| Public Talks
Dynamical Systems, Spectra and Geometry
Dmitry Jakobson, McGill University (slides)
The Laplace operator (or Laplacian) is a differential
operator that arises in celestial mechanics, heat
and wave propagation, and quantum mechanics. Eigenfunctions
of the Laplacian describe vibrations of a string
or a drum, pure states in quantum mechanics (atoms
and molecules), and many other phenomena.
We will discuss how eigenfunctions behave when their
energy (or their frequency) is increasing; the simplest
example is provided by the functions sin(nx), when
n increases. Certain properties of eigenfunctions
of Laplacians on surfaces depend on the properties
of motion along geodesics (geodesic flow) on the
surfaces; geodesics are the shortest paths from
one point on the surface to another (light travels
along geodesics). We will also discuss how curvature
of the surface influences the geodesic flow.
Dynamical Systems and Periodic Orbits
Ralf Spatzier, University of Michigan
In science we often consider systems that
change with time. This could be the solar system,
or a population of viruses or the weather. Dynamical
systems theory in mathematics considers such systems
abstractly. Of particular importance are periodic
orbits. These are positions or states which return
to themselves time and again. For example the earth
revolves around the sun year after year. We will give
precise mathematical formations of these concepts,
discuss examples and questions like: When do periodic
points exist? How many periodic points are there?
Counting periodic points is closely related to entropy
which is a measure of the speed of "mixing" of a system.
I will touch on this fundamental idea which is one
of the most important in dynamics.
Panel Discussion: From Number Theory to Chaos
and Back. Where Do Mathematical Ideas Come From?
Have you ever wondered what mathematicians do? Or
how they approach a problem? Do they set out the try
to win a Fields Medal? How do mathematicians know
what problems are important? These questions and more
will be answered during the panel discussion.
Panellists: Dmitry Jakobson (McGill University), Elon
Lindenstrauss (Hebrew University), Peter Sarnak (Institute
for Advanced Study), Ralf Spatzier (University of
Michigan) Moderator: Zach Paikin (Munk School of Global
Affairs, University of Toronto)