Graduate course on Fractals, Geometric
measure theory
and intersection properties of Brownian motion
Go HERE
for new web page for Brownian motion and their intersection properties
course.
Organizational meeting: October 30 at 10:15 in Room S4.
This course will consist of 2 parts and will be given
during the 2nd and 3rd reading periods.
The first half can be studied on its own for 4 points. The second half
can be studied on
its own for 4 points provided one attends the first
two weeks of the first half where the
necessary background on
Hausdorff dimension will be given.
FIRST HALF
The first half will deal with fractals, Hausdorff dimension and
other notions of dimension. This area is called
geometric measure theory.
A tentative outline of the topics which will be covered in this first half are:
- Hausdorff dimension
- Capacitarian dimension
- Frostman's theorem linking these two dimension concepts
- Theory of self-similar sets (which will allow us to compute
the Hausdorff dimension of the von Koch snowflake,
the Sierpinski gasket and the Sierpinski carpet).
- Minkowski and packing dimension and packing measures
- Dimensions of product sets
- Density theorems for Hausdorff measures
- Behavior of Hausdorff dimension under orthogonal projections and
- Behavior of Hausdorff dimension under
intersections with lower dimensional planes
SECOND HALF
The second half of the course will deal with the geometric aspects of
Brownian motion (the fundamental stochastic process) in Euclidean
space such as, which sets the Brownian motion can intersect,
the Hausdorff dimension of various related sets, the self-intersection
behavior of Brownian motion and (if time permits) relationships to
potential theory. A tentative outline of the topics which will be
covered in this second half are:
- Brownian motion
- Kakutani's criterion of which sets are hit by a Brownian motion
- Hausdorff dimension of Brownian motion paths in R^n
- Hausdorff dimension of the zero set of Brownian motion in R
- Self-intersection properties of Brownian motion paths
(where the critical dimension turns out to be 4)
- Solution of the Dirichlet problem using Brownian motion
- Relationship between packing dimension and Brownian motion paths
Prerequisites: For the first half, integration theory
and for the second half, some elementary probability.
Kursexaminator: Jeff Steif (steif@math.chalmers.se)
Kurslitterature: There will be no specific book that I will
follow but for the first half,
I will have two books on reserve in the library:
- Fractal Geometry: Mathematical Foundations and applications by
Falconer
and
- Geometry of sets and measures in Euclidean spaces by Mattila.
and for the second half of the course, there will be some notes.
Last modified: Tue Jan 22 16:52:41 MET 2002