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.