Committee: Sarnak, Mirzakhani, Sorenson
Special Topics: Algebraic Number Theory, Representation Theory of
Compact Lie Groups
Date: April 29, 2008


Algebra

[S]: I give you two matrices over a field, how would you tell if they
are conjugate or not? What theorem are you using? State it? How does
it apply to this situation? Why is k[T] a PID? If two matrices are
conjugate over algebraic closure of the field does that mean that they
are conjugate over the base field too?
[C]: What is the Galois group of a finite field? What is a generator?
How many elements does a finite field have? What can you say about the
multiplicative group? Prove it.
[M]: Give a prime ideal in k[x,y].
[S]: Why is it prime? What is the variety it defines? What is Nullstellensatz?
[C]: What is an irreducible variety? Give an example of a non-irreducible one.
[S]: Prove Hilbert Basis Theorem. (This was because I mentioned it at some point,
don't know why!!)


Complex Analysis

[M]: Suppose I give you an analytic function, what can you say about
it's zeros? (what do you mean?)
[S]: Say in a disk of radius R?
(I said something about Jensen, and then realized, they didn't say
anything about the order.)
[S]: She never said anything about the order!
(I then told them the answer when the function if finite order)
[S]: Zeros of an analytic function on the unit disk?
[S]: How would you prove Jensen?
Then somehow (I don't exactly remember how, may be Sarnak asked
directly or I said something about it) we started talking about Picard
and Uniformization.
[S]: Prove Picard. ("Small Picard")
[S]: You are using something there. ("Uniformization")
[S]: State it. What do you mean by universal cover? Is it topological
or analytic? How would you prove it?
Then we went back to C again.
[S]: Suppose I give you two open sets in C when are they conformally
equivalent?
How many parameters? Do the moduli spaces have complex structure? What is a Green's
function? When are two annuli conformally equivalent? Prove it.

And the nightmare, real analysis.

Real Analysis

[S]: What is Fourier Transform? (on R) What can you say about the image?
(At this point I told them Riemann-Lebesgue, but forgot to tell about
continuity)
[S]: Is this transform defined a.e or everywhere? Is it continuous?
Prove it. What did you use there?
(This lead to an endless 5 min discussion about dominated convergence
theorem in which I completely blanked!)
[S]: State the theorem. What is dominating what?
[S]: I was going to ask you a harder question but now I won't.
(I asked what the harder question is)
[S]: Is the image of the Fourier Transform onto?
(No! At least I got this right!!)
[S]: Why?
(I told him about the delta "function" and he asked me if I knew a
theorem from functional analysis, Banach-Steinhaus!!)
[S]: Tell me a function that is in L2 but not in L1. (I don't remember
where this fits into..)
[M]: What is a measurable function? Is the composition of two
measurable functions measurable?
[S]: Are all subsets of R measurable? Give me a non-measurable subset.

There was a break at some point in the middle of real analysis, and
then we moved to number theory.

[C]: What is the ideal class group?
[S]: Why is it finite?
(I gave a classical proof, then Sarnak asked if I could prove this
analytically, we spent some time about L-functions trying to give an
elementary analytic proof and then moved on)
[C]: What is the Decomposition group? How does a prime split in an
extension? do everything for a cyclotomic extension.
[S]: Q(sqrt(-163))
(PID!)
[S]: How would you show that?
(I started telling him about Minkowski bound and then the most
surprising question of the exam came:)
[S]: How would Gauss proceed?
(I said quadratic forms, and then we started talking about quadratic
forms, equivalences, SL(2,Z) action....... which finally lead to bound
on the class number)
[C]: Can you say something about the Hilbert Class Field.
[S]: How would one compute Hilbert Class Fields? CM theory, torsion
points on elliptic curves, Kronecker Weber.

And finally representation theory

[M]: Do finite groups count as Lie groups.
[S]: By all means, go on.
[M]: What are the irreducible representations of SL(2,Fq)?
(Where did that come from!! I started with parabolic induction, and
then told them that these were not all, we didn't go much further than
this and on the way somewhere, they asked about the dimension of the
induced representation from Borel, and conjugacy classes in SL(2,Fq))
[S]: Everything about SU(2): Irreducible representations, Characters,
Weyl Integration Formula, Haar measure on the torus..
[S]: Talk about the general theory: Roots, Weights, Conjugacy of tori,
Weyl Character Formula, Weyl Integration Formula
[S]: Give a higher rank example.


The exam lasted about 2 and a half hours and I should add that the committee was very nice!