I plan to lead a
research
oriented focus group on circle packing. Circle packing were first
introduced by Koebe (1930), forgotten for along time,
re-discovered by Andreev (1970) and got their publicity by
Thurston (1980's).
Circle packing are very useful in understanding various aspects of
conformal and low dimensional topology.
We will start our study
with the following deep theorem (Koebe-Andreev-Thurston):
Given
any sensible combinatorics, there exist a circle packing with tangency
pattern matching those combinatorics.
The sensible combinatorics turn out to be those of a triangulation
of a surface. The subtle point is that the combinatorics actually
determines the geometry in which the packing lives.
After understanding this, we will pretty much follow the
group interests. Topics for further research may include:
Discrete analytic Functions
Approximation of conformal maps by circle mapping
maps
Creating new discrete analytic maps using circle
packing...
and much more. This is a field which leads quickly to open
questions and so far the techniques in the field have been a
beautiful mix of combinatorics, conformal maps, differential geometry
and topology. If you can program and enjoy it, you will have many
opportunities to have fun and do
mathematics.
Prerequisites
While the very first
definitions in this field are simple, the theory becomes quite involved
and tricky very quickly.
You definitely need to be beyond your series of Linear Algebra and
Calculus. Exposure to Differential geometry and\or complex analysis
and\or programming is highly desired. Equally important for me (and
probably even more) is how self motivated and self starter you are.
In order for you to see if the subject is of interest to you and for me
to get a gauge of your compatibility with what I require, please
do the following:
1. Have a look at the Gallery ;
if the pictures are something you would like to understand, read a
little bit
2. If you are still interested, and willing to put 2-3
hours per week during the Fall and next Spring, come and see me on
Tuesday, 8.21 between 4:30 --
5:30 pm.
3. We will have an organizational
meeting right after I finish talking with each one of you.
4. If you can not make it at the time slot above and interested, please
send me a detailed e-mail describing your mathematical background.
PLEASE
NOTE : I will not give
an introductory talk on the subject. Following steps 1 and 2 may give
you a
much better feel of what the subject is about and whether you are
motivated to study this.
Grading
To be determined later.
Texts
We will follow
closely
Kenneth Stephenson, Introduction to circle
packing, Cambridge University Press 2005.
Depending on the
group progress, we may read research papers as well.