Oliver Knill | Talk given October 19, 2000 at Montana State University Bozeman |

This is a personal list of problems, I currently like to think about (Oct. 2000). Except of 5),8),10),13) (to which I don't know whether they have been asked before), these are "old" problems. Statements after GUESS: answer chosen when I would be forced to give an opinion. Statements after SOURCE: a starting point in the literature. Clicking on a thumbnail picture reveals the corresponding transparency. Image source to 9) http://turnbull.dcs.st-and.ac.uk/history. Image source to 2) NASA. The other transparencies were made using Mathematica 4.0 and the raytracing software Povray 3.0 both running on the Linux operating system. |

9) KOLMOGOROV PROBLEM. Is there a Hamiltonian system with a
smooth invariant torus, on which the induced dynamics is mixing?
KNOWN: Situations with weakly mixing tori. No mixing can happen on two-dimensional tori. GUESS. Yes, there should exist examples in higher (d>3) dimensions. SOURCE. According to Arnold, this is a question of Kolmogorov which motivated KAM theory. (See O. Knill, Weakly mixing invariant tori of Hamiltonian systems, 1999) SLIDE: Kolmogorov. Picture taken from Mac Tutor. |

14) MATHER THEORY NEAR INTEGRABLE SYSTEMS.
Are there quasiperiodic global minimals for metrics on the torus
which are close to a flat three dimensional torus? (A geodesic is a global minimal
if for any two points on the geodesic, the piece between them is a minimal solution.)
KNOWN. Not necessarily for metrics far away from integrable situations (Hedlund examples). Yes in two-dimensions (Mather theory). GUESS: Beside the KAM transition, there should be a transition, where Mather theory breaks down. SOURCE: "J. Moser, Selected topics in the calculus of variations, ETH, 1988". SLIDE: A piece of a geodesic orbit on a torus. |