Originally Posted 01-16-2012
This semester I am supervising an MA student. This is not a full-blown MA thesis; instead, the student has chosen the “non-thesis option” for his work, which requires him to do 2 cerdit hours worth of research work and write a paper. I talked to a few of my colleagues, and it seems that the requirements are a bit nebulous, but certainly don’t include new work.
I talked with the student and he is interested in geometry, has a background in art (some computer animation), and is curious about fractals. Also, he did a summer project with me a while back, and learned a lot about the geometry of the hyperbolic plane. So, I pulled my copy of Indra’s Pearls down off the shelf…
Now, I haven’t read the book before. (One of my vices is buying math books that I want to read “someday.”) Easy selling points here were that the book is full of pretty pictures, and one of the co-authors is my academic grandmother.
I figure that if a student is going to do some research, they have to make something. In this case it is not a new theorem, or a new argument for an old theorem, but instead a picture. Rather, I am going to ask him to make some pictures of limit sets of Kleinian groups.
If all goes well, he will actually write a bunch of routines in Python/Sage/Cython for making such pictures that can be used for a variety of groups. I don’t know if this has been done before (I suspect it has), but I don’t really care. The student will be doing something of academic value, and it will help me learn some stuff that I really should have finished absorbing by now.
So today I read several chapters of the book to get rolling. It starts off very gently. The book is written for a mathematical layman, really, so they start with chapters introducing complex numbers, symmetry, groups, and Moebius transformations of the Riemann sphere. There is pseudo-code weaved in, too, so that a reader can get going with some basic drawings.
It looks like the rest of tonight’s reading will be about Schottky groups. I haven’t seen anything new (to me), yet, but several of these basic concepts are things that I find myself explaning to students often, and I might adopt one or two turns of phrase.
The goal I set for the first week was to try to understand how the group is essentially the set of isometries of hyperbolic 3-space. I am sure that will keep him occupied for this week. Next we can try to draw pictures of the action of individual Moebius mappings
Oh, if you want, here is a link to the book.