Celebratio Mathematica

“Have you ever thought of this?” is how An­drew Gleason of­ten pre­ceded the for­mu­la­tion of some idea, or some ques­tion, to his math­em­at­ic­al col­leagues and stu­dents. Usu­ally we had not (thought of the idea) and even if we had, we would not have ex­pressed it in as cla­ri­fy­ing, or as en­ti­cing, a man­ner as he did. The “this” could range quite broadly: ideas re­lated to trans­form­a­tion groups and his fam­ous solu­tion to Hil­bert’s Fifth Prob­lem, to meas­ure the­ory, pro­ject­ive geo­metry, Hil­bert Spaces, or to com­bin­at­or­ics, to graph the­ory, to cod­ing the­ory, or — and this was also one of Gleason’s many great loves — to the teach­ing and per­fec­tion of math­em­at­ic­al skills at any level (how to treat meas­ur­ing when teach­ing first-graders; re­form­ing the teach­ing of Cal­cu­lus; and sa­vor­ing the latest Put­nam Com­pet­i­tion ex­am ques­tions). Quite a span.

An­drew Gleason’s own early edu­ca­tion had a sig­ni­fic­ant geo­graph­ic­al span. He gradu­ated from high school in Yonkers, New York, hav­ing also taken courses in Berke­ley, Cali­for­nia. His un­der­gradu­ate years at Yale were spent largely tak­ing gradu­ate level courses. When Andy gradu­ated in 1942, he joined the U.S. Navy as a mem­ber of a group of 8–10 math­em­aticians work­ing to crack en­emy codes.

In 1946 Gleason came to Har­vard, hav­ing been elec­ted as a Ju­ni­or Fel­low of the So­ci­ety of Fel­lows. To be such a Fel­low in those days meant that one could achieve an aca­dem­ic ca­reer without hav­ing a PhD. Andy nev­er did pur­sue a doc­tor­al de­gree; he spent his time in the So­ci­ety of Fel­lows de­vel­op­ing a broad math­em­at­ic­al cul­ture and think­ing about Hil­bert’s Fifth Prob­lem.

At the end of his three-year fel­low­ship, Gleason was ap­poin­ted As­sist­ant Pro­fess­or of Math­em­at­ics in the Har­vard De­part­ment of Math­em­at­ics. Soon there­after, he took a two-year leave of ab­sence from Har­vard to re­turn to the U.S. Navy to serve dur­ing the Korean War (mid-1950 to mid-1953). After this, Gleason re­turned to Har­vard where he spent the rest of his aca­dem­ic ca­reer. He was ap­poin­ted Pro­fess­or of Math­em­at­ics in 1957.

Gleason mar­ried Jean Berko in 1959. Jean Berko Gleason, a prom­in­ent psy­cho­lin­guist, has had a dis­tin­guished aca­dem­ic ca­reer as Pro­fess­or in the De­part­ment of Psy­cho­logy at Bo­ston Uni­versity. The Gleasons have three daugh­ters: Kath­er­ine, Pamela, and Cyn­thia.

Gleason was named the Hol­lis Pro­fess­or of Math­em­aticks and Nat­ur­al Philo­sophy in 1969 (the old­est en­dowed chair in the sci­ences in the United States). He be­came a Seni­or Fel­low of the Har­vard So­ci­ety of Fel­lows in 1977 and was Chair of the So­ci­ety of Fel­lows from 1989 to 1996. Gleason re­tired from Har­vard Uni­versity in 1992.

His math­em­at­ic­al con­ver­sa­tions, his sem­in­ar dis­cus­sions, his writ­ing, and his lec­tures had qual­it­ies most cher­ished in a math­em­atician: he was com­pre­hens­ible, clear and to the point; his for­mu­la­tions had a scin­til­lat­ing pre­ci­sion, and they were al­ways de­livered with en­thu­si­asm and wide-eyed won­der. One of his col­leagues once summed up this say­ing: “When he touched a thing, he made it shine.”

Our late col­league Raoul Bott once joked that Andy lacked the es­sen­tial Hun­gari­an tal­ent for be­ing ab­sent when an im­port­ant ad­min­is­trat­ive task needed to be done. One of us re­coun­ted this in a talk at Andy’s me­mori­al ser­vice, adding:

Andy served Har­vard, his de­part­ment, the So­ci­ety of Fel­lows, and the math­em­at­ic­al pro­fes­sion with gen­er­os­ity and skill. Al­though he held strong opin­ions, he nev­er im­posed them on oth­ers, and nev­er made any­one feel small if they didn’t pos­sess his bril­liance. In fact, I nev­er heard Andy raise his voice, either in con­ver­sa­tion or in a meet­ing. That’s not to say he wasn’t con­vin­cing — it was his vis­ion of a small fac­ulty, train­ing the best gradu­ate stu­dents and amp­li­fied by the en­ergy of out­stand­ing un­der­gradu­ates, that defines the math­em­at­ics de­part­ment we have at Har­vard today.

Gleason’s best known work is his res­ol­u­tion of Hil­bert’s Fifth Prob­lem. Dav­id Hil­bert, slightly over a cen­tury ago, for­mu­lated two dozen prob­lems that have, since then, rep­res­en­ted cel­eb­rated mile­stones meas­ur­ing math­em­at­ic­al pro­gress. Many of the ad­vances in Hil­bert’s prob­lems ini­ti­ate whole new fields, new view­points. Those few math­em­aticians who have re­solved one of these prob­lems have been re­ferred to as mem­bers of the Hon­ors Class.

Hil­bert fash­ioned his “Fifth Prob­lem” as a way of of­fer­ing a gen­er­al com­mo­di­ous con­text for the then new the­ory of Sophus Lie re­gard­ing trans­form­a­tion groups. Nowadays, Lie’s the­ory is the main­stay of much math­em­at­ics and phys­ics, and his kind of groups, “Lie groups,” con­sti­tute an im­port­ant fea­ture of our ba­sic sci­entif­ic land­scape.

One can think of a trans­form­a­tion group as a col­lec­tion of sym­met­ries of a geo­met­ric space. Some spaces ad­mit in­fin­itely many sym­met­ries: think of the circle, which can be ro­tated at any angle. The grand prob­lem fa­cing Sophus Lie is how to deal with these in­fin­ite groups of sym­met­ries. Can one use the meth­ods of Cal­cu­lus ef­fect­ively to treat the is­sues that arise in con­nec­tion with these in­fin­ite trans­form­a­tion groups?

At the In­ter­na­tion­al Con­gress of Math­em­aticians held in Cam­bridge in 1950, Andy pro­posed a pos­sible meth­od to ar­rive at an (af­firm­at­ive!) an­swer to this ques­tion, in the con­text pro­posed by Hil­bert. Andy em­phas­ized the cent­ral role played by the one-para­met­er sub­groups in the pic­ture. The fol­low­ing year he proved a key res­ult about max­im­al con­nec­ted com­pact sub­groups, and the year after that, us­ing res­ults of Mont­gomery & Zip­pin, and Yamabe, Andy clinched things, and showed that the an­swer to Hil­bert’s ques­tion an­swer is “yes.” An ex­tremely im­port­ant ad­vance.

The depth of Andy’s work is ex­traordin­ary, as is its breadth: from his com­puter ex­plor­a­tions very early in the his­tory of ma­chine com­pu­ta­tion (a search prob­lem in the \( n \)-cube) to solv­ing a con­jec­ture of our late col­league George Mackey (about meas­ures on the closed sub­spaces of a Hil­bert space) to the in­tric­a­cies of fi­nite pro­ject­ive geo­metry and cod­ing the­ory, to the re­la­tion­ship between com­plex ana­lyt­ic geo­metry and Banach al­geb­ras. Gleason was also one of the rare breed of math­em­aticians who did not stay on just one side of the pure math­em­at­ics / ap­plied math­em­at­ics “di­vide.” In fact, his work and at­ti­tude gave testi­mony to the ten­et that there is no es­sen­tial di­vide. In­deed, the ideas and math­em­at­ic­al in­terests that Andy nur­tured in his ap­plied work for the gov­ern­ment, which was a pas­sion for him throughout his life­time, con­nects well with his pub­lic work on fi­nite geo­met­ries, and his love for com­bin­at­or­ics.

Andy’s in­terest in the train­ing of math­em­aticians and in ex­pos­i­tion and teach­ing in gen­er­al, led him to edit, with co-au­thors, a com­pen­di­um of three dec­ades of Wil­li­am Low­ell Put­nam math­em­at­ic­al com­pet­i­tion prob­lems, to write a bold text for­mu­lat­ing the found­a­tions of ana­lys­is start­ing with a grand and lu­cid tour of lo­gic and set the­ory, and also to en­gage in the im­port­ant pro­ject of K-12 math­em­at­ic­al edu­ca­tion, and to re­form ef­forts in the teach­ing of Cal­cu­lus.

The found­ing idea be­hind the vari­ous math­em­at­ic­al edu­ca­tion ini­ti­at­ives with which Andy was in­volved—either the pro­grams for early math­em­at­ic­al edu­ca­tion that were re­ferred to (by both de­tract­ors and pro­moters) as New Math, or the pro­grams for teach­ing Cal­cu­lus (provid­ing syl­labi and texts that came to be re­ferred to as the Har­vard Con­sor­ti­um) — was to present math­em­at­ics con­cretely and in­tu­it­ively, and to en­er­gize and em­power the stu­dents and teach­ers. The es­sen­tial mis­sion of the Cal­cu­lus Con­sor­ti­um was and is Andy’s credo that the ideas should be based in equal parts of geo­metry for visu­al­iz­a­tion of the con­cepts, com­pu­ta­tion to ground it in the real world, and al­geb­ra­ic ma­nip­u­la­tion for power.

This relates to Andy’s gen­er­al view: that a work­ing math­em­atician should have at his or her dis­pos­al a toolkit of ba­sic tech­niques for ana­lyz­ing any prob­lem. He felt that all good prob­lems in math — at any level — should weave to­geth­er al­gebra, geo­metry, and ana­lys­is, and stu­dents must learn to draw on any of these tools, hav­ing them all “at the ready.” Andy em­phas­ized this in the Math­em­at­ics De­part­ment’s dis­cus­sion re­gard­ing the struc­ture of the de­part­ment’s com­pre­hens­ive qual­i­fy­ing ex­am for gradu­ate stu­dents. He also loved to think about ex­am prob­lems that ex­hib­ited this uni­fy­ing call upon dif­fer­ent tech­niques; for ex­ample, he would work out the prob­lems of the (un­der­gradu­ate) Put­nam Com­pet­i­tion ex­am, year after year, just for fun.

Andy had many hon­ors. He re­ceived the New­comb Clev­e­land Prize from the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence for his work on Hil­bert’s Fifth Prob­lem. He re­ceived the Yueh-Gin Gung and Dr. Charles Y. Hu Award for Dis­tin­guished Ser­vice to Math­em­at­ics — the Math­em­at­ic­al As­so­ci­ation of Amer­ica’s most pres­ti­gi­ous award. He was pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1981–1982), a mem­ber of the Na­tion­al Academy of Sci­ences, the Amer­ic­an Academy of Arts and Sci­ences, and the Amer­ic­an Philo­soph­ic­al So­ci­ety.

Re­spect­fully sub­mit­ted,

Be­ne­dict Gross
Dav­id Mum­ford
Barry Mazur