#### 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.