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.