Celebratio Mathematica

I first met Andy in 1956, when he taught sopho­more ab­stract al­gebra at nine in the morn­ing (even on fall foot­ball Sat­urdays). He agreed to let me audit his course and sub­mit home­work pa­pers.

It took me sev­er­al years and two gradu­ate courses to real­ize how de­cept­ive a lec­turer he was. The proofs scrolled by. You could read his writ­ing. He lit­er­ally dot­ted his i’s and crossed his t’s. I know; I re­cently found the purple dit­toed hand­writ­ten lin­ear al­gebra notes he wrote for us in the spring of 1957. Strangely, those notes were some­times subtly hard to study from. Now I know why. He took such care pre­par­ing and searched so hard for eco­nomy and el­eg­ance that the rough places were made plain. The hard parts didn’t seem so in the seam­less flow, so it could be hard to find the crux of a proof. George Mackey once told me it was good that one of his teach­ers (I choose to for­get who) was dis­or­gan­ized be­cause it forced him (George) to mas­ter the ma­ter­i­al for him­self. What Andy’s style proves is that dis­or­gan­iz­a­tion may be suf­fi­cient but is not ne­ces­sary.

Andy was the read­er for my un­der­gradu­ate thes­is on mul­ti­pli­city the­ory for ei­gen­val­ues of bounded self-ad­joint op­er­at­ors on Hil­bert space. In those days (per­haps still) each seni­or was set a spe­cial ex­am on the thes­is top­ic. One ques­tion on mine asked me to ap­ply my the­or­ems to the mul­ti­plic­a­tion op­er­at­or \( g \mapsto fg \) for \( g \) in a Hil­bert space \( L^2 (\mu ) \). The func­tion \( f \) was a cu­bic poly­no­mi­al, and, \( \mu \) was Le­besgue meas­ure on \( [0, 2] \) with an ex­tra atom of weight 1 at 1. For­tu­nately, I’d thought of put­ting an ex­ample like that in the thes­is, so I knew how to do the prob­lem. What mattered was where the cu­bic was \( 1 : 1 \), \( 2 : 1 \), or \( 3 : 1 \). But my an­swer seemed not to need the atom at 1. When I asked Andy later about that, he gently poin­ted out how he’d care­fully con­struc­ted his cu­bic with a loc­al max­im­um at 1, so there was a set of pos­it­ive meas­ure on which the cu­bic was \( 2 : 1 \). I missed that, be­cause when find­ing the crit­ic­al points I cal­cu­lated \( 2 \times 3 = 12 \). He gra­ciously said only that I’d spoiled a good prob­lem.

When I chose Andy as a doc­tor­al thes­is ad­visor I had neither a top­ic nor a dir­ec­tion. I thought I was an ana­lyst and thought he was one and knew him, so I op­ted to try to work with him. I struggled with \( p \)-ad­ic groups for a year, go­ing nowhere. But I did have one idea about a way I might prove the Radon–Nikodym the­or­em for meas­ures on lat­tices like those that come up in quantum mech­an­ics. The idea didn’t work, but I did man­age to say some new things about meas­ures on Boolean al­geb­ras even while the gen­er­al­iz­a­tions to lat­tices eluded me. Andy en­cour­aged that play and said after a while that what I was work­ing on was in fact my thes­is. He told me he liked it bet­ter when his stu­dents found top­ics than when he had to sug­gest one.

In the spring of 1964 I thought my thes­is was done. I found the cent­ral the­or­em in Feb­ru­ary, wrote it up, and sent it off to Andy. When I tele­phoned to ask what he thought of it, he asked if I needed my de­gree in hand to ac­cept my new job at Bryn Mawr. When I said “no” he said, “Work on it an­oth­er year.” I know that if I’d said “yes” he’d have ac­cep­ted what I’d writ­ten. But then I’d have had a thes­is with just a the­or­em. The cent­ral mech­an­ism for pro­du­cing ex­amples and counter­examples show­ing the the­or­em was sharp came later that spring. Moreover, I think the idea was his, al­though I didn’t give him due cred­it then. So Andy was right to care about the qual­ity of the work and to ask for the ex­tra year. The thes­is was bet­ter and bet­ter writ­ten and ready for pub­lic­a­tion soon after the de­gree was awar­ded — and my year-old daugh­ter got to go to my com­mence­ment. (He and Jean sent her a Raggedy Andy when she was born.)

Eight­een years later Andy em­ployed her as a paint­er. That’s how I learned how he ap­plied lo­gic out­side math­em­at­ics. She saw him eat­ing break­fast hur­riedly one day — pea­nut but­ter spread on bread right out of the freez­er. He said the nu­tri­tion­al value was the same.

I was telling Andy once about a bijec­tion I’d found for count­ing per­muta­tions with par­tic­u­lar cycle struc­tures. He was in­ter­ested and had some fur­ther ideas and ref­er­ences. When he sug­ges­ted a joint pa­per [1] I jumped at the chance to earn a Gleason num­ber of 1. When I wanted to say something nu­mer­ic­al about the asymp­tot­ics which called for \( \Gamma (1/3) \), I looked up nearby val­ues in a table and in­ter­pol­ated. In re­sponse to a draft I sent Andy he wrote back:

There is one not ter­ribly im­port­ant thing where I can’t check you. You ob­tain \[ \frac{3^{1/6} e^{\pi \sqrt{3}/18}}{\Gamma (1/3)} \approx 0.6057624. \] With my hand cal­cu­lat­or I found \( \Gamma (1/3) \approx 2.678938543 \) (of which at least 8 fig­ures ought to be right) and hence the above num­ber comes out 0.6065193. Hand cal­cu­lat­ors make sub­stan­tial er­rors in ex­po­nen­tials, so I really don’t know which is right.

Andy’s “With my hand cal­cu­lat­or I found…” is a little disin­genu­ous. There’s no \( \Gamma \) key on the cal­cu­lat­or — he pro­grammed the com­pu­ta­tion. Today Math­em­at­ica quickly finds \( \Gamma (1/3) \approx 2.67893853471 \) with twelve sig­ni­fic­ant fig­ures, so Andy’s in­tu­ition about eight was right.

Over the years I had lunch with Andy of­ten, sampling Chinese, Vi­et­namese, and In­di­an food in Cam­bridge and nearby towns. Over lunch once, think­ing about geo­metry, he told me he’d give a lot for “one good look at the fourth di­men­sion.” Any math­em­at­ic­al top­ic, at any level of soph­ist­ic­a­tion, was fair game. I’d tell him why I thought the con­ven­tion for writ­ing frac­tions was up­side down; he’d tell me he was think­ing about the found­a­tions of geo­metry or the Riemann hy­po­thes­is. Of­ten in the past year I’ve wanted to ask him about something that came up in my teach­ing or while edit­ing these es­says and was stunned anew by the real­iz­a­tion that I couldn’t ever do that again.

Andy was a prob­lem solv­er more than a the­ory build­er. He liked hard prob­lems, like Hil­bert’s Fifth, about which you can read more be­low. Oth­ers less deep in­ter­ested him no less. I think he even en­joyed the prob­lems in spher­ic­al tri­go­no­metry and nav­ig­a­tion on the ex­ams he took to main­tain his nav­al com­mis­sion while in the re­serves.

Once he set out to dis­cov­er which reg­u­lar poly­gons you could con­struct if you add the abil­ity to tri­sect angles to the tasks avail­able with Eu­c­lidean straightedge and com­pass. His an­swer, in “Angle tri­sec­tion, the hep­tagon, and the triskai­deca­gon” [2]: just the \( n \)-gons for which the prime fac­tor­iz­a­tion of \( n \) is of the form \( 2^r 3^s p_1 p_2 \cdots p_k \), where the \( p_i \) are dis­tinct primes great­er than 3, each of the form \( 2^t 3^u + 1 \). His proof de­pends on the ob­ser­va­tion that these are pre­cisely the primes for which the cyc­lo­tom­ic field has de­gree \( 2^t 3^u \) and so can be con­struc­ted by a se­quence of ad­junc­tions of roots of quad­rat­ics and of cu­bics, all of whose roots are real.

You solve such a cu­bic by tri­sect­ing an angle, be­cause when the cu­bic has three real roots (the cas­us ir­re­du­cib­il­is), find­ing them with Card­ano’s for­mula re­quires ex­tract­ing the cube root of a com­plex num­ber. To do that you tri­sect its po­lar angle and find the cube root of the mod­u­lus. For the par­tic­u­lar cu­bics that come up in the con­struc­tion of these reg­u­lar poly­gons, the mod­u­lus is the \( 3/2 \) power of a known quant­ity, so a square root com­putes the cube root.

Andy’s solu­tion to that prob­lem al­lowed him to in­dulge sev­er­al of his pas­sions. The pa­per is full of his­tor­ic­al ref­er­ences, in­clud­ing the co­rol­lary that the abil­ity to tri­sect angles doesn’t help you du­plic­ate the cube. That re­quires solv­ing the oth­er kind of cu­bic. He cites (among oth­ers) Plemelj, Fer­mat, Euler, and Trop­fke and con­cludes with a quote from Gauss’s Dis­quisi­tiones Arith­met­icae.1 The “triskai­deca­gon” in the title, where most of us would be sat­is­fied with “13-gon”, ex­em­pli­fies Andy’s love of lan­guage. He had lots of ideas he nev­er got around to pub­lish­ing. I won­der if he wrote this pa­per in part just so he could use that word.

Andy loved to com­pute too. About his con­struc­tion of the triskai­deca­gon he writes:

After con­sid­er­able com­pu­ta­tion we ob­tain \[ 12 \cos \frac{2\pi}{13}= \sqrt{13}-1 + \sqrt{104-8\sqrt{13}} \cos \frac{1}{3}\arctan \frac{\sqrt{3}(\sqrt{13} +1)}{7-\sqrt{13}}. \]

Math­em­at­ica con­firms this nu­mer­ic­ally to one hun­dred decim­al places. I don’t think there’s soft­ware yet that would find the res­ult in this form.

I first ex­pli­citly en­countered Andy’s pas­sion for pre­ci­sion of ex­pres­sion when in gradu­ate school he told me that the prop­er way to read “101” aloud is “one hun­dred one” without the “and”. That pas­sion stayed with him to the end: when he was ad­mit­ted to the hos­pit­al and asked to rate his pain on a scale of 1 to 10, he’s re­puted to have said first, “That’s a ter­rible scale to use….”

Andy told me once that he knew he wanted to be a math­em­atician just as soon as he out­grew want­ing to be a fire­man.2 He suc­ceeded.

In the es­says that fol­low you’ll find more about Andy’s math­em­at­ics and more stor­ies. I’ll close here by quot­ing some that aren’t in­cluded there.

Per­si Di­ac­onis writes about Andy’s le­gendary speed:

Andy was an (un­of­fi­cial) thes­is ad­visor. This was il­lu­min­at­ing and de­press­ing. My thes­is was in ana­lyt­ic num­ber the­ory, and I would meet with Andy once a week. A lot of the work was tech­nic­al, im­prov­ing a power of a log­ar­ithm. I re­mem­ber sev­er­al times com­ing in with my cur­rent best es­tim­ates after weeks of work. Andy glanced at these and said, “I see how you got this, but the right an­swer is prob­ably…” I was shocked, and it turns out he was right.

Jill Mesirov de­scribes a sim­il­ar ex­per­i­ence:

I re­mem­ber quite clearly the first time that I met Andy Gleason. I was work­ing at IDA in Prin­ceton at the time, and Andy was a mem­ber of the Fo­cus Ad­vis­ory Com­mit­tee. The com­mit­tee met twice a year to re­view the work be­ing done, and I had been asked to give a present­a­tion of some work on speech I had done jointly with Melvin Sweet. I worked hard on the present­a­tion, and de­signed it to give some idea of how we were led step by step to the an­swer. The ground­work was laid for re­veal­ing each in­sight we had gained, but in such a way that it should come as a sur­prise to the audi­ence and thus make them ap­pre­ci­ate the sense of dis­cov­ery we had en­joyed as we did the re­search and solved the puzzle ourselves. Need­less to say, I hadn’t coun­ted on Andy’s “in­fam­ous” speed!

Twice I care­fully led the audi­ence through some twis­ted trail to end with the ques­tion, “So, what do you think we tried next?” Twice, be­fore the words had be­gun to leave my mouth, Andy was say­ing, “Oh, I see, then you want to do this, this, and this, after which you’ll ob­serve that…” While I ap­pre­ci­ated his quick grasp of the is­sues, I was be­gin­ning to see my care­fully laid plans fall­ing by the way­side. There­fore, as I was reach­ing the next cres­cendo, and I saw Andy lean­ing for­ward in his seat, I turned around, poin­ted my fin­ger at him and shouted, “You, be quiet!” He smiled, and left me to lead the rest of the crowd through the rev­el­a­tions.

Fi­nally, Vic­tor Man­jar­rez, a gradu­ate school con­tem­por­ary of mine, of­fers this sum­mary:

In the late fifties and early six­ties I took gradu­ate al­gebra and a read­ing course from An­drew Gleason. Whenev­er we spoke at meet­ings in later years I was struck by how un­fail­ingly po­lite he al­ways was. The Eng­lish word “po­lite” (marked by con­sid­er­a­tion, tact, or cour­tesy) evokes the French “po­litesse” (good breed­ing, ci­vil­ity), and the Greek “po­lites” (cit­izen — of the math­em­at­ics com­munity and the world), all of which An­drew Gleason ex­em­pli­fied to the fullest. This, of course, in ad­di­tion to his amaz­ing eru­di­tion.