Date: May 4 2006
Time: 1:00pm - 2:30pm
Committee: Stein (Chair), Yang, Ramin
I think the committee was kind and didn't really push me for anything really
difficult. As you will see, the questions are very standard, and they didn't
mind if I made some silly mistakes along the way. They also let me choose
whatever order I want to take the subjects.
Differential geometry (Yang)
Define Gaussian curvature. Why is it intrinsic?
Talk about the relation between curvature and topology of a manifold. (I
mentioned Bonnet-Meyers and Synge's theorem in even dimension.) Why is the
Synge's theorem that you stated not true in odd dimensions? Do you know
Gauss-Bonnet formula? (Oh, that was what he was asking for when he meant
'curvature and topology'!) Do you know its generalization to higher
dimensions?
Algebra (Ramin)
Talk about Galois theory. Give an example of an extension that is not
normal. Compute the Galois group of x^3 - 2 over the rationals. (He was
kind enough to choose the simplest example that one can compute!) Talk
about solvability by radicals. Why is S_5 not solvable? Why is A_5 simple?
Complex analysis (Stein)
What do you know about the zeros of an entire holomorphic function on the
plane? (The growth of the number of zeros inside a disc is given by the
order of growth of the entire function.) How would you prove a result like
that? (Jensen's formula) Proof of Jensen's formula? (Factorize the
function.) Write down an entire function of order 1 that has precisely the
integers as its zeros. (\sin \pi z). Is it unique, or is it unique up to
something? Can you characterize them all? (Hadamard factorization)
Name one entire function whose zeros are precisely the non-positive
integers. (1/Gamma(z)). Write down its factorization. Now for the zeta
function. How do you analytically continue that? What is the functional
equation for the theta function? Is there any zero of the zeta function on
the line Re z = 1? Do you know the growth of the zeta function there? (I
can't write down the estimate off-hand, but he was kind enough to let it go.)
Harmonic analysis (Stein)
Tell me something about the singular integrals. You started out with an
operator that is a priori bounded on L^2, and this is one of the big business
in the subject. Do you know any theorem that will guarantee that a singular
integral operator is bounded on L^2? (I wanted to say T(1) theorem, but I
quickly backed down since I didn't really know it; then I mentioned another
theorem in the translation invariant setting, which says that if you start
with a kernel that is smooth outside the origin and satisfies a differential
inequality there, and if you get a uniform bound whenever you test the
distribution kernel on any renormalized bump functions, then the Fourier
transform of the kernel is a bounded function.) Prove the L^2 boundedness
theorem that you just mentioned. Can you give some specific examples of the
relation between singular integrals and PDE? (Riesz transforms) Do you know
something about the L^2 boundedness of a singular integral operator in a
setting that is not translation invariant? Say something about Heisenberg
group? (It happens that the only example that I knew off-hand on the
Heisenberg group is the Cauchy-Szego projection, whose L^2 boundedness is
trivial, so he changed the topic.)
Tell me something about BMO functions. Is (log|x|)^2 in BMO? What is the
relation between the Hardy H^1 space and BMO? Do you know the John-Nirenberg
inequality and its relation to the exponential class? State the atomic
decomposition for H^1. Is every BMO function locally in L^p for some p? Can
you prove the boundedness of the Calderon-Zygmund integrals from L^{\infty} to
BMO in an elementary way? Do you know anything about compensation compactness?
The exam ended there. There was no real analysis question, as Ramin suggested
that it is a subset of harmonic analysis. They bursted into laughter as I left
the room. (Ramin jokingly suggested that they were going to discuss how to fail
me :p) Luckily they quickly changed their minds, and congratuated me outside
Stein's office. The exam lasted for 1.5 hours.