Terry Tao's generals

My examiners were Stein, Kleinermann and Rudnick.  (Kleinermann came in
about five minutes late).

The first questions asked were about Harmonic analysis.  They asked me what
I knew, and I said basically singular integrals and functional analysis -
no real harmonic theory at all.  So they examined me on singular integrals
instead, essentially.

Give examples of singular integrals (Calderon-Zygmund).  What theorems do
you know about them, how to prove boundnedness.  (I stated a T(b) theorem,
which nobody seemed very acquainted with, and said with some brief
explanation that knowing boundedness on L^2 - L^2 was sufficient to prove
L^p-L^p for 1<p<\infty).

Since my T(b) theorem was martingale based, Stein asked me to give a
definition of Martingales.  I gave the sigma algebra, purely measure
theoretic version, but stumbled a bit on the definition of conditional
expectation.  I think Stein was satisfied by my explanation of conditional
expectation at the end, though.

Stein then asked me what other boundedness theorems I knew.  I couldn't
think of any concrete ones off hand, but I mumbled about a convolution
operator being bounded if its Fourier transform was bounded and had some
smoothness condition, but I couldn't recall any details.

Some questions about the boundedness of the Hilbert operator, an easy way
of proof.  I said the Fourier transform was bounded, and that should do it
for L^2 - L^2.  They seemed to be hinting at generalizations, and
eventually I clicked on to Riesz potentials, but again I didn't know enough
results, only that they were bounded.  (For example: if \Delta u = f and f
is in L^2, what do you know about u?  I guessed it was in some sort of
Sobolev space).  I did state though that all second derivatives of u would
have L^p norms bounded by that of the Laplacian.  Kleinermann asked me for
an elementary explanation of this, but I couldn't think of one.

A couple definitions of H^1.  I got the atomic definition right, and the
Riesz potentials, but got the limit of harmonic-functions definition wrong.
And I had no idea how to prove any of them were equivalent.. so they
abandoned this after a while, but not until they made me realize that the
latter definition I had gave was actually that of L^1, not H^1.

Then they asked me about fundamental solutions.  I sluffed the exponents in
the denominator a few times, so they began asking me about homogeneity.
Eventually, with a lot of prodding, I got the exponents of the Newtonian
and Cauchy potentials correctly.

Kleinermann asked me about the connections between PDE's and singular
integrals, but I didn't know any more than vague generalities.

Then they switched to Analytic number theory.  Surprisingly, I fared much
better on this topic!

Rudnick (mainly) asked: give several forms of the Zeta function, give one
that allows analytic continuation (I used theta functions), prove the
modularity of the theta function, and thus derive (give or take a number of
pi's and Gammas) the functional equation.  What do you know about the
poles, zeroes.  (I gave the De Vallee-Pouisson (sic?) zero-free region).
State a form of the prime number theorem.  Why li(x) instead of x/logx?
Give an explicit formula for \psi(x).  Throughout, I didn't have to back up
anything with messy calculations: Rudnick always stopped me when I tried.

Talk about primes in arithmetic progressions.  Give an elementary proof of
the infinitude of primes of the form 4n - 3.  What algebraic tools are used
in Dirichlet's proof of infinitude of primes in arithmetic progressions?
(It took a while before I understood, "Oh, you want me to talk about
characters.")  I gave the basic run-through of how it is sufficient to
prove L(1,\chi) is non-zero.  (With Rudnick's help, this was very smooth.)
Then they asked how Dirichlet got an explicit formula for this when \chi
was a real character.  I was going to write a messy (but finite) expression 
involving sines and logs, but then I realized that they were talking about 
the class number formula.  (I said carelessly though that "this was a 
disgusting way to do it", since I was still thinking about the sine-log
formulas.  Then they made a comment that "This would put thousands of
people out of work", or something like that.)  Anyway, I stated what the
class number was (vaguely), but didn't give out an explicit equation (again
Rudnick waved me off when I tried this), and they seemed satisfied.

They asked me what else I had prepared, and I said some stuff on the circle
method and the large sieve.  So Rudnick asked me a few desultory questions
on the sieve, and I stated one formulation, but didn't know of any useful
applications.

Then Rudnick asked a few algebra questions only and it was over.  Talk
about Galois theory.. construct a field extension of order S_n.
(polynomials over symmetric polynomials).  Construct one of order A_n.  I
got the index and the order mixed up, but Rudnick corrected me, and so
eventually it came down to finding a non-symmetric polynomial whose square
was symmetric.  I couldn't guess it, so Rudnick said "product of
differences", and I said, "Oh, the Pfaffian" and they were satisfied.

"What else do you know about A_n?" (I said it was simple for n>=5).  What
other simple groups do you know.  (The monster?  But I backed down quickly,
saying I knew not how to define it.)  Then the discussion meandered a bit
as the professors talked among themselves whether the group of order 168
(which I mentioned) was sporadic.

After this, they decided to pass me, though they said that my harmonic
analysis was far from satisfactory. :(  They didn't ask any real or complex
analysis, but I guess from my handling of the special topics they decided
that wasn't necessary.  Besides, we were almost getting snowed in.

The exam lasted 2 hours.