Celebratio Mathematica

Irving Kaplansky

Algebra  ·  U Chicago

Irving Kaplansky:
Some reflections on his early years

by Nancy E. Albert

Dur­ing the fifties and six­ties, I knew Irving Ka­plansky as a fam­ily friend. Forty years later, he provided con­sid­er­able as­sist­ance to me in my work on the bio­graphy of my late fath­er, A. A. Al­bert. In the course of our cor­res­pond­ence and through my in­ter­views with some of Ka­plansky’s former stu­dents and col­leagues, I gained a few fresh in­sights in­to Ka­plansky, the man and the math­em­atician.

These re­flec­tions cen­ter on three areas: math­em­aticians who in­flu­enced him, events that helped to shape his math­em­at­ic­al ca­reer, and his per­son­al style.

I will refer to him as “Kap,” since I nev­er heard any­one ex­cept his wife refer to him oth­er­wise. Kap was born in Toronto after his par­ents emig­rated from Po­land. The fam­ily re­mained in Canada dur­ing his youth, and he en­rolled in col­lege at the Uni­versity of Toronto after high school.

You might say that Kap was blessed with good for­tune due to a series of events oc­cur­ring at crit­ic­al junc­tures in his ca­reer. When he entered col­lege, Toronto was not ne­ces­sar­ily a sought-after des­tin­a­tion for as­pir­ing math­em­aticians. However, Toronto’s Math­em­at­ics De­part­ment stood to gain from the fact that top math­em­aticians such as Richard Brauer began flee­ing Hitler’s Ger­many in the fall of 1933. In a rather per­verse way, math­em­at­ics on the North Amer­ic­an con­tin­ent be­nefited from the Nazi re­gime’s “Law for the Re­or­gan­iz­a­tion of the Civil Ser­vice” re­quir­ing re­mov­al of Jews from civil ser­vice po­s­i­tions in Ger­man uni­versit­ies.

After spend­ing a year at the Uni­versity of Ken­tucky and a year at the In­sti­tute for Ad­vanced Study, Brauer ac­cep­ted an as­sist­ant pro­fess­or­ship at Toronto in the fall of 1935. An­oth­er new pro­fess­or, Don­ald Coxeter, joined Toronto’s fac­ulty in 1936. Kap men­tioned in an email that the ar­rival of the two pro­fess­ors “raised Toronto to a new level of mag­nitude”.

In 1938, by the end of his un­der­gradu­ate ca­reer, Kap had already dis­tin­guished him­self as a mem­ber of the win­ning team in a newly es­tab­lished in­ter­col­legi­ate com­pet­i­tion. He took home the Wil­li­am Low­ell Put­nam prize, which has since come to be re­garded as the apex of math­em­at­ic­al achieve­ment for un­der­gradu­ates.

Fresh out of col­lege, Kap traveled to Chica­go for an in­ter­lude that would in­flu­ence his bud­ding ca­reer. He de­scribed it to me as fol­lows: “The sum­mer long event at Chica­go in 1938 was an al­gebra pro­gram dom­in­ated by Ad­ri­an [Al­bert]…. That sum­mer put a stamp on me that las­ted a life­time.”

A. A. Al­bert, a/k/a “A cubed,” had or­gan­ized the sum­mer pro­gram, which began with a ground­break­ing Con­fer­ence on Al­gebra that in­tro­duced cut­ting edge con­cepts to a new Amer­ic­an audi­ence. Saun­ders Mac Lane, a young­er math­em­atician spend­ing the year at Chica­go after an in­struct­or­ship at Cor­nell, as­sisted Al­bert in or­gan­iz­ing the con­fer­ence.

Be­sides Al­bert and Mac Lane, Kap met a num­ber of math­em­at­ic­al su­per­stars at the con­fer­ence. Be­sides his pro­fess­or, Richard Brauer, and con­fer­ence or­gan­izers Al­bert and Mac Lane, speak­ers in­cluded L. E. Dick­son, Nath­an Jac­ob­son, Emil Artin, So­lomon Lef­schetz and Oscar Za­r­iski. Later that sum­mer, Kap at­ten­ded a class giv­en by Al­bert as well as a course called Con­tinu­ous Groups giv­en by Jac­ob­son.

In the fall, Kap pro­ceeded to gradu­ate school back at Toronto, where he re­ceived his M.A. in 1940. Brauer taught at Toronto con­tinu­ously between 1935 and 1940, that is, from Kap’s early years as an un­der­gradu­ate math­em­at­ics stu­dent un­til he com­pleted his Mas­ter’s de­gree. In a 2003 email, Kap ac­know­ledged, “I was con­sid­er­ably in­flu­enced by Brauer.”

In 1940, Kap re­ceived a Put­nam Fel­low­ship and be­came Mac Lane’s first doc­tor­al stu­dent at Har­vard Uni­versity. With­in one year, he had com­pleted his dis­ser­ta­tion at Har­vard and earned his doc­tor­ate un­der Mac Lane.

Kap’s first pri­or­ity upon earn­ing his doc­tor­ate in 1941 was seek­ing a uni­versity po­s­i­tion. As Ivan Niven ob­served in a his­tor­ic­al ac­count of math­em­aticians dur­ing the first half of the twen­ti­eth cen­tury, be­fore World War II young Jew­ish math­em­aticians be­came aware that anti-Semit­ism pre­vailed throughout the na­tion’s top re­search uni­versit­ies. Jac­ob­son re­called that, when he had coveted a Har­vard fac­ulty po­s­i­tion in the mid- to late-1930s, Har­vard’s per­man­ent math­em­at­ics fac­ulty in­cluded no Jew­ish mem­bers what­so­ever. Be­cause Kap’s fam­ily was of Jew­ish ori­gin, it was highly un­likely that he could have re­ceived an of­fer for a ten­ure-track po­s­i­tion at Har­vard in 1941. The in­flu­en­tial Har­vard pro­fess­or, George Birk­hoff, had well-pub­li­cized repu­ta­tion for anti-Semit­ism (al­though he non­ethe­less main­tained friendly re­la­tions with both Za­r­iski and Al­bert).

Des­pite that han­di­cap, Kap was for­tu­nate in one re­spect — Har­vard had a non-ten­ure track po­s­i­tion, the Ben­jamin Peirce In­struct­or­ship. Kap glided eas­ily from Put­nam Fel­low to Ben­jamin Peirce In­struct­or in 1941. Al­though Kap knew that the in­struct­or­ship was geared to short-term ap­point­ments, he man­aged to re­tain the po­s­i­tion un­til 1944.

In 1944, a new op­por­tun­ity arose as part of the U.S. war ef­fort. Pres­id­ent Roosevelt es­tab­lished OS­RD, the Of­fice of Sci­entif­ic Re­search and De­vel­op­ment, to co­ordin­ate all gov­ern­ment-sponsored sci­entif­ic ef­forts dur­ing World War II. Kap’s former doc­tor­al su­per­visor took over the reins of an OS­RD re­search pro­ject at Columbia Uni­versity known as the Ap­plied Math­em­at­ics Group, and Kap se­cured a po­s­i­tion at the pro­ject.

The pro­ject had three ad­vant­ages. First, it offered young math­em­aticians, re­gard­less of their re­li­gion or eth­nic ori­gin, a good job at a time when jobs in the field were scarce. Second, it kept Ph.D.math­em­aticians and gradu­ate stu­dents out of the draft (this af­fected Kap dir­ectly, as he had gained his U.S. cit­izen­ship in 1940). And most im­port­antly, it en­abled the na­tion’s mil­it­ary de­fense es­tab­lish­ment to tap the ex­traordin­ary abil­it­ies of re­search math­em­aticians like Kap to aid in win­ning the war. Their con­tri­bu­tions were crit­ic­al to the war ef­fort.

Kap shared a room in New York with a young­er math­em­atician, Daniel Zel­in­sky, who had in­ter­rup­ted his gradu­ate stud­ies to work at the pro­ject. In con­ver­sa­tions I had with Zel­in­sky, he de­scribed the war work at Columbia as “re­search on aer­i­al gun­nery”. Kap, on the oth­er hand, men­tioned that he had been strug­gling with a math­em­at­ic­al ques­tion arising in “aer­i­al pho­to­graphy” when Al­bert, who was then serving as As­so­ci­ate Dir­ect­or of a sis­ter re­search pro­ject at North­west­ern Uni­versity, guided him over the pit­falls of that sub­ject.

This brings me to the ques­tion of Kap’s per­son­al style. In an email, I asked him to cla­ri­fy the nature of his work at Columbia — was it aer­i­al gun­nery or aer­i­al pho­to­graphy? His reply con­sisted of two words, “aer­i­al pho­to­graphy.” Kap’s re­sponse to the ques­tion was pre­cise and to the point, with no sur­plus ver­biage.

After the war, Har­vard failed to is­sue an in­vit­a­tion for Kap to re­turn as a fac­ulty mem­ber. In a 2004 email, he ex­plained, “My po­s­i­tion at Har­vard was strictly tem­por­ary: a Ben­jamin Peirce In­struct­or­ship. So — at the end of World War II I needed a job.”

Al­bert had a good eye for spot­ting math­em­at­ic­al tal­ent, and lost no time in see­ing to it that the Uni­versity of Chica­go offered Kap an in­struct­or­ship that would lead to ten­ure. The ad­di­tion of Kap demon­strated a slight thaw in Chica­go’s un­of­fi­cial freeze on hir­ing Jew­ish fac­ulty mem­bers — un­til then, Al­bert had been the sole Jew­ish mem­ber of Chica­go’s per­man­ent math­em­at­ics fac­ulty. This signaled the be­gin­ning of a trend by top Amer­ic­an uni­versit­ies to re­cog­nize the past and po­ten­tial con­tri­bu­tions of Jew­ish math­em­aticians who had aided this na­tion’s war ef­fort. Ex­clu­sion­ary ad­min­is­tra­tions gradu­ally gave way to mer­ito­cra­cies.

Five dec­ades after the fact, Kap shared his feel­ings with me about re­ceiv­ing the Chica­go ap­point­ment. He con­fided, “Was I ever happy!” However, in 1945, with char­ac­ter­ist­ic verbal eco­nomy, he kept those feel­ings to him­self. Per­haps he hoped to avoid jinxing his good for­tune by broad­cast­ing it, or per­haps he shied away from dis­cuss­ing it out of mod­esty. In any case, he breathed nary a word about it to his room­mate, Zel­in­sky, as they packed up their things at the close of Columbia’s war re­search pro­ject.

Zel­in­sky and his new bride, Zelda, soon got wind of the news. They were sur­prised to en­counter Kap on the train headed for Chica­go in 1945. Zel­in­sky was re­turn­ing to com­plete his doc­tor­ate un­der Pro­fess­or Al­bert’s su­per­vi­sion. Now, he learned that his former room­mate would soon play a new role as one of his teach­ers. It was an awk­ward mo­ment, but the two men quickly over­came it.

After tak­ing an ad­vanced class in ring the­ory at Chica­go from Kap, Zel­in­sky de­scribed Kap’s math­em­at­ic­al style as “smooth and el­eg­ant,” and ad­mired his “nat­ur­al, con­cep­tu­al” ap­proach to math­em­at­ic­al re­search. Kap and Zel­in­sky re­mained good friends over the years.

Once more, due to a lucky con­flu­ence of events, Kap joined the fac­ulty at a pivotal time, just a year or so ahead of an im­port­ant shift in the math­em­at­ic­al land­scape at Chica­go. Al­bert set the ball in mo­tion as Act­ing Chair in the fall of 1946. He went to Dean Wal­ter Bartky to com­plain about the De­part­ment’s tend­ency to ap­point pro­fess­ors from the ranks of former Chica­go gradu­ate stu­dents. As a res­ult, he man­aged to en­gin­eer the ap­point­ment of an out­stand­ing math­em­atician named Paul Hal­mos whose 1942 book, Fi­nite Di­men­sion­al Vec­tor Spaces, had re­ceived rave re­views for math­em­at­ic­al ex­pos­i­tion.

Then, Chica­go’s Chan­cel­lor Robert Maynard Hutchins took up the cause by re­cruit­ing Mar­shall Stone, the son of the U.S. Su­preme Court’s Chief Justice. At the time, Stone was well en­sconced at Har­vard, and was in no hurry to come to Chica­go.

Hutchins ori­gin­ally offered Stone a dis­tin­guished ser­vice pro­fess­or­ship in the Math­em­at­ics De­part­ment and noth­ing more. That did not sit well with Stone, who wanted a free hand in shap­ing a de­part­ment that had weakened dur­ing the war. Hutchins ca­pit­u­lated by up­ping the of­fer to in­clude chair­man­ship as well as the dis­tin­guished ser­vice pro­fess­or­ship. Stone ac­cep­ted be­cause he be­lieved that the situ­ation at Har­vard was stag­nat­ing and he saw the op­por­tun­ity to put his stamp on the De­part­ment at Chica­go, which had five va­can­cies due to re­tire­ments and resig­na­tions.

Stone took over with a series of new ap­point­ments that began in 1947 with the ap­point­ments of An­dré Weil, Ant­oni Zyg­mund and Mac Lane. Stone, who played a rather hero­ic role in this saga, felt ob­liged to threaten the Uni­versity’s Vice-Pres­id­ent that he would resign if the ap­point­ment of S. S. Chern was not ap­proved. The former Har­vard pro­fess­or won the battle, and Chern joined Chica­go’s fac­ulty in 1949. Stone also ad­ded two ex­cep­tion­al ju­ni­or math­em­aticians, Irving Segal and Ed­win Span­i­er. As a res­ult of these de­vel­op­ments, Kap be­came part of the most vi­brant, strongest math­em­at­ics de­part­ment in the coun­try at the time. For a young math­em­atician, work­ing in the midst of such stel­lar math­em­aticians must have af­forded Ka­plansky an amaz­ing op­por­tun­ity to stim­u­late and height­en his cre­ativ­ity. Kap was al­lowed to bloom and prosper in a new era of col­legi­al­ity where race and re­li­gion were no longer bar­ri­ers to a syn­ergy of tal­ents.

Kap en­joyed a repu­ta­tion as an out­stand­ing teach­er. Sev­er­al of the math­em­aticians I in­ter­viewed found his teach­ing style to be en­ga­ging and en­thu­si­ast­ic. They de­scribed his present­a­tions as “slick and flashy” — a mark of high praise. Zel­in­sky re­called that Kap em­ployed drama in class — he would lull the stu­dents in­to be­liev­ing that a prob­lem could not be solved and then sud­denly sur­prise them by demon­strat­ing a pat­tern lead­ing dir­ectly to a solu­tion.

I first re­call see­ing Kap dur­ing the early 1950s, al­though I may have met him a few years earli­er. My fam­ily had moved in­to a sprawl­ing East Hyde Park apart­ment about a mile from Chica­go’s cam­pus. I can pic­ture Kap at de­part­ment­al math parties, seated at the pi­ano in our din­ing room, sur­roun­ded by fac­ulty mem­bers such as Shi­ing-Shen Chern and Paul Hal­mos with their wives. Kap would be play­ing show tunes and Tom Lehr­er songs from memory, and he was al­ways the life of the party.

He was one of the most dash­ing mem­bers of the math­em­at­ics fac­ulty, with his ath­let­ic build and win­ning, dimpled smile. Here, I must di­gress to de­scribe his wife, the vi­va­cious Rachelle (“Chel­lie”), whom I re­col­lect as be­ing one of my fa­vor­ite people. She seemed young­er than most of the fac­ulty wives, in part be­cause of her care­fully coifed, but flow­ing, black tresses. She had a lyr­ic­al voice, a cer­tain soph­ist­ic­ated el­eg­ance, and a kind, pleas­ant man­ner.

She earned a repu­ta­tion with­in the Math­em­at­ics De­part­ment as a most charm­ing, cap­able host­ess who brought fac­ulty mem­bers to­geth­er in a way that fostered their math­em­at­ic­al re­search. She be­came an as­set to her hus­band’s math­em­at­ic­al ca­reer and to the De­part­ment as a whole. When he be­came de­part­ment­al chair­man in 1962, she was ideally suited for the role of chair­man’s wife. Folks that I in­ter­viewed still re­tain pleas­ant memor­ies of the de­part­ment­al gath­er­ings that she or­ches­trated at the Ka­plansky home.

Un­like many Amer­ic­an math­em­aticians, Kap did not per­ceive a sharp di­cho­tomy between his pure and ap­plied math­em­at­ics. For ex­ample, in the early 1950s, Kap par­ti­cip­ated in sum­mer­time ap­plied math­em­at­ics re­search pro­jects in Cali­for­nia. The top-secret pro­jects were giv­en the ac­ronyms SCAMP (South­ern Cali­for­nia Ap­plied Math­em­at­ics Pro­ject) and ALP. The ori­gin of the name “ALP” is un­clear — re­search­ers jok­ingly re­ferred to it as “Ad­ri­an’s Little Pro­ject.” The os­tens­ible pur­pose of these pro­jects was to con­duct re­search in nu­mer­ic­al ana­lys­is, but they fo­cused on crypto­logy.

Kap saw these pro­jects through a dif­fer­ent lens. He re­por­ted, “At SCAMP and ALP we were told that the ul­ti­mate ob­ject­ive was crypt­ana­lys­is but the prob­lems we worked on were pure al­gebra (and pretty in­ter­est­ing), fac­tor­iz­a­tion of poly­no­mi­als, groups giv­en by gen­er­at­ors and re­la­tions, etc.”.

Sim­il­arly, in ref­er­ence to a clas­si­fied Air Force re­search pro­ject at Bowdoin dur­ing the sum­mer of 1957, he de­scribed it as fol­lows: “Fif­teen or so of us simply had fun all day do­ing math­em­at­ics we loved. I am not com­pet­ent to say wheth­er we con­trib­uted any­thing sig­ni­fic­ant to crypt­ana­lys­is.”

Kap and his col­leagues at the top-secret ap­plied math­em­at­ics pro­ject did, however, con­trib­ute to pure math­em­at­ic­al re­search. With char­ac­ter­ist­ic mod­esty, Kap neg­lected to men­tion that he and I. N. Her­stein pro­duced some sig­ni­fic­ant res­ults in group the­ory that sum­mer by show­ing that ABA-groups are ne­ces­sar­ily solv­able. He de­scribed the im­pact of the pro­ject as fol­lows: “[T]here was an im­port­ant con­sequence with­in pure math­em­at­ics. …  [B]ecause of Bowdoin … [Daniel Goren­stein] fell in love with group the­ory. Ul­ti­mately he be­came the gen­er­alis­simo of the army of group the­or­ists that clas­si­fied fi­nite simple groups — a ma­jor achieve­ment of 20th cen­tury math­em­at­ics.”

M.I.T.’s Richard Schafer re­called par­ti­cip­at­ing in the Bowdoin pro­ject. He re­por­ted that Kap kicked off the sum­mer with a fast, cha­ris­mat­ic talk to par­ti­cipants. But Kap’s present­a­tion was so brief, they were all left gasp­ing for more!

One thing that emerges from the notes Kap sent to me is that he had a joie de vivre, and that joy re­volved around pure math­em­at­ic­al re­search. Even after be­com­ing quite ill, his son Steven re­por­ted that he con­tin­ued to fill up note­books with his math­em­at­ics. Be­sides math­em­at­ics, the love of his fam­ily filled his life. He re­mained wed­ded to the lovely Chel­lie and they had three won­der­ful chil­dren — Steven and Alex, whom he de­scribed as “com­puter ex­perts,” and Lucy, whose mu­sic­al ca­reer pleased him a great deal. He was ex­cep­tion­ally proud of all of them. And I might add that, when he first replied to me in 2002, he soun­ded quite happy that his 10-year old grand­son loved math­em­at­ics!