Student: Jacob Tsimerman
Examiners: Okounkov, Lieblich, Sarnak (chair)
April 27, 2007
Special Topics: Algebraic Number Theory, Algebraic Geometry
Length: 3 hours

They asked me what I wanted to start with, I said Algebra.

[O]: Talk about the representation theory of compact lie groups 
[O]: How do you know you have a finite dimensional representation?
(they wanted to hear "compact operator")
[L]: Classify division algebras over R
[o]: Do you know a lie group that has no faithful finite dimensional representations?
(I said double cover of SL2(R), but didn't know the proof, so he walked me through it)
[S]: State Riemann mapping theorem. Do you know Uniformization theorem?(yes) Find a
meromorphic function on a compact riemann surface.

I first proved that there are lots of meromorphic functions as long as the 
first cohomology of the sheaf of regular functions is finite dimensional. What ensued
was Sarnak leading me through an analytic proof of that fact, which ended up being my
real analysis segment. some of the questions that came up were as follows"

[S]: Suppose f:R/Z-->R is differentiable,f(0)=0 and the L2 norm of f' is finite.
Bound |f|_infinity.
[S]: Suppose now f is on (R/Z)2, f(0)=0, and both its derivatives have finite L2 norms.
Can you still bound |f|_infinity?(no)
[S]: What if you have similiar bounds on second derivatives? (The key is to expand in a
fourier series and apply cauchy's ineq)

[S]: Talk about rational canonical form. If B is in GL_n(C), can you solve e^A=B?
(yes, jordan canonical form)
[O]: What if B is in SL_n(R)? (I guessed no, and after a hint from Okounkov we looked
through the conjugacy classes in SL_2(R) and proved that if B=(-1,1)(0,-1) it's impossible.
This was quite neat).

[S]: Suppose you have a bounded injective holomorphic map f:D-->C, f(0)=0, f'(0)=1. Can
you bound f? (I didn't know how)
[S]: Ok, suppose f is harmonic on D, |f| bounded by 5 on the top half of the boundary,
by 4 on the bottom. How well can you bound f in the disk?

I first used cauchy's integral formula to get a bound, but Sarnak said that I could do
much better. Eventually I figrued out to use green's function. 

[O]: Can doubly connected regions be conformally mapped to tori? (yes, gave a proof
using uniformization)
[S]: Can you prove it without uniformization? (gave another proof using solution to
dirichlet problem)

There was now another analysis discussion of how to solve the dirichlet problem. 

[S]: If two tori are biholomorphic, prove ratios of radii are the same. (I gave a proof
using harmonic conjugate of ln|z|)

Okounkov wanted me to glue up the torus somehow to get an algebraic curve and prove it
that way, but we decided to take a 10 min break. 

[S]: is every complex manifold algebraic? (no, some complex tori aren't)
[O]: Which ones? (Need Riemann billinear relations, for instance). How do you get
meromorphic functions?
I then (with help) wrote down some theta functions and somewhere in there I stated
Abel-Jacobi theorem.


[S]: Define the genus of a curve in every way you can.
[O]: Why is this equal to the number of "handles".
[L]: Do you have an algebraic curve of every genus? (Yes)
[L]: What about non-hypereliptic? (Yes, dimension of the moduli space, or nonsingular
degree 4 curves in P2)
	
Okounkove then showed me a proof that the dimension of the moduli space is 3-3g

[L]: Are there seperable polynomials of any degree over any field?(yes)
[L]: How many? (I said the word "recursively", and he was happy)
[L]: I guess I'm supposed to ask you if you know serre duality (I said yes, he didn't
pursue it)
[L]: If you have two maps to P1 from an algebraic curve, prove they must agree at a point
(Maps to P1 are rational functions, so divide)
[L]: Damn, that works. Ok, what is the class group of P^1XP1? Why? When is a divisor very
ample? So you could also use intersection theory.
[S]: Let's do some number theory. Prove finiteness of class group. Dirchlet's unit theorem.
What's class field theory? Class number formula? Frobenius elements? Compute some class groups.

We then had a discussion about quadratic fields having class number 1, and the exam was over.