Antoine's generals
04/20/2016, 2:00-4:00pm
Special topics: Riemannian Geometry and PDE
Examiners:
[M] Fernando CodÃ¡ Marques (chair)
[C] Peter Constantin
[K] Nicholas Katz
I was asked to choose the first topic and decided to begin with
Algebra [K]
1. Tell me about the structure theorem for finitely generated modules over a PID. How is it related to the Jordan canonical form?
2. Tell me about Galois theory. Guess if X^n-X-1 is irreducible or not, and what its Galois group is (answer: yes and S_n, but it is a result of Selmer and is difficult to prove).
3. Tell me about finite fields.
4. Take a polynomial with integer coefficients irreducible in Q. Do you think it has to be irreducible in F_p at least for some primes p? Consider the extension Q(\sqrt{2},\sqrt{3},...,\sqrt{11}) of Q and say something about its Galois group (action and order). What happens to the minimal polynomial of \sqrt{2}+\sqrt{3}+...+\sqrt{11} over Q in F_p?
This part went smoothly thanks to the reports of the previous years and some luck in my guesses. Katz didn't ask for complete proofs but mostly for oral explanations. The last question confused me and I could only follow his hints. He eventually explained that the minimal polynomial of \sqrt{2}+\sqrt{3}+...+\sqrt{11} over Q is reducible modulo p for any prime p.
PDE + Real/Complex Analysis [C]
1. Define convergence in measure.
Give an example of a sequence of functions converging a.e. but not in L^1.
Suppose f_n and f are L^1. When does the integral of f_n converges to the integral of f?
Consider an operator whose kernel has a singularity in x^(-a) at 0; give sufficient conditions to define it on L^1, L^2.
Talk about the Hilbert transform.
2. If a sequence of harmonic functions converges uniformly in any compact of a domain, what can you say about the limit?
If a continuous function satisfies the mean value equality, does it have to be smooth?
If u is harmonic, what properties are satisfied by |u|?
3. State the strong maximum principle. What is your favorite application of this result?
What happens if you have a harmonic function in R^n minus the unit ball, equal to 1 on the unit circle and going to 1 at infinity?
If u is harmonic, can you control its oscillation?
4. Consider u_t + u.u_x=0, where u is real valued. Solution, charcteristics, singularities?
5. What is the difference between of the behaviors of solutions to the wave equation in odd and even dimensions?
Say the initial data is smooth compactly supported. Give some control over some norms of u.
This part began quite unfavorably since I forgot the definition of convergence in measure and barely knew anything substantial about the Hilbert transform. Constantin moved on quickly and the few questions about elliptic PDEs helped me stay focused. I answered the last two questions only with general knowledge, but Constantin gave me hints whenever I needed it and did not insist when I did not know something.
"Riemannian Geometry" + Real/Complex analysis [M]
Give your favorite proof of Liouville theorem.
Does there exist a bounded superharmonic function in R^2? Prove it. (the answer is no)
Do you need "bounded"? (only bounded below is enough)
And in R^n, n bigger than 2? A smooth one? (yes)
What is the difference between the two cases?
Do you think there exists a positive superharmonic function in the hyperbolic plane? Bounded harmonic? Prove it. (the answer is yes)
Just when I started to wonder if finally I would have questions about Riemannian Geometry, Marques decided it was enough and ended the examination. This last part consisted of a long question about superharmonic functions. Even if I knew the trick to prove the non-existence of positive superharmonic functions in R^2, I was getting unreasonably hungry and it took me a while to set up the proof correctly, but Marques patiently helped me to keep moving forward.
As other people mentioned, one useful thing to do during the preparation is to work on explicit, basic examples and not merely on the theory and advanced exercises: it will spare you some embarrassing moments of hesitation at the blackboard. Except for the Algebra part, the gap between the questions and what I actually prepared was bewildering at times, so try to discuss with your committee before the examination and delimit clearly your preparation. Fortunately, the committee was very friendly and generous when helping me.
Good luck!