Jonathan Luk's Generals
5/21/2008 1:03pm
Examiners: Rodnianski (R, chair), Alexakis (A), Grushevsky (G)
Duration: 185 minutes
Topics: Partial Differential Equations, Differential Geometry
Remark: I am sure that I have confused the order of questions. Nevertheless, I have
tried to follow the order as closely as possible and to include some of my thoughts
so as to let the reader know that I have been given many hints in the form of
intermediate questions. On the other hand, I would like to save space by not
including standard answers.
Algebra
A: Now everyone is here.
R: The victim is here. Let's start with the basic subjects. Do you have a preference?
Me: Not really. Let's do alphabetical order and start with algebra.
G: Why is everyone looking at me? So what did you study?
G: Consider Q[x,y]. Is it a PID? Prove it.
G: First look at C[x]. If two ideals have the same zero set, what can you say?
Me: Generating polynomial has the same linear factors.
G: Now C[x,y]. Can you say ideals are generated by at most 25 elements?
Me: It's certainly finitely generated by Hilbert Basis. Then I cooked up .
G: Or you can use .
Me: Oh, then using homogeneous polynomials, clearly I can have many generators.
G: What do the zero sets (he defined this) for ideals in C[x,y] look like?
Me: Can clearly be points. Look at some examples and say it seems like it cannot be two
dimension.
G: Then try to prove if the zero set of a polynomial contains a ball, then it's the whole
plane.
G: Define Radical. Prove it is an ideal.
G: The ideal of all polynomials vanishing on the zero set of I is rad(I). Prove one
inclusion and I'm happy with algebra.
R: That's it? Let's do some linear algebra. Suppose you have a symmetric matrix.
Me: I can diagonalize with orthogonal matrices. It has real eigenvalues. Do you want me
to prove it?
R: Sure.
Complex Analysis
A: State Louiville.
R: I've a follow up question on that.
A: I also have a follow up question on that! Suppose you have an entire function whose
growth is bounded by log.
Me: Showed using the same argument as Louiville that it must be constant.
R: What about harmonic function?
Me: Solve for harmonic conjugate and take exp.
R: Without using complex?
Me: We can use the gradient estimate of log.
R: Do something simpler!
Me: Oh, just use mean value equality.
G: Do you know about the Dirchlet problem on domains in C?
Me: Explained Perron's method.
A: What is the sup bounded?
Me: Maximum principle.
G: If you have the ball and I give you zero boundary values.
Me: Must be zero by maximum principle.
G: What about upper half plane?
Me: Can be y.
G: Are 0 and constant.y all the examples?
Me: Tried to prove this, unsucessfully. (It's actually not true)
A: Wait. Let's back up. If you map this y to the ball. Why won't you contradict maximum
principle there?
Me: It's not continuous up to boundary at one.
R&A&G: Use the map from half plane to disk to solve the previous problem.
Me: (With much hint mainly from A) It corresponds to holomorphic function zero on the
circle, sigular at one.
A: Can a harmonic function with one sigularity be wild at that point?
R: (As a hint) Think of a cheap way to construct harmonic function.
Me: Oh, I certainly can take Re(exp(1/z))
G: This is a good point to ask about classifying singularities.
R: Can you give a constructive way (instead of Perron's) to solve Dirichlet problem?
Me: I don't know unless it's the ball. I know energy method but it's even less
constructive.
R: Well, the problem was particularly about domains in C.
Me: (Still didn't understand the point) I still only know the ball and the half plane
for domains in C!
A&G: So what do you know about domains in C and the ball?
Me: Oh, use Riemann mapping. So if the boundary is nice, you can map to the ball and
solve it there.
G: Ok, state Riemann Mapping Theorem.
A: You mentioned nice boundary. What is sufficient?
Me: (Thinking C1 should be good, but decided to be safe) C2
R: Polygons?
G: Piecewise C2!
Real Analysis
R: Let's do real. Define l^p. Inclusion? What about L^p. Inclusion?
G: If you have a bounded operator on l^1. Does it have an eigenvector?
Me: Shift operator.
G: What about L^p?
G: What adjective can you add to the front of operator so that it has an eigenvector?
Me: Made random guesses...
R: NO! Let's say you are going to cheat on this question, what would you do?
Me: No idea.
R: Do finite dimensional operators have eigenvectors?
Me: Certainly. So I believe compact would work.
G: Define compact operator.
Me: Defined. And said it has eigenvectors by spectral theroem and realized that it's
actually the first step towards spectral theorem. Then I tried more, unsuccessfully.
R: Let's not prove it. Just tell me why you would believe so.
Me: An operator is compact iff it can be approximated by finite dimensional operators.
A: Just for culture. If you have smooth compactly supported functions? What do you
know about its Fourier transform?
Me: Schwartz maps to Schwartz.
A: Define Schwartz. How do you prove that?
R: What about L1? Why?
Differential Geometry
R: Let's do special topics. What are your special topics?
A: Wait, do you need a break first?
Me: Just give me half a minute for a cup of coffee (which I brought with me).
A: Define second fundamental form. Why does it depend only on the vectors at the
points?
Me: Spend way too much time on this simple question.
R: (As a hint) What is the second fundamental form?
Me: (Didn't understand the question until A said "say it in 3 words") Symmetric 2
tensor. Then proved that.
A: If I have a graph in R3, Can you tell me what's the second fundamental form at a
point where the gradient is zero?
A: Then wrote down what it is when gradient is not zero.
R: You should know what that is without having him write that down!
Me: (Being guided to say) The laplacian of the induced metric.
A: What's the mean curvature? What's a minimal surface?
A: Wrote down the minimal surface equation. Prove that if u is C3, then u is C
infinity. He then decided to take this back before I attempted.
A: Can you show that all compact minimal surfaces in R3 are points? (*)
Me: Showed the coordinate functions are harmonic with respect Laplacian of the
surface.
A: Wait, but do you have maximum principle for this Laplacian?
Me: Wrote out the Laplacian in local coordinates and showed it's elliptic. Then
proved the weak max principle and said I can prove the strong one via Hopf lemma.
A: Isn't the weak one sufficient here?
Me: Maybe, but I can only do it with the strong one.
A: I had another proof of (*) in mind. I give you a hint towards that. Say if you
have a surface enclosing another, touching at only one point, what can you say about
their mean curvatures.
Me: The one in the inside is larger. Then used this to prove (*).
R: Suppose you have a manifold with negative sectional curvature. What can you say?
Me: Covered by Rn.
R: Define negative curvature and prove what you said.
R: Can you give a class of manifolds on which all harmonic 1 forms vanish.
Me: Nonnegative Ricci with strictly positive Ricci at a point.
R: Prove.
A: You missed a word in your statement.
Me: Closed manifold.
R: State volume comparison, supposing you have sectional curvature bounds.
A: Is sectional curvature nessecary?
Me: Ricci is enough.
R: State Uniformization.
A: You missed one word again.
Me: Closed!
A: You said conformal, what do you mean? Do you know how the sectional curvature
transform under conformal change of coordinates?
Me: No.
A: How would you compute it?
R: (After I explained.) In theory, yes. In practice, you look it up in a book.
R: What is sectional curvature in two dimensions?
Me: It determines the full curvature tensor. (Didn't quite understand the question
because I defined sectional curvature earlier!) So I mentioned it's the Gauss
curvature.
Partial Differential Equations
R: Who wants to ask a question? (Everyone looked at him.)
R: Consider Laplacian u = e_ij D_i D_j u, where e is small and smooth. Suppose on
the boundary of a bounded domain it has boundary value in Lp. Prove u is in W2,p.
Me: First tried to extend boundary function and change to a problem with zero
boundary condition but an Lp term on the RHS.
R: (Before I completed) Let's say you can do it.
Me: Freeze the coordinate and then use singular integral theory to estimate the
second derivatives of the Newtonian potential. Worked out in details.
R: But when the domain is not R, the function is not equal to that potential you
wrote down!
Me: Oops... but the difference is harmonic and I know how to estimate that.
R: What are the names that are attached to the singular integral theorem that you
used?
Me: Calderon-Zygmund.
R: Now consider the heat equation on the torus. What can you say about the the
long term behavior of the solutions?
Me: You want me to prove the maximum principle?
R: No!
Me: Oh, it tends to harmonic functions.
R: What are the harmonic functions?
Me: Constants!
R: I want you to tell me something about convergence rate.
Me: Consider the Fourier seriers and see that it converges as exp(-t).
A: What about on a closed manifold that's not a torus?
Me: I can't use Fourier series...
R: What's special about Fourier series?
Me: Oh!! I can consider the eigenvectors of the Laplacian! So the convergence
rate is exp(-y t), where y is the smallest eigenvalue!
R: (to A) Ask a question.
A: You ask one more and I'll ask.
R: Consider wave equation in R3. u(0) is smooth. Is that enough initial condition?
Ok, let's say Dt u (0) is also smooth. Prove global boundedness.
Me: Showed the energy identity.
R: (Interrupted) Say how you can prove that first.
Me: Proved. Then use that and Sobelev embedding. Said I need s>3/2.
R: 3/2 is not enough. You can't get the same bound for all time.
Me: True. Tried to write down Klainerman-Sobolev estimate.
R: (Interrupted before I even wrote down the LHS) Let's back up. State Sobolev
embedding.
R: Now say you have the L2 norm of the first and second derivatives (but not the
function itself), can you bound the sup norm?
Me: I can bound L3.
R: Should be L6, you subtracted wrongly. That's ok, you can then interpolate.
R: (to A) Now your turn.
A: I think we should let him go.
They told me to wait outside. After a few minutes, Alexakis was the first to walk
out, first looked very seriously, then smiled and congratuated me.
Reflections
The atmosphere was very nice. They certainly gave more hints than what I can
include here! Except in algebra when I was still quite nervous, I actually felt
comfortable working on problems that I haven't seen. I tried to ease my nervousness
by offering proofs of theorems I knew well, but they often didn't want to see them.
My advice is to bring a bottle of water, bring coffee, always think out loud and
always ask for clarification!