Philip Gressman
8 May 2002, 1 p.m., Fine Hall 802 (Chair's office)
Committee: Stein (Chair), Chang, Ellenberg
Special Topics: Harmonic Analysis, PDEs
The committee let me choose which subject I wanted to start with. I went
with harmonic analysis.
(S) What are the symmetry properties of the Fourier transform?
I talked about decomposing L^2 into spaces whose angular parts are
spherical harmonics.
(S) How does the F.T. act on these subspaces?
They are invariant; the radial part is given by integrating against
a Bessel fn.
(S) Give an equation for Bessel fns. What is the asymptotic behavior?
(S) Where else do Bessel fns. arise? Is it the F.T. of any measure?
How does the F.T. of the uniform measure on the sphere decay?
I didn't know the precise rate of decay, but the committee wasn't
too concerned.
(S) State some Calderon-Zygmund type boundedness theorem.
I did so and was asked to explain the proof. I talked about the
weak-(1,1) bound and interpolation, which prompted the next questions.
(S) What do you mean by weak-(1,1)?
(S) State the C-Z decomposition and sketch the proof.
(S) What kind of condition do you need on the kernel of a C-Z operator?
(C) What is an example of such an operator?
(C) What about Laplace u = f? Which operators do you use to show that u
has all weak second derivatives in L^p if f is in L^p?
Next we moved on to algebra.
(E) What do you know about representations of finite groups?
(E) What is an irreducible representation? Can you give an example?
(E) Why can you decompose representations of finite groups into
irreducible ones?
(E) Can you give a group and a representation which is reducible but
can't be decomposed?
(E) What is the key underlying property here?
I said (after much assistance) compactness and a finite Haar measure.
(E) Talk about the ideals in C[X,Y]. Can you make some maximal ideals?
I really floundered around on this one. Along the road to solution
we discussed such things as the quotient of a ring by a maximal
ideal being a field, etc.
(E) Whad do you know about the quarternions? How do you know they're a
division algebra?
(E) If you join i,j,k to some other field besides R, what do you get?
I eventually saw that in the case of a finite field, you get a ring
with zero divisors.
(E) What do you know about division algebras over algebraically closed
fields? I said "nothing" and instead talked about the division
algebras over the reals instead.
(E) What are the quarternions topologically?
In retrospect, I think they were trying to get me to talk about
Lie groups and Lie algebras, but I didn't follow at that time.
After algebra, the committee decided to continue with PDEs.
(C) What are the nice properties of harmonic functions?
(S) What is the difference between the weak and strong maximum principles?
(C) What is Harnack's inequality for positive harmonic fns?
(C) What is the idea behind Schauder theory? How much regularity do
you gain?
(C) What is the Sobolev embedding theorem?
(C&S) What's the best space you can map into in each case?
(C) When is the embedding compact?
Next Stein moved to complex analysis. He started off by asking me about
a theorem whose name I didn't recognize (and I can't recall now). I told
him I wasn't familiar with it, so he asked a different question.
(S) What is the Weierstrass product theorem? What are those extra
factors for? Give an example where the product wouldn't converge.
(S) What is that function you wrote down?
I had written down the product for 1/Gamma(z) since I figured they'd
ask me about the Gamma function anyway.
(S) What if we want zeros on all integers (but z=0). Do we need the
convergence factors this time? (No, if we take a symmetric limit.)
What do you get? (sin up to a factor of z)
(C) What is a normal family?
(S) What is a sufficient condition in complex variables to guarantee
you have a normal family? (i.e. Montel's theorem)
(S) What theorem uses normal families in its proof?
(S) State the Riemann mapping theorem.
(E) What does it mean for a domain to be simply connected?
Comments: I probably spent more than half of the exam on the algebra
section (similar to Jose's experience). The committee was very helpful,
willing to clarify questions if necessary or change directions if I
wasn't getting it. Several times (during the algebra section) I said
things which weren't true, but the committee didn't seem to mind as long
as I eventually saw why I was wrong. The exam lasted 1 hour and 10 min.