Turning Points

Dan Rockmore

In the early 1990s I struck up a friendship with Charles Tresser at IBM Yorktown Heights. He was very excited that there was in his midst someone who knew some group theory, because he was sure he was thinking about a problem with groups, permutation groups in fact. The problem boiled down to thinking about discretixed versions of unimodal maps - basically permutations such that when you graph the points (1,s(1)),...,(n,s(n)) and connect the dots, you get a graph with one peak. After playing around with these for a bit I realized that they had a wonderful algebraic structure, mimicking, but different from related work related to shuffling. After computing many, many examples I came up with a conjecture: that lumping together all the permutations whose graph had a fixed number of "turning points" would give rise to a commutative algebra. I tried mightily to prove this for a while, and became kind of obsessed with it - all to no avail. After some time I became friendly with Peter Doyle, then at UCSD. Peter is an expert on the mathematics of shuffling and one of the most creative people I have come to know. I invited myself out to La Jolla and managed to get Peter interested in this problem. We talked and talked, worked and worked, and then Peter saw the light. His write-ups of a solution included his own reworking of the way in which he would think of and notate permutations. It was a beautiful experience to come to the realization that I had stumbled on to something new, but the most beautiful part of it was engaging with the creative minds of two new friends, Charles Tresser and Peter Doyle. Brilliance is a beautiful thing to behold.