Monday, May 3rd, 2004
2:30pm - 3:30pm, in Science 2-065

Greg Huber

UMass Boston

Variations on a theme by Hofstadter: Q theory

Abstract: In 1979, Hofstadter introduced and briefly discussed a chaotic, recursively-defined function which he called Q: $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$. Over the past 25 years, a number of mathematicians have studied this function and a handful of variants of it, and though some statistical details about Q's type of chaos have been observed, no one has managed to prove a single fact about Q -- not even that it -is- a function! In my talk, I'll describe a new avenue of (mostly empirical) which looks at a family of variants of Q, and which borrows some ideas from other disciplines. It turns out that this family of functions, on the collective level, exhibits an amazing degree of regularity, and yet within the framework of that family-level regularity, there are very weird irregular patterns unlike any seen before. I'll discuss what is currently known and unknown about this new family of functions, presenting the ideas mainly through a series of computer-generated graphs which display the tantalizing eccentricities that have been recently uncovered.