Martin KRUSKAL(Sept. 28, 1925-Dec. 26, 2006) and Peter HAMMER(Dec. 23, 1936-Dec. 27, 2006), TWO GIANTS OF DISCRETE MATHEMATICS

By Doron Zeilberger , Rutgers University

Friday, Dec. 29, 2006.

The random variable X(t):="number of deaths of giants at week t" is far from uniform, or else, how can we explain that this week (and, like the rest of my department, I found about them at the same day, today) two Giants of Discrete Math passed away.

Martin Kruskal was, officially, not explicitly a discretian, but his seminal work, with Robert Miura, John Greene, and Clifford Gardener, was a beautiful piece of differential algebra. (And do not be fooled by the word "differential", it is just as bit as discrete as the word "algebra", since it can be viewed formally for a symbolic function u=u(x,t) (e.g. one satisfying the KdV equation) and using combinatorial rules like ( ux )x=uxx etc.). Furthermore, it revolutionized Exactly Solvable Models that lead to lots and lots of gorgeous combinatorics. Martin was also the co-guru, with John Conway, of Surreal Numbers, and his approach to them was even more discrete than Conway's. And of course, what made Martin a household name among magicians, and the general public, is the nifty Kruskal card trick that is a lovely application of a discrete-time Markov Chain.

Martin was only implicitly a discrete mathematician, but Peter Hammer was the personification of Discrete Mathematics. While he probably didn't create the phrase, he definitely made it a legitimate part of mathematics, by founding, back in 1971 (and editing it to this day!) the famous journal of that name. It was Peter's great vision that Discrete Math is more than Combinatorics, and is really a culture and a mentality, so different than the then-mainstream-culture of continuous mathematics. So, in addition to his many technical contributions, Peter will always be remembered as a brave pioneer of discrete-lib, once so scorned, and now so hot.

Let me end with a personal note. I always liked and admired Peter, but was never aware that he directly influenced my professional development. Call it hindsight if you will, but it was the shock treatment of his death notice that made me realize that I owe him much more than I thought. When I was eighteen-years-old, in 1968, right before my mandatory service in the Israeli army, I spent a few months working at the Weizmann Institute of Science under the mentorship of Joe Gills, Pinchas (Phillip) Rabinowitz, and Nira Dyn. I also audited a graduate class that a brilliant young professor, who just fled from Romania, by the name of Peter Hammer, was teaching, about pseudo-Boolean functions. It was my first exposure to advanced mathematics, and I remember loving it. Then I forgot all about it. It only occurred to me today that my love of the discrete (and dislike of the continuous) probably is due, in large part, to the inspiring and gentle preaching of Peter, and because it was implicit and almost subliminal "propaganda", I didn't realize, until today, its importance on my own professional development.

I think, Peter, that I once told you, a few years ago, that you were my first math professor, and that I really liked your course, but I didn't tell you (since I didn't know it myself) how important it was for me.

But, as soon as I meet you again, in the Discrete part of, I will tell you. And, until then, if our trivial terrestrial internet is not blocked by the heavenly firewall, and should you google yourself, you might see this page and get my very belated thanks.

Thanks, Peter, for being responsible, in no small part, for my becoming a discretian!

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