Andy Booker
May 12, 1999
1:00pm-3:30pm
Committee: Sarnak (chair), Fefferman, Stalker
Topics: analytic number theory, singular integrals.
Sarnak asked me which general topic I'd like to start with. I said
complex analysis, so he motioned to Stalker to ask the first question.
Complex Analysis and Analytic Number Theory (I)
-----------------------------------------------
Do you know a variant of the Phragmen-Lindelof theorem? I knew the
statement of one version, but that wasn't good enough. We worked out
how to prove the theorem I had quoted step by step. I was given
millions of hints along the way. That was it for general complex
analysis, as Stalker's next question was aimed towards number
theory...
Consider the series \sum_{n>=0} (-1)^n (2n+1)^{-s}.
Where does it converge?
(I answered that it converges for Re s > 1 and can be analytically
continued to Re s > 0 by partial summation.)
Can it be continued beyond that?
(Yes, it satisfies a functional equation.)
How do you show that?
(I talk about Poisson summation twisted by a primitive character, which
leads to the functional equation for the theta function.)
What is its value at one? (pi/4) Why?
(I said it is arctan 1.)
Can you do it with a contour integral?
(I wrote down an integral with the csc function.)
Can you arrive at the value at 1 in some other way?
(I mumbled class number formula, much to my distress...)
How would you prove the class number formula?
(I talked about the zeta function of a quadratic extension. It soon
became clear that I don't understand this stuff very well, but Sarnak
didn't seem to mind. He said he could see that I'm an analyst at
heart... :)
Do you know another proof that L(1,\chi) doesn't vanish (for any \chi)?
(I explained why it doesn't vanish for complex characters and
gave the de la Vallee Poussin argument for real ones.)
Why doesn't zeta(s) have any zeros on Re s = 1?
I think Sarnak wanted to get back to general topics, so we switched to
algebra. But more number theory later, he promised.
Algebra
-------
Talk about group representations.
What's special about using C (complex numbers) in the definition of
group algebra? Is it possible to work over other fields?
What goes wrong if the characteristic of the field divides the order of
the group?
How would you work out the orders of the irreducible representations of
D_n?
Why is the the sum of squares of dimensions equal to the order of the
group?
What are some other finite subgroups of O(3)?
Do you know what Frobenius reciprocity is?
What is rational canonical form?
What is a PID?
What's an example of a UFD that is not a PID? Why?
Is k[x] a PID? Why?
How does the rational canonical form relate to the expansion you just
wrote down? What is the module?
How do you identify the minimal polynomial and characteristic polynomial?
There were some more questions in algebra, but I don't remember them
all... anyways, nothing out of the ordinary. At this point, Sarnak
motioned to Fefferman to do real analysis.
Real Analysis
-------------
Write down a function which is everywhere differentiable, but whose
derivative is not continuous.
If you have a function which is a.e. differentiable with derivative 0,
must it be constant? Construct a counter example.
Define convex function.
What do you know about them?
How smooth must a convex function be?
Does it have a second derivative?
Is there a convex function with second derivative a.e. 0 but which is
not linear in any interval?
These questions took only a few minutes, after which Fefferman said
he was happy with real since singular integrals is one of my topics.
So we took a five minute break.
Singular Integrals
------------------
Do you know the Whitney extension theorem?
(No.)
Fefferman started to explain it, but Sarnak broke in, saying we should
start with something simpler.
Consider \sum_{n\ne m} a_n a_m / (n - m) for a in l^2(Z).
What can you say about it?
(It's l^2 bounded.)
Sarnak laughed and said I answered too quickly, and must have seen the
question on a previous exam (which is true, Adrian Banner was asked the
same question).
How would you prove L^2 boundedness of the (continuous) Hilbert
transform?
(It's an L^2 multiplier.)
How about L^p boundedness?
(First show weak (1,1), then L^p boundedness follows by interpolation and
duality.)
What does weak (1,1) mean? How do you prove it?
(Calderon-Zygmund lemma.)
What is special about the bad part of the function?
(Integrates to zero over cubes in the decomposition.)
Does the above example remain l^2 bounded if n-m is replaced by its
absolute value? Why not?
(After some prodding I computed the Fourier transform of the kernel.)
Can you turn 1/|x| into a distribution?
(I thought this was a little vague, but with committee's help I
eventually understood what Fefferman was getting at. You define it in
the natural way and subtract off the logarithmic singularity at 0.)
What do you need to do to check that this defines a distribution?
(Continuity.)
So you didn't really need to use the Schwarz class here. Do you know a
weaker condition under which the above would make sense?
Now what if we replace |n-m| with |n-m|^(1+it)?
Fefferman said he was happy, and perhaps we should move on to more
analytic number theory. Sarnak first asked me about Sobolev spaces and
PDEs, but gave up when I said I don't know anything about them.
Analytic Number Theory (II)
---------------------------
How can we count (asymptotically) the number of solutions to
x_0^2 + x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1
with |x_i| <= N?
(I wrote down the generating function and the Fourier integral for the
answer.)
What do you expect the answer to be?
(Sarnak led me through a probability argument to show it should be about
N^3.)
So, now how do you proceed?
(I explained roughly the Hardy-Littlewood method.)
How do we get a nontrivial bound on the minor arcs?
(I mentioned Weyl's inequality, and Sarnak made me explain why it's
true.)
Why doesn't circle method work for Goldbach?
After answering this question we entered into a discussion on lower
bounds of L^p norms of the generating function for Goldbach, which I
found very interesting. This was the last question.
General Comments
----------------
The most important thing is to remain calm and not get too worked up
over the exam. If you're like me, you will make many mistakes. (Don't
expect to be able to think very clearly at the board!) But, at least in
my case, the committee was used to this, was very patient, and didn't
mind giving hints when necessary.