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.