Ricardo Saenz's generals

Date: May 14, 1998
Time: 1:00 pm
Place: 802 (Stein's office)
Topics: Harmonic Analysis and Partial Differential Equations
Committee: Stein (C), Fefferman and Browder


Algebra

B: Prove that the center of a group of order p^r (p=prime) is not trivial.
B: What is the Jordan canonical form?
B: In what kind of field every matrix can be written in this form?
B: What can we say in the case of a quadratic form? I said that the matrix
   can be diagonalized.
B: What else? -I didn't quite understand this question at first, but I
   finally got to the point that after a change of variables the diagonal
   elements can be put as 1, -1 or 0.

Complex Analysis

S: Give an example of a holomorphic function whose zeros are (only) the
   nonpositive integers. -I said that 1/gamma function, and gave the product
   expansion.
S: Why does this product converges?
S: Now an example of a function whose zeros are all the integers. -Sin (Pi x)
S: What is the relation between this function and the gamma function?
S: How can you prove it? -I said that one uses the functional equation of
   the gamma function, and that was enough. They didn't want to see the
   details.
F: Do you know an integral form of the gamma function?
S: Where does this integral converges?
S: How can you extend this function (starting from the integral form) to the
   complex plane?
S: State the Riemann mapping theorem? -They did't ask me for the proof, but
   Stein asked if a I knew the original proof by Riemann (this was based in
   PDE, and one of my topics was PDE!). I said that I haven't seen it before,
   and he explained to me what his ``proof'' (Riemann's proof wasn't precise)
   was.
B: (Kind of an extra question) How would you proof the Fundamental Theorem of
   Calculus?
Me: ???
F&S: Do you mean of algebra?
B: Yes! Of algebra! -I said using Liouville's theorem.
B: Do you know a proof using topology? -He was kidding, and didn't ask me for
the proof.

Harmonic Analysis

Stein and Fefferman asked me what I have studied.

F: If you know that Laplacian(u) = f, how can you calculate all the second
   derivatives of u? -I gave the expression using the Riesz transforms.
After this question, I said a few things about boundedness of singular
integrals, differentiability, and continuity. During this discussion, Stein
asked me what can we say of a function whose all L^p derivatives exists.
In this case the function is almost everywhere equal to a continuous function.
S: Is the Hilbert transform of a continuous function continuous? -No
F: Do you know any example? -I didn't remember an example of this. Fefferman
   then help to find one, after some trial and error (Fefferman started
   wondering if he himself would pass the exam).
S: You mentioned spherical harmonics (I did when I was asked what I have
   studied). What is the relation between spherical harmonics and Bessel
   functions?
S: Write an expression for the Bessel function (anyone you know). -I didn't
   remember any!! But Fefferman said he didn't remember either, and so I was
   safe. Stein gave the integral form.
S: What is the behavior of this function as x->infinity? -I said it was of the
   order of x^(-1/2)?
S: Do you know any way to prove this? -I said by using stationary phase
   methods.
S: What is stationary phase?

PDE

Again, they (F&S) asked me what I have studied.

F: What do you know about the wave equation? -I said that one can conclude
   that waves ``travel with finite velocity''.
F: Can you prove it? -I prove it with an energy argument. Stein said that he
   didn't know this argument, and Fefferman commented that I was ``clearly
   influenced by Klainerman'' (and I was, in the sense that he recommended me to
   read Evans book, where I had found that proof).
F: Why in the wave equation we need two initial conditions (u and D_t u at
   t=0), while at the Laplace equation only one (u at t=0)? -I wondered a
   little bit and said that one can calculate D_t u (t=0) from the Laplace
   equation.
F: HOW? -Of course, I didn't remember how (that's why I wondered a bit!). He
   gave some hints until the answer came (more Fefferman's answer than mine).
S: Do you know any ``nonsolvability'' theorem? -I gave the Lewy example. I said
   that if L = Lewy operator, and Lu(x,y,t) = f(t) near zero (with some
   suitable regularity conditions), then f must be necessarily real analytic
   near zero (as a function of t).
F: ??? -I gave an outline of how to prove it, and Stein told Fefferman that
   the argument I gave was Lewy's original argument.
S: What if f is analytic? -We apply Cauchy-Kowaleskaya.

That was it. Stein asked me to live the room so that they can discuss my exam,
and after a few seconds the door was opened and ... Stein told me that I have
passed.

Remarks:
1) The committee was very nice, and they help me to get the answers that I
   didn't know right away (or at all).
2) There wasn't any ``real analysis'' question, but, after all, harmonic
   analysis and PDE are real analysis.
3) The exam was about 1:30 long.
4) Believe it or not, the algebra questions that I wrote above were all the
   algebra questions.
5) The books I used for algebra, etc. were the standard texts. For harmonic
   analysis: Stein&Weiss' Fourier Anlysis, Stein's Singular integrals and
   Stein's Harmonic Analysis (only parts of it). For PDE: Folland and Evans.