Research Seminar in Mathematics (MTH G450 - 30757)
Organized by Professor Jonathan Weitsman
Guest Speaker: Richard Schwarz
Brown University
Title: Billiards on the Penrose Kite
Date: Tuesday, October 7, 2008
Time: 1:00 p.m.
Location: 509 Lake Hall
List of talks:
Pretalk: 1 p.m. - 2:00 p.m. by Richard Schwarz, Outer billiards, tilings, and the modular group
Abstract: I will talk about dynamical systems defined by
piecewise isometric maps. (The most well-known examples of these
maps are interval exchange transformations.) The main example of
interest to me is outer billiards. I will define this dynamical
system and explain how it gives rise to interesting and
not-well-understood tilings of the plane. As preparation for my
main talk, I will also say a few words about hyperbolic geometry,
the modular group, and Diophantine approximation. In the first
talk, you will not see the connection between outer billiards
and the modular group, but I will explain this in the next talk.
Break 2:00 p.m. - 2:30 p.m.
Research talk: 2:30 p.m. - 3:30 p.m. by Richard Schwarz, Outer billiards on kites I
Abstract: Outer billiards was introduced in the 1950's by
B.H. Neumann and popularized in the 70s by J. Moser as a toy model
for celestial mechanics. One of the central problems in the subject
has been whether or not one can have an outer billiards system with
unbounded orbits. This question is analogous to the question about
the stability of the solar system. In this talk, I will explain my
solution ("yes") to the Moser-Neumann question about unbounded orbits.
The solution relates outer billiards to such topics as quasi-crystals,
Diophantine approximation, odometers, and the modular group. I
will illustrate the main ideas with computer pictures from my
program, Billiard King. Everything is written up in my recent
book, http://www.math.brown.edu/~res/Papers/kite.pdf
Department tea, 3:30 p.m. - 4:00 p.m.
Second research talk by Richard Schwarz, 4:00 p.m. - 5:00 p.m., Outer billiards on kites II: the revenge
Abstract: In this talk I will go more deeply into some
of the ideas behind my solution to the Moser Neumann problem.
The main idea is that outer billiards in the plane has
a nice compactification, where it is a kind of "irrational slice"
of a higher dimensional compact dynamical system. I will try
to explain this point of view, and its consequences.
Discussion at 5:00 p.m. followed by dinner with the speaker.
Video Clips: Clip 1
Previous Talks:
Fall 2008:9/16, 9/23,
Started: 23 September 2008 Last modified: 30 September 2008
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