pi­an­ist. Joan Baez got her start there. And the South Amer­ic­an pro­ject went on just as planned: the equip­ment ar­rived, the man­u­fac­tur­ing plant opened and the part­ners now have a net­work of busi­nesses op­er­at­ing there. There are oth­er ven­tures one could men­tion, but the most fam­ous is Renais­sance Tech­no­lo­gies, es­pe­cially the Medal­lion Fund, ar­gu­ably the most suc­cess­ful hedge fund ever.

Through the Si­mons Found­a­tion, Jim has also been act­ive as a phil­an­throp­ist in a wide vari­ety of fields. The Found­a­tion’s con­tri­bu­tions to the math­em­at­ic­al sci­ences alone are con­sid­er­able: it es­tab­lished the Si­mons Cen­ter for Geo­metry and Phys­ics (at Stony Brook), and has sup­por­ted pro­grams and in­sti­tu­tions as var­ied as Math for Amer­ica, the Afric­an Math­em­at­ics Pro­ject, gradu­ate and postdoc­tor­al fel­low­ships in math and com­puter sci­ence, chal­lenge grants to SUNY Stony Brook and MSRI, the con­struc­tion of MSRI’s Chern Hall, Brookhaven Na­tion­al Labor­at­ory, and the en­dow­ing of sev­er­al uni­versity chairs.

With his wife, Mar­ilyn, Si­mons cre­ated the Avalon Park and Pre­serve in Stony Brook un­der the aus­pices of the Paul Si­mons Found­a­tion (es­tab­lished in memory of their son, Paul), and the couple has made sig­ni­fic­ant con­tri­bu­tions to aut­ism re­search (SFARI).

The Si­mons’ phil­an­thropy has an in­ter­na­tion­al scope as well: the Nick Si­mons In­sti­tute in Nepal, foun­ded in memory of their son Nick who had traveled widely in the re­gion, sus­tains a rur­al health care ini­ti­at­ive that was launched in 2006.

Jim re­tired from Renais­sance Tech­no­lo­gies in 2010 and is yet again work­ing on math, hav­ing em­barked on a series of joint pa­pers with Den­nis Sul­li­van.


[1] J. Si­mons: “On the trans­it­iv­ity of holonomy sys­tems,” Ann. Math. (2) 76 : 2 (September 1962), pp. 213–​234. Based on the au­thor’s 1962 PhD thes­is. MR 0148010 Zbl 0106.​15201 article

[2] J. Si­mons: “Min­im­al vari­et­ies in rieman­ni­an man­i­folds,” Ann. Math. (2) 88 : 1 (July 1968), pp. 62–​105. Rus­si­an trans­la­tions were pub­lished in Tse­lochislennyye po­toki i min­im­al’nyye poverkh­nosti (1973) and Matem­atika 16:6 (1972). MR 0233295 Zbl 0181.​49702 article

[3] S.-S. Chern and J. Si­mons: “Char­ac­ter­ist­ic forms and geo­met­ric in­vari­ants,” Ann. Math. (2) 99 : 1 (January 1974), pp. 48–​69. MR 0353327 Zbl 0283.​53036 article

[4] J. Chee­ger and J. Si­mons: “Dif­fer­en­tial char­ac­ters and geo­met­ric in­vari­ants,” pp. 50–​80 in Geo­metry and to­po­logy (Col­lege Park, MD, 1983–1984). Edi­ted by J. Al­ex­an­der and J. Harer. Lec­ture Notes in Math­em­at­ics 1167. Spring­er (Ber­lin), 1985. MR 827262 Zbl 0621.​57010 incollection