1/17

Runge’s Approximation Theorem

4.4, 5.4

For those of you who were in the class last semester, Zakeri treats Runge’s Theorem in sections 9.3 and 9.4.

1/19

Inhomogeneous Cauchy-Riemann Equations

6.1-6.2

Please consider which topic you’ll be presenting later in the semester and discuss this with me by, say, 1/31.

1/24, 1/26

Interpolation: Mittag-Leffler and Weierstrass Theorems

6.3.1-6.3.3

1/31

Riemann surfaces: definition(s) and examples

7.1-7.2

The implicit function theorem and Riemann surfaces defined by polynomials on C^2 isn’t covered in the book. At least not in the same way.

2/2

Holomorphic functions on Riemann surfaces

7.4.1-2

2/7, 2/9

Holomorphic mappings between Riemann surfaces. Riemann-Hurwitz Therem

7.4.3, 7.4.5

2/14, 2/16

Holomorphic maps and meromorphic functions on tori

7.4.4

Zakeri’s book has a nice, longer treatment of the Weierstrass P-function in Section 9.2

2/21

Analytic continuation and the monodromy theorem

9.1.1-2

2/23

A compactness theorem for Schlicht functions

See e.g. Forster S27.6

Zakeri’s book has a proof different from the one I gave.

2/28

Normal exhaustions of simply connected Riemann surfaces

See Forster S23

The needed fact (Lemma 10 in 9.4) is basically just asserted in Shaw & Stanton. Proving it is actually pretty involved.

3/2

(Sub)harmonic functions and the Dirichlet problem revisited

9.3

3/7

Uniformization

9.4

3/9

Consequences of uniformization

3/21-23

Conformal metrics (and a fix concerning Green’s functions)

9.4

My take on metrics is pretty different than Shaw and Stanton. More complement than parallel.

Note that I struck one problem from the current hwk set.

3/28

Differential forms on Riemann surfaces

See my own classnotes (including an appendix) for a summary here.

3/30

Guest lecture: Richard Birkett’s thesis defense.

4/4

A finiteness Theorem for compact Riemann surfaces

11.4

My argument is substantially different (and somewhat lower tech) that Shaw and Stanton.

4/6

Existence of meromorphic functions on compact Riemann surfaces

11.1-2

The main analytic result I rely on concerns solving Laplace’s equation. I derive results about solving the dbar equation as corollariess. For Shaw and Stanton, the dbar results come first.

4/11-13

No class

I’m at a conference in Canada this week. Will make these lectures up later.

Note that the final exam is now posted (see the hwk folder). I might tweak it later but will let you know if that happens

4/18

Clay presents. More on 1^st de Rahm and Dolbeault cohomology groups.

4/20

Khoi presents. Mittag-Leffler Theorems on compact Riemann surfaces

11.2.3

Please read the Riemann-Roch section in my notes in order to get familiar with the notation I’m using.

Riemann-Roch Theorem

11.3