Pin YU's Generals
2006-May-03,
Committee: Klainerman (Chair), Chang, Conway
Topic: Differential Geometry and PDEs (1h50m)
ALGEBRA:
Conway: Describe Jordan canonical form and the uniqueness.
I spent lots of time on this simple question with mentioning the classification of
finitely generated modules over a p.i.d.
Conway: how to diagonalize a hermitian matrix.
Again, I found this easy question became too difficult. They gave me some hints.
I didn't get them. Then Chang asked me to state the following theorem:
Chang: Spectral Theorem for a compact operator.
I gave the statement.
Chang: Spectral Theorem for a general operator.
I state the version for bounded self-adjoint operator.
Klainerman: If the operator is not self-adjoint, what happens?
I didn't know. After general, I realized that some thing similar was true for normal
operator.
Chang & Klainerman: How to compute the first eigenvalue for a compact operator.
I wrote down the variational approach. They were happy with this answer and asked me
to use this to prove the diagonalisation of a hermitian matrix. I tried without any
success. But finally I proved it by myself.
Klainerman: How to use ODE to prove the uniqueness of Jordan Canonical form?
No Idea. He skipped this question. (Maybe we should look at the singularities of the
solution.)
Conway: State Sylow's theorem.
Conway: Talk about the group of order 15.
I spent two minutes to show there was only one.
Conway: What's a solvable group?
I gave the definition then we move on complex analysis.
COMPLEX ANALYSIS:
Klainerman: Talk about some theorems which hold in both complex analysis and harmonic
function theory.
Mean value property. Maximal principle. Liouville theorem.
Chang: Prove them.
Klainerman: Prove Estimate estimate the prove Liouville.
I started to prove it in a wrong way. Then he stopped me and gave a smart hint.
Chang: What's the gradient estimate on manifold?
I gave Yau's gradient estimate on a Riemannian manifold with positive Ricci curvature.
Conway: How to use Liouville theorem in complex analysis to prove Liouville for harmonic
function.
First find a harmonic conjugate v for the harmonic function u, consider exp(u+iv).
Chang: State and prove the theorem of Rouche.
I proved it by some tedious computation. They asked me the idea of the prove. I said
computation. Then they talked a little about winding number.
Conway: Classify all singularities.
DIFFERENTIAL GEOMETRY:
Klainerman: Your favorite theorem in gemeotry.
I spent 30 seconds then picked Guass-Bonnet because I thougt I was able to prove this one.
Klainerman: What's the version for a polygon?
I wrote down the formula for a geodesic polygon and Chang said I missed some term. Then
I added the additional term of geodesic curvature.
Klainerman: Generalizations?
I gave them the formula for hypersufaces.
Chang: Exact formula for dimension 4?
I said I did not quite remember the constants but I showed the Weyl tensor term and \singa_2
term.
Chang: Curvature for hyperbolic space.
I wrote down the Christoffel symbols and the definition of the curvature tensor. They asked
me some question on the definitions then moved on next question.
Chang: What's your favorite comparison theorem.
I spent again 30 seconds to find one with a easy proof. I choosed Bishop-Cheeger-Gromov's
volume comparison. They were not interested in the prove.
Chang: Do you know the relation between scalar curvature and Ricci curvature.
She wanted d R=2div Ric. I was not able to prove it at that time. But I said that second
Bianchi was the crucial point in the proof. They were happy with this answer. Klainerman then
gave some comments on the proof.
Chang: How to use it to get Einstein's equation?
I wrote down the field equation in general relativity. Chang said she wanted another one. Then
I said some basic points about Einstein manifold.
Klainerman: Talk about RHS of Einstein's equation in general relativity.
I said divergence free because it was an energy-momentum tensor.
Klaimernan: Derive the conservation law for a energy-momentum tensor.
He asked me to write down all the formula in local coordinate.
Klainerman: What's a Killing field and the relation between Killing field and energy-momentum
tensor.
I computed the deformation tensor.
Chang: What's the Killing field for spheres.
I gave the isometry group and said Killing was just the corresponding Lie algebra.
Klainerman then suggested to look at real analysis but Chang thought it was not necessary because
I got PDEs. Then, finally PDE began:
PDEs
Klainerman: Classify different types of non linear PDEs.
I didn't know the answer then said semilinear and quasilinear.
Klainerman: Examples? I wrote down some equations.
Klainerman: How to prove well-posedness theorem for a nonlinear PDE.
He suggested me to use the simplest example 3-dim Euclidean space:
\box u = u3 with initial data in H2 * H1
I preferred to use Hille-Yosida. But he didn't like this absbract approach and said suppose
people here didn't know Hille-Yosida. I guessed we had to use energy estimate and fixed point
argument because that was his favorite tool! He gave me some suggestions and asked me which kind
of Sobolev inequalities might be useful here. Finally, when I was thinking how to finish the
proof, he told me that everything was ok.
Good luck to everyone!!