May 18, 2006
Examiners: Sergiu Klainerman, Janos Kollar, Robert Gunning
Topics: PDE, Harmonic analysis
Time: roughly 3 hours. 

Preamble: I got there five minutes early. Kollar arrived a few minutes
after I did. Gunning came right on time. We stood outside waiting, me
fidgeting and extremely nervous. Finally Klainerman shows up and let
us into the office.

Klainerman and Gunning each pulled up a chair. I stood. Kollar
proceeded to layout a foam padding and reclined on his back,
explaining he has lower back problem.

They asked me what topic I want to start with, having really no
preference, I told them to start with Harmonic. Klainerman then
decided that prior to Harmonic we should ask a bit about real, which
he deferred to Gunning.

Real Analysis.

Gunning: Suppose we have a sequence of integrable functions on the
interval [0,1], tell me about convergence.

Me: Fatou's lemma. 

Gunning: What would be a converse that would "make
sense" for Limit inferiors? Is it true?

Me: You change the sign in the inequality, and it is not true.

Gunning: Counter example?

Me: The sequence of functions f_n defined by n times the
characteristic function of 1/n. The liminf of the integrals is 1, but
the integral of the liminf is 0.

Gunning: How about pointwise convergence?

Me: I stated Egarov's theorem and stated that outside a set of epsilon
measure, the convergence would be uniform.

Gunning: But if a sequence converges in L1, must the sequence
converge pointwise?

Me: Yes? (saw Gunning shake his head) No...

Gunning: Give an example.

Me: (after much thought) The sequence of functions f_n defined by f_1
= 1. f_2 is characteristic function of [0,1/2]. f_3 is that of
[1/2,1], f_4 is characteristic of [0,1/4] and so on.

Gunning: Does it contradict Egarov's theorem?

Me: hmmmmm perhaps Egarov's theorem requires a.e. convergence?
(apparently I left that part out in the initial statement.)

Gunning: How about the other way around, if a sequence of functions
converges, does it necessarily converge in L1.

Me: No. You need to assume uniform convergence.

Gunning: okay, let's change the subject, how about fundamental theorem
of calculus for Lebesgue measures.

Me: give the theorem in both directions (for absolutely continuous
functions the derivative exists a.e. and integrates to it).

Gunning: what is the corresponding theorem for measures.

Me: Give Radon-Nikodym.

Klainerman: What is the differentiability/continuity of Lebesgue
integrable functions.

Me: I had no idea what he was asking about. After getting cues from
Klainerman a bit, I finally realized he was asking about Lebesgue
Differentiation Theorem. Then he asked me for a proof. While I was
hestitating, he told me to consider the Maximal function. So we talked
a bit about that: the definition, the fact that it is L1->weak L1,
and L^p to L^p for p up to infinity. Then he asked me for a proof ot
the L.D.T. using the maximal function, but I still have no idea. So he
asked me whether the theorem is obvious for continuous functions, I
said yes, and sketched the proof. He then asked about how to show that
C0 is dense in L1, and, for some bizarre reason, I blanked out
(which will happen quite a lot during the next three hours). So he
hinted me at C^infty functions, and finally I caught on and gave the
mollifier construction. At which point, we moved to 

Harmonic/PDE. Part I

Klainerman: Time for some Harmonic analysis. What can you say about
Calderon-Zygmund theory?

Me: I defined a C-Z operator, and said that it would take L^p to L^p,
1<p<infty. Surprisingly, he did not ask me to prove that [I was so
very prepared to give the proof... and this won't be the first time
that they just conveniently not ask about stuff I actually know. It is
like they can smell fear or something]. Klainerman also was not
completely satisfied with my statement of the theorem, though I am
pretty sure I reproduced it almost word-for-word compared to the one
in his notes. I still don't know what he was objecting to.

Klainerman: What is C-Z theory useful for?

Me: Littlewood-Payley square function estimates [he didn't care about
that], Miklin-Hormander [yes, it is important, but not what he was
looking for].

Klainerman: Let me rephrase, do you know any results in PDEs that
result from C-Z?

Me: Can't think of one off the top of my head.

Klainerman: Really? What about regularity estimates for elliptic
theory?

Me: Oh! That. So I wrote down the Laplace equation (triangle u = f) on
R^n, gave its fundamental solution. And said that if f is in L^p then
u is in W2,p.

Klainerman: define W2,p

Me: I cheated a bit and used the form without intermediate derivatives
(since we are on R^n, we have a Sobolev multiplication theorem that
gives me that if he wanted to know).

Klainerman: right, hum, the L^p term would have some difficulties, but
how about just the homogenous part, i.e. show that the Hessian of u is
L^p

Me: I cheated again and appealed to Miklin Hormander, by showing that
Hessian times inverse laplacian is a zero-order differential operator
and hence is L^p bounded.

Klainerman: how about is physical space.

Me: stumbled a bit. Finally realized that he just wanted me to take
the derivatives inside and hit the Newton potential, so that we end up
with something that satisfies the equivalent C-Z formulation that the
kernel outside of a compact set decays like x^-n and its derivative
decays like x^-n-1.

So far so good, and he told Kollar to ask about algebra.

Algebra

Kollar: start with the x-y plane, Put on it the square with end-points
on 1's (which I suppose meant that the corners have coordinates (+/-
1, ±1). What is its symmetry group? (D_4) How many elements does it
have (8). What is the structure of it? (C_2 product C_4) Is it simple?
(No, C_4 is a normal subgroup). Give the elements explicitly.
(rotation by 90 degrees and reflection about x axis). Is it Abelian
(no, and I showed the yxy-1 = x3 if x is the rotation and y the
reflection).

Kollar: Okay, now, since you used x and y for the elements we'll
change the name of the plane to the u-v plane (oops!). What are the
polynomials in u and v that are fixed under the action of D_4.

Me: this one I got really stuck on. I had no idea how to do it, and he
pretty much hand-held my way through the problem using Eigenvectors
and so forth. I feel extremely dumb. Basically we look at the
eigen-basis of the infinite dimensional vector space of polynomials
under the operations of x and y, and find the basis for the subspace
fixed by both of them. I made a lot of stupid mistakes along the way.

Kollar seems satisfied after that. I was disappointed that he didn't
ask about Galois theory of Sylow theorems, two subjects that I was
extremely well prepared for. Though I am also happy he didn't ask too
much about ring and module theory, even when I know for a fact I would
have no problem proving Jordan/Rational canonical forms or structure
theorems. In hindsight I seemed to have spent too much time worrying
about Algebra and not enough worrying about everything else.

Complex Analysis, part I

Gunning: While you still have that box on the board, how about
conformal maps from there to the upper half plane?

Me: Oh crap. Schwarz-Christoffel. I gave the integral representation,
but I can't say anything more about it.

Gunning: What freedom do you have for the images of the corners?

Me: I don't know.

Gunning: well, consider conformal self-maps of the upper-half plane to
itself.

Me: after much prodding and that Gunning hinted at the reflection
principle, I finally realized that it must be a fractional linear
transformation that takes the real to the reals (and hence has real
coefficients) and takes i to something with positive imaginary part
(and so if the transformation is written as (az+b)/(cz+d) we need
bc-ad>0). [On hindsight, that is completely, utterly, obvious.]

And here comes the really humiliating parts.

Harmonic/PDE. Part II

Klainerman: tell me about the Maximal principle.

Me: gave the strong one.

Klainerman: how does this relate to the maximal principle in complex
analysis?

Me: I gave some other things and he wasn't too happy. Finally I
figured out he only wanted that the real and imaginary parts of an
analytic function each is a harmonic function.

Klainerman: explain ellipticity. (simple.)

Klainerman: How about the Dirichlet problem?

Me: I wrote out the problem, and mentioned that on special domains
(half plane or disk) we have Poisson's formula.

Klainerman: What makes those two special.

Me: No idea. 

Klainerman: How about for arbitrary domains.

Me: I blanked out. For a whole 5 minutes. I was just about to say
somthing about energy methods when Klainerman told me that "If you
don't know just say so, no need to waste our time." And he was
obviously unhappy about my really, really despictable performance on
PDEs.

Klainerman: that was rather unacceptable. What do you actually know?

Me: So I told him that maybe we can switch to harmonic again and do
Littlewood Paley theory.

Klainerman: okay, fine, give us a short lecture on what you know about
Littlewood-Paley theory.

Me: I gave the basic construction, wrote down the properties (cheap
littlewood-paley, square function estimate, finite band, almost
orthogonality, and Bernstein. Didn't get a chance to do commutator
estimates before he interrupted).

Klainerman: But this is just a tool, what is it good for?

Me: Started doing Sobolev multiplication estimates. (fg) in H^s can be
bounded be f in H^r and g in H^t if s<r, s<t, r+t>0, and s<r+t-n/2.

Klainerman: but this is trivial!

Me: Is it? Only for s =r = t >n/2 it is trivial. But I don't think it
is trivial when one member is less than n/2. Klainerman acquieced, so
I sketched a proof using the tricotomy formula.

Klainerman: Talk about the Berstein inequality in the case the left
hand side is L^infty. 

Me: I started talking, he hinted me that he want to know about Sobolev
inequality, so I stated that f in H^{n/2+epsilon} embeds into L^infty.
So he commented on the additional sharpness of the Littlewood Paley
version, and I said that Littlewood Paley needs always an epsilon
derivative to allow summing, so never that sharp.

Klainerman: Okay, but do you know any good applications of this theory
to PDEs.

Me: I don't know (well, actually I do know about harmonic maps, but I
don't know it well enough if they asked me to give regularity for
harmonic maps.)

Klainerman: that isn't very much, is it? What else do you know?

Me: Riesz-Thorin-Stein, Marcinkewicz.

Klainerman: Ah, interpolation theorems.

But he doesn't seem too interested in a proof of either. So we left
that subject. Klainerman asked if Gunning wants to ask more about
Complex, so Gunning started one last question:

Complex, Part II

Gunning: If you have a punctured disc, and a function analytic on it,
what is its behaviour at the puncture?

Me: classify singularities, explain the method of classification
(multiply locally by z^alpha and take limit).

Gunning: intrinsic property of essential singularity?

Me: Weierstrass-Casaroti: image of neighborhood is dense in C.

Gunning: a proof?

Me: I started out well, but some how forgot how to do it half way.
Gunning and Klainerman prompted me a lot. And finally I managed.
Extremely embarassing.

Somehow Klainerman decided that my PDEs were too poor, and decided to
ask me some stuff about ODEs instead.

ODE

Klainerman: tell me about a system of first order operators.

Me: I was just about to write down the equation (actually stalling for
time because I can't recall Cauchy Kowalevski), when he interrupted:

Klainerman: I'll give you the equation. Say x' = Ax, where A is some
matrix.

Me: The general form of the solution: x = exp(At)x_0

Klainerman: what does the exponent of a matrix mean (write out the sum
in terms of taylor expansion), what happens if it is diagonal (the
exponential hits the diagonal parts directly), what about it general?

Me: I stumbled. After some prodding I wrote down the Jordan form and
he asked if that would help me calculate the exponential, and I
couldn't quite figure it out, except that the blocks each exponentiate
by themseles.

Klainerman: (gave up on that line) what can you diagnolize?

Me: I blanked out for a sec, and after a hint, I said "Hermitian"
matrices. Gave the definition. He askedme to prove that it is
diagnolizable. I completely forgot how to do it. So Klainerman told me
to think about the a vector orthogonal to an eigenvector, and it
clicked, so I finished the proof.

Then they asked me to step outside. And after 10, 15 minutes, they
congratulated me while Klainerman pulled me aside and reprimanded me
for my extremely poor performance, but also consoled me not to be too
hung-up on this exam since as long as I learned my lesson from it, it
shouldn't affect too much my future prospect as a mathematician (he
cited Terrey Tao's exam as an example.)

Overall, I did really poorly. I was kicking myself the whole time
while waiting for them to discuss, and was, to a certain extent,
surprised when they passed me.