Jose Luis Rodrigo
18 October 2000, 9 a.m. , Fine Hall 301 (Chairman's Office)
Committee: Fefferman (Chair), Hagelstein, Browder.
Special Topics: Harmonic Analysis, PDE
They asked me in which order I wanted to be tested. I decided to start
with Real. (both Fefferman and Hagelstein)
- (H) Can you state Radon-Nikodym Theorem?
- (F) State Lebesgue Differentiation Theorem. How do you prove it?
I proved it using the maximal function and so they asked my to define
it and state some properties.
- (H) Where do you use Lebesgue Diff. Thm in Stein's book?
- (H) What's a Calderon-Zygmund decomposition? They asked me to give the
whole proof.
- (F) What's a singular integral? Why do we care about them? Give some
examples.
- (F) In the examples I mentioned the Hilbert transform and they asked
my to write it down and prove some properties. I proved a more general
theorem about singular integrals, which could be applied to the Hilbert
transform. I had to talk a little bit about Marcinkiewicz operator.
At that point Fefferman ask me which of the other topics I wanted to
continue with. I said I didn't mind and F. asked Browder to start with
algebra.
- Can you give me a polynomial whose Galois group is Z/3Z? I had to
talk about the Galois group of x^n-1.
- Can any group be realized as a Galois group?
- What theorems do you know about finite groups? Sylow theorems. Ok,
state them.
- Suppose you have a group of order a power of a prime p. This group is
acting on a set and p doesn't divide the cardinal of that set. Prove
that there is a fixed element. I wrote down the class equation and he
asked my to sketch the proof.
- What can you say about free abelian groups?
Browder was satisfied with that and they moved into complex variables.
- (H) Can you map the plane into the disk? Which domains can you map
into the disk? Prove Riemann's Theorem.
- (F) What can you say about domains that aren't simply connected?
- (F) Can you map an annuli into another? I said in general no, the
quotient of the radii has to be the same. Prove it. I came out with
a proof they didn't know using Schwarz reflection (Fefferman was
quite pleased). The good thing about this proof is that you don't have
to write down anything on the blackboard...
- (F) Talk a bit more about Schwarz reflection.
- (F) What can you say about a function that grows like sqrt(z) for
large z?
Then Fefferman decided to move into PDE. I have forgotten some of the
questions, but most of the questions were about the Laplacian
(Fefferman's favorite operator, I guess). Most of the questions also
involved some Harmonic Analysis. Fefferman asked basically all of these
questions.
- Suppose Laplacian of u = f. What can you say about u? Using Fourier
analysis? (Riesz transform)
- Fefferman gave me a general second order elliptic operator (only
involving second derivatives). He gave me a function and asked me if I
could say if there were solutions or not without computing. (Use the
max. principle to see that it couldn't. It was attaining the min. at the
origin.)
- Ok, state the maximum principle. How do you prove it?
- What can you say about the regularity of a solution of that
operator=f?
I had to talk about Sobolev spaces...
How do you prove it? Again the Riesz transform.
- Suppose a function u is divergence free and curl(u)=w for a given
(smooth) w. What can you say about u. How can you calculate it.
I had seen part of this studying fluid mechanics. If you take Fourier
transforms you can get a linear system and it becomes a problem of
linear algebra. With Fefferman's help I discussed a bit more about the
properties of the solution.
The exam lasted 1 hour and 15 minutes. The committee was really helpful
giving me hints and clarifying some vague questions. I spent around 30
minutes doing algebra, something I didn't expect.