Alex Peng
January 14, 1999, 2pm

Committee: Sarnak (c), Sinai, Ellenberg

Special topics: Theory of Probability, Analytic Number Theory

    Sarnak began by asking me which of the three general subjects that
I'd like to begin with.  I said complex analysis. Sarnak then
gestured for Sinai to pose the first question. Sinai asked me what
an entire function was, which I defined following on the
definition of a holomorphic function. Sarnak interjected to ask if
I knew of any regularity condition on the function. To which I
responded with the standard proof of existence of power series
representation. Sinai was satisfied at this point, but Sarnak
pressed on with another question in the same vein--if we know f to
be entire, and that |f(z)|<|z^100| what can I say about this
function? I blanked out for a while before their persistent hints
prodded me to the idea of dividing and then applying Liouville's
theorem. Sarnak then remarked that since we were in this entire
function mood, we might as well do another one--f entire, nowhere
vanishing, and |f(z)|<exp(|z|). This time it didn't take me too
much time to blurt out Weierstrass factorization. But he wanted
something more elementary unless I could prove Weierstrass to him.
So I said that I could probably take log, get a Harmonic function,
and maybe use Harnak's inequality (which turned out to be not such
a good idea). Amused, Sarnak said that he had always thought that
it should have been called Sarnak's inequality. Eventually, we
reduced the problem to an entire function whose real part is
bounded, which then by little Picard is found to be a constant.
Sinai cut in at this point to say that he had something simple to
ask the integral from -inf to inf of exp(imx)/(1+x^2). I did the
standard contour integral bit, and they were satisfied. Sarnak sat
pensive for a moment before asking if I could state the
uniformization theorem for Riemann surfaces. I said no. "What
about Riemann mapping." This I did, but he wasn't a bit interested
in the proof. It however provided him a perfect transition to
Dirichlet's problem for simply connected regions. I said that you
solve it for the unit disk with Poisson kernel and the rest with
Riemann mapping. There then came a little digression as Sinai
interjected to pose the question of solving the Dirichlet problem
using probability. Apparently he had Wiener measure in mind. I
said I had no idea.  Sarnak's next question was if I knew how to
solve the Dirichlet problem, can I prove Riemann mapping from
that. I didn't know anything to say except "maybe using Green's
function." They didn't get anything else out of me in this line.
    The floor was then yielded to Ellenberg for Algebra. I defined a
principal ideal domain for him. His next question was for two
examples, one of a PID, one not. Inexplicably, I started raving
about rings of quadratic integers with class number greater than
1... Sarnak cut me short and said that the first advice is always
to say the simplest thing that suffices, but since I brought these
items up, they might as well test me on them. I was able to define
ideal classes and the class group all right, but got a bit hung up
on why the class number is finite. I said that I could link it to
the value of the L function at 1, or even more directly, binary
quadratic forms, at least in the imaginary case. We spent another
five minutes here before they dug to the bottom of my knowledge
here. Now Sinai decided to chime in and asked if I knew what Lie
Groups are. I said that I knew how to define them. Sarnak laughed,
"I would hope so since you've been to my lectures the whole term,"
and then added, "not that I ever defined it." Sinai asked for an
example, and then an example of a compact Lie Group, and finally,
if I knew something about SU2 and SO3. I said that there was a
double covering. Despite my efforts to get them to ask me
questions about it, Sarnak wasn't interested but instead asked
Ellenberg to pose questions about representation
theory--definition, definition of an irreducible representation,
regular representation, the orthogonality theorem for irreducible
characters, and why representations can be decomposed into direct
sums of irreducible representations.  At each instance they were
satisfied with a quick answer or a brief sketch (for example, to
show that a representation of a finite group is conjugate to a
unitary representation, Sarnak said he was happy once he heard me
say take an average over the group.)
    Next up was real analysis. Sinai wanted to know something about
Laplace method, and Sarnak indicated that maybe I was more familar
with stationary phase--I wasn't. The only thing that I could come
up with at the spot was integration by parts, which Sarnak said
was in fact a good idea. He then moved on to Lebesgue's theorem,
which I stated. They seem to be in a definition mode, and asked me
to define practically everything that came out of my
mouth--absolute continuity, Lebesgue measurable, L1, etc. The last
question in real analysis was the dual of Lp spaces, again no
proof.
    At this time, Sarnak suggested that we take a three minute break.
It turned out to be more like six or seven minutes--Sarnak wandered off,
Ellenberg went to get some coffee, and Sinai held a conversation in
Russian with a student.
    After the break, Sarnak resumed by saying, "now let's hear about
the theory of probability." Sinai said that he had only three
questions--state, in the most general form that I know, the law of
large numbers, central limit theorem, and ergodic theorem for
Markov chains. I began to state the Law of Large numbers
(Kolmogorov I), at which point, Sinai said it was not correct.
Sarnak commented that as I stated, these random variables can all
be the same. Again, it seems that I was blanking out something,
and it took Sarnak's saying that there is another "i" word before
identically distributed to jolt me to say independent. For the
following five minutes or so, I was having a great difficulty
convincing Sinai that I knew what independence meant and that we
need more than pairwise independence for the theorem (partly
because I had become so frazzled and feared to say anything wrong,
and then feared to say anything at all). Central limit theorem
went much better, other than a minor slip of the pen with
normalization. I stated the theorem for Markov chain, and it led
to a question about stationary distributions, and why they are
unique. Without thinking too much, I began giving a proof that I
had prepared which uses a coupled Markov chain. Sinai reminded me
that I could assume finite state. So I changed tact, and gave the
proof from his book via contraction mapping. Sarnak was happy now.
"What about Birkhoff's ergodic theorem?" he then asked. This I
stated, and he conceded that the proof was difficult, but wondered
if I could give an outline. I had read this before, but couldn't
recall anything. What about von Neumann's version? He and Sinai
then entered into a short discourse about who proved which version
first, etc. The proof came back to me in bits and pieces, Fourier
coefficients, Weyl sum... but not the most import fact, which is
the unitary nature of the operator in L2.
    It was already past 4:30, and Sarnak said that he'll start at the
end of Analytic Number theory-- how come Vinogradov was able to
prove his theorem for three primes, but we can't use it to prove
Goldbach. I had more or less anticipated this, so I said the
problem was with the minor arcs in the application of the circle
method. For three primes, one can use Parseval's to treat the
integral of the exponential sum, which is not available for two.
His next question was quite unanticipated--what about the L1 norm
of the exponential sum--do I know of a lower bound. The only thing
that I could think of to say was that it must be some power of n,
because it didn't work. He said I could use information from the
L4 norm, and L2 norm and then the fact the Lp norms are convex. I
could only just nod. The final question was in the same area, the
maximum of the exponential sum when the irrational involved is far
away from rationals with small denominators. He was finally
satisfied when I explained why the method of passing to a sum over
a progression and then bilinear forms didn't work if the sum is
twisted with a character (because it's multiplicative with norm 1,
which is exactly when it fails).
    Sarnak was then eager to end it, and asked me to wait outside for
a minute or so before telling me that I passed. A couple things
were pretty clear during the exams. If they sense that you know
something pretty well, then they'll move on to something else
right away--which means that it's not necessarily a bad sign if
you spend a lot of time on something that you are hung up on.
That's simply their strategy. From my description, you can
probably tell that we didn't go over any of the standard proofs,
but instead spent much more time on problem solving and
definitions. I don't suppose there is anything to be done about
the former, but in retrospect, I would have been a lot less
frazzled, if I had gone through all the important definitions
carefully. We may deal with concepts all the time, but to define
them well under pressure may be something else (and there is the
added pressure sometimes, as I thought that I would look like
quite an idiot if I couldn't define the basics). And finally,
Sarnak's admonition about sticking to the simple answers
notwithstanding, it may not be a bad idea to spew out things at
random, especially if you are out of better ideas.  They seemed to
be more interested in finding out what I knew than what I did not.