Website

The optimal paper Moebius band. Suppose you take a 1 x L strip of paper, twist it around in space, and tape the length-1 ends together to make a Moebius band. If L is very large this is easy and if L is very small this is impossible. What is the cutoff? In this talk I will prove that you can do it if and only if L>sqrt(3), and moreover if L is near sqrt(3) the Moebius band you make must be very close to a certain limiting example that is shaped like an equilateral triangle. This answers a question that goes back to a paper of W. Wunderlich in the 60s, and more specifically confirms the 1977 conjecture of B. Halpern and C. Weaver about this. The proof is pretty elementary and I will explain it all in the talk.

Outer billiards on kites. Outer billiards is a billiards-like dynamical system that moves points around in the plane on the outside of a convex shape. B. Neumann introduced this game in the late 1950s and in the 70s J. Moser (somewhat) popularized the game as a toy model for planetary motion. One of the central questions about outer billiards, called the Moser-Neumann problem, is the question of whether one can pick a convex shape and a point with an unbounded orbit. In this talk I will sketch some ideas in my solution of this problem: Outer billiards has unbounded orbits with respect to any irrational kite — i.e., a quadrilateral having a diagonal of symmetry whose other diagonal divides the shape into two irrationally related areas. My proof was inspired by computer experimentation, and I will show computer demos which illustrate the rich combinatorial phenomena that underly the unbounded orbits result.

Five Points on a Sphere. Thomson’s problem asks which configurations of points on the sphere (considered as electrons) minimize the total electrostatic potential. Computer simulations done by physicists Melnyk, Knop, and Smith in the 70s suggested that the triangular bi-pyramid minimizers the potential with respect to a power-law of exponent s provided that s<15.048077… and then there is a phase transition so that the answer becomes a pyramid with square base. (Thomson's problem concerns s=1.). In this talk I will sketch my computer-assisted proof of this conjecture, showing demos of the programs running and doing their business. I should say that this work has not been published (after some years) and the computer-assisted nature may prevent it from ever being published. So, you could interpret this talk as a good story but perhaps a cautionary tale about computer-assisted proofs.