Celebratio Mathematica

Math­em­aticians are con­ven­tion­ally meas­ured by the depth and cre­ativ­ity of their con­tri­bu­tions to re­search. On these grounds alone Ka­plansky is a tower­ing fig­ure. But an­oth­er, per­haps com­par­ably im­port­ant, way to con­trib­ute to the ad­vance­ment of math­em­at­ics lies in the build­ing of hu­man ca­pa­city, in the form­a­tion of pro­duct­ive young re­search­ers, through teach­ing, ment­or­ing, and writ­ten ex­pos­i­tion. In this re­gard, Ka­plansky, with an as­ton­ish­ing fifty-five doc­tor­al stu­dents (among whom I count my­self), and 627 math­em­at­ic­al des­cend­ants, has had a sin­gu­lar im­pact on our field.

Kap was born March 22, 1917, in Toronto, the young­est of four chil­dren, shortly after his par­ents had emig­rated to Canada from Po­land. His fath­er, hav­ing stud­ied to be a rabbi in Po­land, worked in Toronto as a tail­or. His moth­er, with little school­ing, was en­ter­pris­ing and built up a busi­ness, “Health Bread Baker­ies”, that sup­por­ted (and em­ployed) the whole fam­ily.

Kap showed an early and evolving tal­ent for mu­sic, as he him­self re­counts [e1]:

At age four, I was taken to a Yid­dish mu­sic­al, Die Goldene Kala (The Golden Bride). It was a rev­el­a­tion to me that there could be this kind of en­ter­tain­ment with mu­sic. When I came home I sat down and played the show’s hit song. So I was rushed off to pi­ano les­sons. After 11 years I real­ized there was no point in con­tinu­ing; I was not go­ing to be a pi­an­ist of any dis­tinc­tion…. I en­joy play­ing pi­ano to this day…. God in­ten­ded me to be the per­fect ac­com­pan­ist — or bet­ter, the per­fect re­hears­al pi­an­ist. I play loud, I play in tune, but I don’t play very well.

In­deed, Kap be­came a pop­u­lar ac­com­pan­ist and per­former through much of his ca­reer. At one point, to demon­strate the vir­tues of a struc­ture he dis­covered com­mon to his fa­vor­ite songs, he says, “I tried to show that you could [use it to] make a pass­able song out of such an un­prom­ising source of them­at­ic ma­ter­i­al as the first 14 di­gits of \( \pi \).” The res­ult­ing “Song about \( \pi \)” was later giv­en lyr­ics by En­id Rieser and is of­ten per­formed by Kap’s daugh­ter, Lucy, her­self a pop­u­lar folk sing­er-song­writer [e2]. Some more per­son­al fam­ily vign­ettes of Kap can be found be­low in Lucy’s re­min­is­cences of her fath­er.

As a seni­or at the Uni­versity of Toronto in 1938, Kap won the very first Put­nam Com­pet­i­tion, as did the Toronto team. This won him a fel­low­ship to Har­vard, where he earned his Ph.D. in 1941, un­der the dir­ec­tion of Saun­ders Mac Lane. He stayed on as a Ben­jamin Peirce In­struct­or till 1944, when Mac Lane brought him to the Ap­plied Math­em­at­ics Group at Columbia Uni­versity in 1944–45, which was do­ing work to sup­port the war ef­fort. Kap re­counts, “So that year was spent largely on or­din­ary dif­fer­en­tial equa­tions. I had a taste of real life and found that math­em­at­ics could ac­tu­ally be used for something.”

From there Kap moved to the Uni­versity of Chica­go in the fall of 1945, where he re­mained till his re­tire­ment in 1984, hav­ing chaired the de­part­ment dur­ing 1962–67. A year after Kap’s ar­rival, Mar­shall Stone came to Chica­go to dir­ect and build up the math­em­at­ics de­part­ment, ush­er­ing in what some have called “the Stone Age”. Stone made four gi­gant­ic ap­point­ments — Saun­ders Mac Lane, Ant­oni Zyg­mund, An­dré Weil, and Shi­ing-Shen Chern — fol­lowed by waves of tal­en­ted young fac­ulty and gradu­ate stu­dents. Among the young­er col­leagues who greatly in­flu­enced Kap were Irving Segal, Paul Hal­mos, and Ed Span­i­er. Kap’s life style, out­side his fam­ily and mu­sic, was rig­or­ous and aus­tere. He sched­uled classes and meet­ings at (de­fi­antly) early hours of the morn­ing. Daily swim­ming was a lifelong prac­tice; he loved the Lake Michigan shoreline. Lunch was lean in time as well as sub­stance. With stu­dents he was gen­er­ous and in­dul­gent in math­em­at­ic­al con­ver­sa­tion, but en­ter­tained little else.

After Chica­go, Kap, suc­ceed­ing Shi­ing-Shen Chern, be­came the second dir­ect­or of the Math­em­at­ic­al Sci­ences Re­search In­sti­tute (MSRI) in Berke­ley, 1984–1992. Also in 1984, Kap was elec­ted pres­id­ent of the AMS. So we see here a ca­reer tra­ject­ory from a pre­co­cious col­lege stu­dent to a ded­ic­ated, well es­tab­lished and pro­lif­ic re­search­er, to a lead­er of some of the premi­er in­sti­tu­tions of the pro­fes­sion. Along the way, Kap was honored by elec­tion to the Na­tion­al Academy of Sci­ences and to the Amer­ic­an Academy of Arts and Sci­ences, and he was named an hon­or­ary mem­ber of the Lon­don Math­em­at­ic­al So­ci­ety. In 1989 the AMS awar­ded him the Steele Prize, Ca­reer Award.

To un­der­stand Kap’s math­em­at­ic­al ac­com­plish­ments, it is im­port­ant to speak of his stu­dents as well as his pub­lic­a­tions, to dis­tin­guish and com­pare what these two re­cords tell us. Kap’s math­em­at­ic­al work is dis­trib­uted across sev­er­al dif­fer­ent areas of math­em­at­ics. For pur­poses of sur­vey­ing them, I have some­what ar­bit­rar­ily grouped them as fol­lows, the ma­jor areas in bold font:

In this chro­no­lo­gic­al chart I have col­or-coded Kap’s journ­al art­icles, books, and mono­graphs ac­cord­ing to which of these areas they be­long. The data are taken from Math­S­ciNet. Not in­cluded are the nu­mer­ous con­tri­bu­tions to the Prob­lem sec­tions of the Amer­ic­an Math­em­at­ic­al Monthly; Kap re­mained throughout a vir­tu­oso prob­lem solv­er and con­trib­ut­or.

Sev­er­al re­mark­able things stand out from this chart.

• As a fresh Ph.D dur­ing the years of WWII, Kap pub­lished, bey­ond his dis­ser­ta­tion (on max­im­al fields with valu­ation), a small but in­ter­est­ing mix of pa­pers on com­bin­at­or­ics and on prob­ab­il­ity and stat­ist­ics, per­haps in part in­flu­enced by his ap­plied work at Columbia.

• Then, in the dec­ade 1945–54 there is an ex­traordin­ary out­pour­ing of pub­lic­a­tions, pre­dom­in­antly in what we are call­ing to­po­lo­gic­al al­gebra. In fact, in the four years 1948–52, Kap pub­lished thirty-two pa­pers! Some of this may have been back­log from the war years, but it is an as­ton­ish­ing en­semble of cut­ting-edge work in this area. Kap’s gen­er­al in­clin­a­tion was to al­geb­ra­ic­ally ax­io­mat­ize the vari­ous struc­tures of con­cern to func­tion­al ana­lysts, in the pro­gram launched earli­er by Mur­ray and von Neu­mann. Dick Kadis­on [e3] writes in some de­tail about this phase of Kap’s work.

• Kap’s work in pure non­com­mut­at­ive ring the­ory is a per­sist­ent, but re­l­at­ively mod­est theme in his work. One of his most in­flu­en­tial pa­pers, on “Rings with poly­no­mi­al iden­tity”, opened an im­port­ant branch of non­com­mut­at­ive al­gebra. Here he proves the fun­da­ment­al res­ult that a prim­it­ive al­gebra with poly­no­mi­al iden­tity is fi­nite di­men­sion­al over its cen­ter.

• Lie the­ory, in its many as­pects, is an­oth­er im­port­ant strand. This in­cludes work on the clas­si­fic­a­tion of simple Lie al­geb­ras in char­ac­ter­ist­ic \( p \), lec­ture notes on the solu­tion of Hil­bert’s Fifth Prob­lem, and work, partly in col­lab­or­a­tion with the phys­i­cist Peter Fre­und, on graded Lie al­geb­ras, su­per-sym­metry, and re­lated clas­si­fic­a­tion prob­lems. Peter Fre­und writes vividly be­low about their col­lab­or­a­tion.

• Quad­rat­ic (and high­er) forms: This sub­ject, from the be­gin­ning to the end of Kap’s ca­reer, was dear to his heart. This in­terest was first in­spired by his at­tend­ing L. E. Dick­son’s lec­tures in num­ber the­ory and quad­rat­ic forms at Chica­go in the 1940s. It was re­kindled dur­ing the years of his re­tire­ment, when he turned to the arith­met­ic the­ory of such forms, partly in col­lab­or­a­tion with Wil­li­am Jagy. A charm­ing ac­count of a sig­ni­fic­ant piece of this work can be found in the con­tri­bu­tion of Man­jul Bhar­gava be­low.

• In the eyes of many math­em­aticians today, com­mut­at­ive and ho­mo­lo­gic­al al­gebra is the field with which they now most as­so­ci­ate Ka­plansky’s name. Yet we see that its (yel­low) col­or oc­cu­pies re­mark­ably little of the chart of pub­lic­a­tions. How can we ex­plain this para­dox? Well, for one thing, Kap’s pub­lic­a­tions in this area in­clude sev­er­al books and mono­graphs (lec­ture notes), and these con­tain a num­ber of new res­ults and meth­ods that were not else­where pub­lished. This also re­flects the fact that Kap was gen­er­at­ing math­em­at­ics in this rap­idly evolving field more through in­struc­tion than through pa­pers writ­ten in solitude. And so what he was pro­du­cing math­em­at­ic­ally was sig­ni­fic­antly em­bod­ied in the work of the stu­dents who were learn­ing from him.

• We can see this phe­nomen­on in the next chart of Kap’s Ph.D. stu­dents, again col­or-coded by the areas of their dis­ser­ta­tions.

The first thing to no­tice in com­par­ing these two charts is that the “re­l­at­ive masses” of to­po­lo­gic­al al­gebra and com­mut­at­ive al­gebra have been ap­prox­im­ately re­versed, of course with a time shift. In to­po­lo­gic­al al­gebra, Kap was a pi­on­eer and a ma­jor, in­tensely pro­duct­ive, con­cep­tu­al de­veloper of the field. In com­mut­at­ive and ho­mo­lo­gic­al al­gebra, in con­trast, the field was already in rap­id mo­tion, in­to which Kap boldly ven­tured as more of an ap­pren­tice, guid­ing a flock of sim­il­arly un­ini­ti­ated gradu­ate stu­dents and postdocs with him. Ho­mo­lo­gic­al al­gebra was spawned from al­geb­ra­ic to­po­logy. In the hands of Ei­len­berg, Mac Lane, Grothen­dieck, and oth­ers it evolved in­to a new branch of al­gebra, em­bra­cing cat­egory the­ory and oth­er new con­structs. Mean­while, the Grothen­dieck–Serre re­for­mu­la­tion of al­geb­ra­ic geo­metry de­man­ded that its found­a­tions in com­mut­at­ive al­gebra be deepened and ex­pan­ded.

A ba­sic new concept of ho­mo­lo­gic­al al­gebra was that of glob­al ho­mo­lo­gic­al di­men­sion, a new ring-the­or­et­ic in­vari­ant. This turned out to be un­in­ter­est­ing for the most in­vest­ig­ated rings, fi­nite di­men­sion­al (non­com­mut­at­ive) al­geb­ras. On the oth­er hand, a land­mark dis­cov­ery (of Aus­lander–Buchs­baum and Serre) was that, for a com­mut­at­ive no­eth­eri­an loc­al ring \( A \), the glob­al di­men­sion of \( A \) is fi­nite if and only if \( A \) is reg­u­lar (the al­geb­ra­ic ex­pres­sion of the geo­met­ric no­tion of nonsin­gu­lar­ity). This equi­val­ence, and Serre’s ho­mo­lo­gic­al for­mu­la­tion of in­ter­sec­tion mul­ti­pli­cit­ies, firmly es­tab­lished ho­mo­lo­gic­al al­gebra as a fun­da­ment­al tool of com­mut­at­ive al­gebra. However, these de­vel­op­ments were known mainly on a Cam­bridge (MA)–Par­is-ax­is. It was in this con­text that Kap offered a Chica­go gradu­ate course in­tro­du­cing these new ideas, meth­ods, and res­ults, then still very much in mo­tion. Use of these meth­ods led to the gen­er­al proof (by Aus­lander–Buchs­baum) of unique fac­tor­iz­a­tion for all reg­u­lar loc­al rings. Kap’s course, and its se­quels, lif­ted a whole gen­er­a­tion of young re­search­ers (my­self in­cluded) in­to this field. This played out for Kap over the next two dec­ades, with many stu­dents and sev­er­al books to show for it.

In math­em­at­ic­al style, Kap was a prob­lem solv­er of great vir­tu­os­ity. For course goals he sought prob­lems, and the­or­ems of great ped­i­gree, and probed them deeply. His main fo­cus was on proofs (path­ways), more than on the­or­ems (des­tin­a­tions). He sought geodesics, and the most eco­nom­ic (high mileage) means to get there. Proof ana­lys­is led to double-edged kinds of gen­er­al­iz­a­tion/ax­io­mat­iz­a­tion:

• A giv­en proof yields more than claimed. The giv­en hy­po­theses de­liv­er more than the stated the­or­em prom­ises.

• The hy­po­theses can be weakened. We can get the same res­ults more cheaply, and so more gen­er­ally.

The strength of this dis­pos­i­tion was per­haps some­times over-zeal­ous, push­ing to­ward “pre­ma­ture mat­ur­a­tion” of the math­em­at­ics. But it was an ef­fect­ive mode of in­struc­tion, yield­ing power­ful con­cep­tu­al com­mand of the ter­rit­ory covered.

As the re­cord above in­dic­ates, and the testi­mo­ni­als be­low will af­firm, Kap was a gif­ted teach­er, ment­or, and writer. Here are a few of the things he him­self has said in re­flec­tion on this.

I like the chal­lenge of or­gan­iz­ing my thoughts and try­ing to present them in a clear and use­ful and in­ter­est­ing way. On the oth­er hand, to see the faces light up, as they oc­ca­sion­ally do, to even get them ex­cited so that maybe they can do a little math­em­at­ic­al ex­per­i­ment­a­tion them­selves — that’s pos­sible, on a lim­ited scale, even in a cal­cu­lus class.

Ad­vice to stu­dents: “Look at the first case, the easi­est case that you don’t un­der­stand com­pletely. Do ex­amples, a mil­lion ex­amples, ‘well chosen’ ex­amples, or ‘lucky’ ones. If the prob­lem is worth­while, give it a good try — months, maybe years if ne­ces­sary. Aim for the less ob­vi­ous, things that oth­ers have not likely proved already.”

And: “Spend some time every day learn­ing something new that is dis­joint from the prob­lem on which you are cur­rently work­ing (re­mem­ber that the dis­joint­ness may be tem­por­ary). And read the mas­ters.”

When a great math­em­atician has mastered a sub­ject to his sat­is­fac­tion and is present­ing it, that mas­tery comes through un­mis­tak­ably, so you have an ex­cel­lent chance of un­der­stand­ing quickly the main ideas. [He cites as ex­amples, Weil, Serre, Mil­nor, Atiyah.]

… the thing that be­dev­ils the math­em­at­ic­al pro­fes­sion — the dif­fi­culty we have in telling the world out­side math­em­at­ics what it is that math­em­aticians do. And for shame, for shame, right with­in math­em­at­ics it­self, we don’t tell each oth­er prop­erly.

And here is a sampling of how Kap was seen by oth­ers, in­clud­ing some of his stu­dents:

“He was not only a fant­ast­ic math­em­atician but a mar­velous lec­turer, and he had a re­mark­able tal­ent for get­ting the best out of stu­dents.”
— Richard G. Swan

“I knew Ka­plansky in his later years, and also through some of his books. Cheer­ful, gra­cious, and el­eg­ant are some of the words that come to mind when I think of him.”
— Ro­ger Howe

“The math­em­at­ic­al com­munity in In­dia is shocked to have news of the de­mise of Pro­fess­or Irving Ka­plansky. We all feel very sad at this ir­re­par­able loss. Pro­fess­or Ka­plansky was a source of in­spir­a­tion for math­em­aticians around the world. He will no doubt live for all time through his math­em­at­ic­al con­tri­bu­tions. We will miss his per­son­al wit, charm and warm per­son­al­ity.”
— I. B. S. Passi (Pres­id­ent, In­di­an Math­em­at­ic­al So­ci­ety)

“I did know about the work of Emmy No­eth­er and it may have in­flu­enced my choice of area, al­gebra, al­though I think the teach­ing of Irving Ka­plansky was what really in­spired me.”
— Vera Pless

Ka­plansky’s books “have one fea­ture in com­mon. The con­tent is re­fresh­ing and the style of ex­pos­i­tion is friendly, in­form­al (but at the same time math­em­at­ic­ally rig­or­ous) and lu­cid. The au­thor gets to the main points quickly and dir­ectly, and se­lects ex­cel­lent ex­amples to il­lus­trate on the way.”
— Man Keung Siu

“I learnt from his books in my youth, and would not have sur­vived without them. Even today, I ask my stu­dents to read them, to learn the ‘tricks’ of the trade.”
— Ravi Rao

“Ka­plansky was one of my per­son­al her­oes: dur­ing my stu­dent years, I dis­covered his little volume on abeli­an groups and no­ticed that al­gebra too has stor­ies to tell…”
— Birge Huis­gen-Zi­m­mer­mann

Kap as a Thes­is Ad­visor: “I was very young and very im­ma­ture when I was Kap’s stu­dent. I’m deeply in­debted to Kap for put­ting up with me and help­ing me to de­vel­op in my own ec­cent­ric way. I asked Kap for a thes­is prob­lem that didn’t re­quire any back­ground and, sur­pris­ingly, he found one with enough meat in it to al­low me to get a feel­ing for do­ing re­search.

“It wasn’t un­til I had my own thes­is stu­dents that I real­ized how hard it must have been to ac­com­mod­ate my spe­cial needs and help me de­vel­op in my way, not in his way.”
— Don­ald Orn­stein (Kap Ph.D., 1957)

For me Kap’s trans­ition from course in­struct­or to thes­is ad­visor was al­most im­per­cept­ible, since I had be­come deeply en­grossed in his courses on com­mut­at­ive and ho­mo­lo­gic­al al­gebra and ques­tions about pro­ject­ive mod­ules, an ex­cit­ing ter­rit­ory wide open for ex­plor­a­tion, and for which Kap had laid a sol­id ground­work. He did float a few oth­er prob­lems to me, such as the struc­ture of cer­tain in­fin­ite di­men­sion­al Lie al­geb­ras, whose sig­ni­fic­ance I only later came to ap­pre­ci­ate. But I didn’t take that bait then. He was a gen­er­ously avail­able and stim­u­lat­ing ad­visor, of­ten shar­ing prom­ising ideas that he had not yet had time to pur­sue. What I re­mem­ber most of that time is the bril­liance of his courses, and the rich­ness and ex­cite­ment of the math­em­at­ic­al mi­lieu that he had cre­ated among his many stu­dents then. This mi­lieu power­fully amp­li­fied the many math­em­at­ic­al re­sources that Kap had to of­fer. I think that it is fair to say that Kap’s stu­dents are an im­port­ant part of his oeuvre. One could hardly have asked for a bet­ter teach­er and ad­visor.