Celebratio Mathematica

We are two of the fifty-five stu­dents who com­pleted a doc­tor­ate with Ka­plansky between 1950 and 1978. This is an as­ton­ish­ing num­ber. In­deed dur­ing the years 1964–1969, when we were at the Uni­versity of Chica­go, Kap over­saw an av­er­age of three com­pleted dis­ser­ta­tions a year des­pite serving as de­part­ment chair from 1962–1967. His secret, we think, was an ex­traordin­ary in­stinct for pro­duct­ive av­en­ues of re­search coupled with a gen­er­ous will­ing­ness to spend time work­ing with his stu­dents. He also of­ten en­cour­aged stu­dents to run a sem­in­ar, with be­gin­ning stu­dents present­ing back­ground ma­ter­i­al and ad­vanced stu­dents present­ing parts of their theses.

When Evans worked with him, Kap was teach­ing the com­mut­at­ive al­gebra course that was pub­lished soon af­ter­ward by Allyn and Ba­con. As with each course he taught, he filled it with new thoughts about the sub­ject. For in­stance, at one mem­or­able point he ex­per­i­mented to see how much he could de­duce if he knew only that \( \operatorname{Ext}^1(A,B) \) was zero. He man­aged to get pretty far, but even­tu­ally the proofs be­came un­pleas­antly con­vo­luted. So he ab­ruptly an­nounced that hence­forth, he would as­sume the full struc­ture of \( \operatorname{Ext}^j(A,B) \), and the next day he re­sumed lec­tur­ing in his usu­al pol­ished fash­ion. This epis­ode was atyp­ic­al in that he first de­veloped and then cut off a line of in­quiry. More fre­quently, after com­ment­ing on new in­sights of his own, he would in­ter­ject ques­tions for stu­dents to ex­plore and de­vel­op. In his lec­tures he made the role of non-zero di­visors, and hence reg­u­lar se­quences, cent­ral in the study of com­mut­at­ive rings. At one point he gave an el­eg­ant proof, avoid­ing the usu­al fil­tra­tion ar­gu­ment, that the zero di­visors are a fi­nite uni­on of prime ideals in the case of fi­nitely gen­er­ated mod­ules over a No­eth­eri­an ring. Then he asked Evans to try to de­term­ine what kinds of non-No­eth­eri­an rings would have the prop­erty that the zero di­visors of fi­nitely gen­er­ated mod­ules would al­ways be a fi­nite uni­on of primes. One of the ideas in Kap’s proof was just what Evans needed to get the work on his thes­is star­ted.

The year that Epp worked with Kap, he was not teach­ing a course but had gone back to a pre­vi­ous and re­cur­ring in­terest in quad­rat­ic forms. A quint­es­sen­tial al­geb­ra­ist, he was in­ter­ested in ex­plor­ing and ex­pand­ing clas­sic­al res­ults in­to more ab­stract set­tings. Just as in his courses he tossed out ques­tions for fur­ther in­vest­ig­a­tion, in private ses­sions with his stu­dents he sug­ges­ted vari­ous lines of in­quiry bey­ond his own work. In Epp’s case this meant ex­plor­ing the res­ults Kap had ob­tained in gen­er­al­iz­ing and ex­tend­ing H. Brandt’s work on com­pos­i­tion of qua­tern­ary quad­rat­ic forms and try­ing to de­term­ine how many of these res­ults could be ex­ten­ded to gen­er­al Cay­ley al­geb­ras.

Kap typ­ic­ally sched­uled an early morn­ing weekly meet­ing with each stu­dent un­der his dir­ec­tion. For some it was much earli­er than they would have pre­ferred, but for him it fol­lowed a daily swim. He led our ef­forts mostly by ex­press­ing lively in­terest in what we had dis­covered since the week be­fore and fol­low­ing up with ques­tion after ques­tion. Can you prove a sim­pler case? Or a more gen­er­al one? Can you find a counter­example? When one of us ar­rived dis­ap­poin­ted one day, hav­ing dis­covered that a hoped-for con­jec­ture was false, Kap said not to be dis­cour­aged, that in the search for truth neg­at­ive res­ults are as im­port­ant as pos­it­ive ones. He also counseled per­sist­ence in oth­er ways, com­ment­ing that he him­self had had pa­pers re­jec­ted — a mem­or­able state­ment be­cause it seemed so im­prob­able. Hav­ing made con­tri­bu­tions in so many fields and hav­ing ex­per­i­enced the be­ne­fits of cross-fer­til­iz­a­tion, he ad­vised be­ing open to ex­plor­ing new areas. Some of his stu­dents may have taken this ad­vice fur­ther than he per­haps in­ten­ded, ul­ti­mately work­ing far from their ori­gin­al top­ic areas at the Na­tion­al Se­cur­ity Agency, at the Jet Propul­sion Labor­at­ory, and in K–12 math­em­at­ics edu­ca­tion, for ex­ample.

Kap de­rived a great deal of pleas­ure from hav­ing gen­er­ated 627 math­em­at­ic­al des­cend­ants, per­haps es­pe­cially from meet­ing his math­em­at­ic­al grand­chil­dren and great-grand­chil­dren. When one en­countered him at the MSRI bus stop one day and, not know­ing what to say, com­men­ted on the weath­er, Kap re­spon­ded with a smile, “Cut the crap. Let’s talk math­em­at­ics.” They did, and he be­came one of the many stu­dents Kap ment­ored long after he re­tired.