Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA. This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).

246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series. The initial topics I have in mind to cover are

- Elliptic functions;
- The Riemann-Roch theorem;
- Circle packings;
- The Bieberbach conjecture (proven by de Branges); and
- the Schramm-Loewner equation (SLE).
- This list is however subject to change (it is the first time I will have taught on any of these topics, and I am not yet certain on the most logical way to arrange them; also I am not completely certain that I will be able to cover all the above topics in ten weeks). I welcome reference recommendations and other suggestions from readers who have taught on one or more of these topics.

As usual, I will be posting lecture notes on this blog as the course progresses.

[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]

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13 March, 2018 at 10:56 am

AnonymousAnother possible topic is spaces.

15 March, 2018 at 11:02 am

Maths studentI can’t wait for the notes, and will certainly gladly give any corrections I might find.

16 March, 2018 at 4:30 am

Jaikrishnan JanardhananOne topic I am very fond of teaching is the Ahlfors–Schwarz lemma and using to give proofs of the big Picard theorem, the general Montel’s theorem, etc. The book “Complex Analysis: The geometric viewpoint” has a very nice presentation of all this. The book also introduces several invariant metrics which are indispensable in the study of holomorphic mappings in higher dimensions.

18 March, 2018 at 4:57 am

John MangualThere’s a discussion of Riemann Roch theorem in Chapter 21 of Fulton’s “Algebraic Topology” after the chapter on Riemann Surfaces and Algebraic Curves in Part X on Riemann Surfaces. It is also in Chapter 5 of Joe Polchinski’s book on String Theory.

One wishes to integrate objects over a “curve” . I very modestly assume that “cohomology” just means “integrals” and carry on.

I don’t know much about SLE except that it relates conformal mappings and random walk. It gets very complicated. Oded Schramm wrote about many exciting statistical models during his time at Microsoft :-)

20 March, 2018 at 1:52 am

RobAre the notes from 246B available? Thanks :)

20 March, 2018 at 9:56 am

RussellJohn Garnett taught the 246B course, we did the last few chapters of Ahlfors Complex Analysis, Caratheodory’s theorem, the Uniformization theorem, and intro to univalent functions.

20 March, 2018 at 12:49 pm

RobThank you Russell :)

25 March, 2018 at 6:27 pm

AnonymousIt would be interesting to see some elliptic modular forms in the course.