Celebratio Mathematica

Irving Kaplansky

Kap across generations

by Manjul Bhargava

I was a gradu­ate stu­dent at Prin­ceton in the year 1999. And be­ing a stu­dent of al­gebra, I ob­vi­ously knew of Pro­fess­or Ka­plansky, though I knew of him more as a “le­gend” than as a per­son. His name was one that was at­tached to a num­ber of great the­or­ems, some go­ing back to the 1940s. At the time I sus­pect it nev­er oc­curred to me that he might be an ac­tu­al per­son who was still do­ing great math­em­at­ics. While work­ing on my dis­ser­ta­tion, I be­came in­ter­ested in a clas­sic­al prob­lem from num­ber the­ory re­lat­ing to quad­rat­ic forms. (It was not really a prob­lem in the “Ka­plansky style”, or so I thought!) The ques­tion was: When does a pos­it­ive-def­in­ite in­teg­ral quad­rat­ic form rep­res­ent all pos­it­ive in­tegers? (For ex­ample, Lag­range’s Four Squares Form \( a^2+b^2+c^2+d^2 \) gives such an ex­pres­sion — i.e., every pos­it­ive in­teger can be writ­ten as a sum of four square num­bers.) This was a beau­ti­ful ques­tion of Ramanu­jan that Pro­fess­or Con­way taught me about and got me hooked on. After work­ing on the ques­tion for some time, I real­ized that some good head­way could be made provided that one could un­der­stand the clas­si­fic­a­tion of what are known as “reg­u­lar tern­ary forms”. In par­tic­u­lar, I needed to know: How many such reg­u­lar tern­ary forms are there? I did some searches on Math­S­ciNet, and soon enough found a 1997 (!) pa­per by W. Jagy and I. Ka­plansky en­titled: “There are 913 reg­u­lar tern­ary forms”.

Here was the ex­act an­swer to my ques­tion in the very title of a pa­per writ­ten only two years ago! It was quite ex­cit­ing, and I thought to my­self “Surely this is not the same Ka­plansky!,” [sic] but after some re­search I soon dis­covered that it was. I emailed Jagy and Ka­plansky later that week, and heard back from both al­most im­me­di­ately. Kap and Will (Jagy) were also both very ex­cited that their re­cent work had found ap­plic­a­tions so soon. I men­tioned to them that I would be in Berke­ley for a few weeks that sum­mer to learn tabla with my teach­er, and Kap kindly in­vited me to vis­it MSRI while I was there.

Kap asked Dav­id Eis­en­bud, the dir­ect­or of MSRI, to give me an of­fice for the sum­mer, and Dav­id gen­er­ously agreed. That sum­mer turned out to be one of my most pro­duct­ive sum­mers ever. I worked on math­em­at­ics dur­ing the day and played tabla by night. Rather than work­ing in my private of­fice, I found my­self mostly work­ing in Kap’s of­fice! We didn’t really work to­geth­er, but rather we worked in­de­pend­ently and then shared what we had dis­covered or learned at vari­ous in­ter­vals throughout the day. Kap, Will, and I dis­cussed and learned vari­ous math­em­at­ic­al top­ics to­geth­er in what were some ex­tremely en­joy­able ses­sions. Kap’s love, en­thu­si­asm for, and unique view of math­em­at­ics were con­stantly evid­ent and al­ways in­spir­ing!

In ad­di­tion, I talked to Kap a lot about oth­er things; we shared com­mon in­terests not only in math­em­at­ics but also in mu­sic, mak­ing it a rather fre­quent top­ic of con­ver­sa­tion. In the pro­cess, I also learned a great deal about Kap’s amaz­ingly reg­u­lar life and his oth­er as­so­ci­ated charm­ing idio­syn­crasies. He brushed his teeth more of­ten than any­one I’ve ever known. And no mat­ter how ex­cit­ing a par­tic­u­lar con­ver­sa­tion or work ses­sion was, if it was time for his daily noon swim, then there was no stop­ping him from run­ning off to the pool! (The same oc­curred when it was time for his chosen 5:14 p.m. end-of-the-day bus from MSRI.) I found my­self chan­ging my own sched­ule to match his work sched­ule bet­ter (in­clud­ing wak­ing up rather early!).

The same sched­ule was ad­hered to the fol­low­ing few sum­mers, as he al­ways gen­er­ously in­vited me back (He would write, “Look­ing for­ward to re­new­ing our ses­sions!,” and there were al­ways new and ex­cit­ing things to dis­cuss; every year I looked for­ward to it.) Un­til the very last sum­mer, when I heard the sad and dev­ast­at­ing news. I’ve since al­ways felt that it was un­fair that I got to know him only to­ward the later years of his life. Of course, deep down I know I should be grate­ful that I got to meet him at all, and to have been one of the lucky ones in my gen­er­a­tion to have had the priv­ilege of know­ing him. He was so en­cour­aging to me al­ways, as a per­son, as a mu­si­cian, and most of all, as a math­em­atician. I will al­ways cher­ish the memor­ies of his en­thu­si­asm, bril­liance, gen­er­os­ity, and friend­ship. I will miss him very much.