Shaoyun’s generals
5/11/2018, 1:30-3:00pm
Examiners
[JP] John Pardon (chair)
[CF] Charles Fefferman
[GC] Gabriele di Cerbo
Special topics: Partial Differential Equations and Symplectic Topology
PDEs
[CF] Define an elliptic PDE.
I gave the definition of classical solution to a second order elliptic PDE.
[CF] Show that the smoothness of coefficients and right-hand side implies smoothness of the solution.
I used the L^2 theory of weak solutions to give the argument. I had to set up everything, such as the definition of weak solution, choice of test functions, etc. I was stopped before going into tedious computations.
[CF] Discuss about how you can extend this result near the boundary.
I chose a problem with homogeneous boundary value, and used the boundary charts and reflection techniques to reduce the discussion to the previous case.
[CF] How about the boundary value does not vanish?
I was guided to use the example of the Poisson problem on unit ball to illustrate the regularity relies on boundary regularity.
It was a surprise that these were all the PDE questions. Nothing about maximum principles, Schauder theory, de Giorgi-Nash-Moser theory, L^p theory or nonlinear theory was asked.
Symplectic Topology
[JP] Define a Hamiltonian vector field and show that the symplectic form is preserved under the flow.
I gave the definition and the standard Lie derivative argument.
[JP] Talk about Gromov compactness.
As this was a pretty vague question, I talked about the convergence modulo bubbling, i.e. the standard elliptic bootstrapping argument.
[JP] What can you say if a priori we have C^0 convergence?
I used the blow-up argument to show that bubbling cannot happen in this case, thus we actually have C^{\infty} convergence.
[JP] Talk about the general transversality result for J-holomorphic curves.
Again I had to set everything up, i.e. the Sobolev spaces, universal moduli space of J-holomorphic curves, linearized operator and Sard-Smale etc. Again I was stopped before going into serious computations.
[JP] Does the choice of Sobolev spaces affect the argument?
As the kernels and cokernels are the same, elliptic regularity shows that it does not matter.
[JP] Quick question: what can you say about Lagrangian submanifolds in the cotangent bundle if they are projected to the 0-section?
They are locally represented by closed 1-forms.
[JP] Consider CP^2 with the standard symplectic structure, and let J be a generic tamed almost complex structure. Given two points in CP^2, is there always a J-holomorphic curve passing through these two points?
I used the invariance of Gromov-Witten invariants to illustrate there is always one such curve.
[JP] Define a stable map and illustrate the conditions.
- I gave the definition using trees and mentioned that one wish to trivialize the automorphism group of constant maps.
The questions were fairly standard. Pardon did not insist on details once I gave the ideas of the proof.
Algebra
[GC] When can you find the square root of a 2*2 matrix?
When it’s invertible. I used Jordan canonical form to give the argument.
[GC] How many square roots does a matrix have?
I stuck for a while on the Jordan canonical forms. After talking about some nonsense and receiving some guidance, I concluded that there are infinitely many even for the identity matrix.
[GC] If you have an invertible 2*2 matrix A with integer coefficients and A^n = I, show that A^12 = I.
Showed that the characteristic polynomial must divide x^12 – 1 and applied the Cayley-Hamilton theorem.
[GC] What are the finitely generated Z-submodule of Q?
- I was trying to use the structure theorem of PIDs to give an argument, then I was guided to used an elementary discussion using Bezout type of argument.
[GC] Talk about finite groups of order pq.
Standard Sylow theorem arguments.
[JP] Consider the field extention F_p (x^{1/p}, y^{1/p}) ovrt F_p (x, y). What are the intermediate fields?
It was hilarious that it took me a while to realize the point of this problem --- the character is p. Then I was asked to show the phenomenon here couldn’t happen in the case of chacteristic zero.
[GC] Talk about your favorite proof of fundamental theorem of algebra.
- I used Liouville theorem. After I finished the proof, Fefferman asked if I knew who gave the first proof then he told me the answer is Gauss.
The algebra questions were not that standard but elementary. The committees were really patient and nice.
Complex Analysis
[JP, GC] Talk about classification of singularities, behavior near essential singularities, and a sufficient condition for removable singularity.
I tried to use the removable singularity theorem for harmonic functions to give a condition, but then realized they just wanted o(1/z).
[JP] What can you say about an entire function with bounded real part?
[JP] Classify simply connected regions in S^2.
[JP] What can you say about a holomorphic map from the sphere to a torus?
I used Riemann-Hurwitz and was asked to prove this. Then di Cerbo asked me what the universal cover of a torus is and I realized that I could lift this map to the universal cover and apply maximum principle.
Real Analysis
[CF] Define the Fourier transform. Talk about its properties (behavior of the image of L^1 functions). How to extend the definition beyond L^1 (Plancherel theorem and the decomposition of functions in L^r, 1 < r < 2)? Extend the Fourier inverse transform to L^2 functions and show that you can say nothing more in the L^2 case.
The analysis part was pretty easy and I could feel that they kind of wanted to end this exam. I was sent out for 5 seconds (really a short time) and they congratulated me for passing the exam.
The committee members were really nice and helpful and they didn’t ask me any hard questions. Perhaps Fefferman considered me as a topologist so he didn’t ask hardcore PDE problems. But Pardon asked many analytical aspects of J-holomorphic curves so it was kind of interesting.