Celebratio Mathematica

Irving Kaplansky

Kap as teacher and mentor

by Lance Small

I was not a stu­dent of Ka­plansky — at least, not in the sense we usu­ally mean in math­em­at­ics. He was, however, my teach­er in a num­ber of courses, un­der­gradu­ate and gradu­ate, and was chair­man of the Uni­versity of Chica­go math de­part­ment when I was a gradu­ate stu­dent. Kap’s “style”, math­em­at­ic­al as well as per­son­al, shone through every­where. Nowadays, most math de­part­ments of­fer a “bridge” course for their ma­jors. This course is de­signed to ease the trans­ition to real, up­per-di­vi­sion math­em­at­ics from (in­creas­ingly) less rig­or­ous cal­cu­lus courses. Chica­go has had such a course for years. In my day, it was Math 261; at present it has the fash­ion­ably in­flated num­ber 26100. Cur­rently, just as it did sev­er­al dec­ades ago, the course cov­ers “sets, re­la­tions, and func­tions; par­tially ordered sets; car­din­al num­bers; Zorn’s lemma, well-or­der­ing, and the ax­iom of choice; met­ric spaces; and com­plete­ness, com­pact­ness, and separ­ab­il­ity.” When I took the course, Kap used notes of Ed Span­i­er on “Set The­ory and Met­ric Spaces”. Span­i­er nev­er got around to writ­ing these notes up as a book. Kap, however, did! Set The­ory and Met­ric Spaces ap­peared in 1972 and con­tin­ues in the AMS Chelsea series. Ka­plansky’s style is as ap­peal­ing to cur­rent stu­dents as it was to us dec­ades ago. I have used the book in my classes for many years. One of my re­cent stu­dents en­joyed the book so much that she bought it as a birth­day present for her en­gin­eer fath­er!

As chair­man, Kap main­tained a keen in­terest in gradu­ate stu­dents and the gradu­ate pro­gram. His sens­it­iv­ity to grad stu­dent-ad­visor dy­nam­ics can be il­lus­trated by the fol­low­ing an­ec­dote. One af­ter­noon at math tea, my ad­visor, Yitz Her­stein, and I got in­to a “dis­cus­sion” on how Kap (of Ca­na­dian ori­gin like Yitz) pro­nounced “sched­ule”. I main­tained that Kap would pro­nounce it with an “sk” as Amer­ic­ans do and Yitz, of course, said that Kap would say “shed­ule”, as Ca­na­dians and Bri­tons do. So, Yitz and I bet a quarter. When Kap ar­rived at tea, Yitz and I bounded up to him and told him of our bet. Kap thought for an in­stant and, then, care­fully pro­nounced “sked­ule” re­mark­ing that fac­ulty shouldn’t take money from stu­dents and that Yitz “should pay up.” However, I only got 15 cents.

Kap’s rhet­or­ic­al flour­ishes are well known; but, some­times they had un­in­ten­ded con­sequences. For my first job, I needed of­fi­cial cer­ti­fic­a­tion that I had com­pleted the Ph.D. A let­ter from the chair­man would suf­fice. Kap wrote such a let­ter con­clud­ing “… and, bar­ring cata­strophe, he will re­ceive the de­gree on June 11….” This was deemed in­suf­fi­cient by a de­part­ment­al ad­min­is­trat­or at Berke­ley who quoted the “bar­ring cata­strophe” re­mark. Kap washed his hands of it and sent me off to the Dean of Stu­dents in the Di­vi­sion of Phys­ic­al Sci­ences for a “really” of­fi­cial let­ter.

Even, at the last mo­ment, dur­ing my fi­nal or­al ex­am, Kap’s style was ap­par­ent. He asked me where would you find a com­mut­at­ive ring with some prop­erty or oth­er. I star­ted to con­struct the ring when he in­ter­rup­ted: “No, no, in what book would you look for it?” I replied, “Nagata” and was off the hook!

Kap’s les­sons and ad­vice re­main fresh to this day. His books and his ex­pos­i­tions are as at­tract­ive to the cur­rent gen­er­a­tion of stu­dents as they were to mine.