I’m currently in Pittsburgh for the 2010 MathFest conference. (My probability students must be heart-broken; I had to cancel a class for the endeavour.) I’ve learned a lot already, and the conference is only half-over. But I didn’t just come to listen; I also came to give a talk.
You can get the talk slides here. (Just right-click and select “Save”.)
If you want more in-depth information on Apollonian Circle Packings, the best place to start is probably a sequence of five papers. (If you want to really know everything, I recommend reading them in the listed order; if your interests are more strictly number-theoretic, then perhaps start with the fourth and jump back to the earlier papers on an as-needed-basis.) The first four are by Graham, Lagarias, Mallows, Wilks, and Yan. The fifth is by Erikkson and Lagarias.
- Geometric Group Theory I
- Geometric Group Theory II
- Geometric Group Theory III
- Number Theory I
- Number Theory II
A reference for Elena Fuchs’ result is here.
Sarnak’s letter to Lagarias (in which is proved the “twin prime conjecture”) is here.
As Morpheus says, time is always against us. These slides were written for a 15-minute talk, and I could easily have given two 60-minute talks on this topic, and a third on the generalizations I’ve been playing with most recently.
This talk was part of a special session on open and accessible problems in number theory and algebra, and I tailored it accordingly. You’ll notice that I’ve written a lot more about questions I haven’t answered than questions I have. My own discoveries were present only “obliquely”. Upon my return to Michigan, I hope to complement these slides with some more posts including material from my papers-in-progress and my freshest thoughts on this subject. So there’s more on the way.
Most importantly (and those of you who saw my talk will know this already), if any part of this interests or intrigues you, contact me. Use comments here, email me, find your way to the Cafe Aroma in Fenton, whatever. Graduates and undergraduates, I’m talking to you. There is a lot of accessible stuff here at a lot of levels and with a lot of flavors. Want to collaborate? I am friendly, and I will always work with students. Just want some more information? Please ask; if I don’t know the answer, I will find someone who does.