I made a complete fool of myself. If I were the examiner, I might not have passed me. It was disappointing and horrible. But I passed.

Here's a transcript of what happened:

Examiners: Sergiu Klainerman (chair), János Kollár, Robert Gunning.

Topics: PDE, Harmonic Analysis

Total exam 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 L^1, 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 L^1.

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 L^1->weak L^1, 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 C^0 is dense in L^1, 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 W^2,p.

Klainerman: define W^2,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 = x^3 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... but how can I compare to Terry...)

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.