Second homework assignment due Friday September 8

Third homework assignment due Friday September 15

Fourth homework assignment due Friday September 22

Fifth homework assignment due Friday September 29

Sixth homework assignment due Monday October 16

Seventh homework assignment due Friday October 20

Eighth homework assignment due Monday November 6

Tenth homework assignment due Monday November 6

Potential projects I wanted to give you all some feedback on the presentations in the last week. It was a very difficult task and I somewhat deliberately didn't give you much guidance, I wanted to see what you would make of it. I think you all realised that preparing and giving a talk of less than 20 minutes is challenging. You have to think very carefully about what you want to say and say only what is necessary to achieve that goal. Adding things that "seem important" just uses up valuable time. Apart from that I was looking for something that I didn't see too much of and that was "getting to the essence of the proof/idea". I wasn't expecting you to understand proofs in detail but I was hoping to see that you had understood at some level what was making it work. For instance in the case of the modular forms there was the result that the numbers of zeros and poles in a fundamental domain are equal. I would have liked to hear something like "by integrating around the boundary as for elliptic functions, but one has to worry about infinity" or words to that effect. Or for the Riemann zeta function, what was never given was any idea of how one analytically extends. This involves an integral transform and the functional equation which never reared its head. For the uniformisation theorem there was no hint of how the combinatorics of a triangulation might yield the 3 different possibilities for a simply connected surface. The quasiconformal topic was harder than the others and perhaps because of that the team working on it had sought some kind of intuitive understanding in the face of overwhelming technical difficulties. In general I thought I would offer some advice on how to quickly get something out of reading a theorem which one has to know on some level. The first job is simply to spend some time thinking about how one might actually prove it oneself. This is very useful and the amount of time spent doing that should depend on how much one needs to know. If it's a serious theorem you won't actually be able to prove it just like that but the next job is to start looking at the proof and comparing it to the ideas you had. It is pretty rare that there will be empty intersection and what intersection there is will help you massively in understanding the proof. You might even get the "aha I almost had it" feeling and go away thinking that you have really got the proof. Anyway, I enjoyed listening to your presentations and I had a good time teaching the course over the course of the semester. I wish you all the best of success in your mathematical careers! Vaughan

Arzela Ascoli on equicontinuous families

Branches and Riemann surfaces for inverse functions and relations.

Weierstrass function inverts elliptic integral

Meromorphic functions with prescribed principal parts

Iterating rational functions on the Riemann sphere

Otherwise grades will be assigned on the basis of a short (2-3 pages) essay on a topic not covered explicity in the course, e.g. "The Riemann Zeta function." Topics may be proposed by students or selected from a list that I will give. The student is supposed to investigate the literature on the topic and the essay will be the result of that investigation. I encourage students to work in groups of at most 3, and will be happy to discuss progress on the essay with a group.