Arie Israel's Generals, Fall 2007
Committee : Fefferman (chair), Yang, Oblomkov
Topics: Elliptic PDE, Differential Geometry
My exam was scheduled for 2 pm. It had to be held in Dr Yang's office, as Dr Fefferman's
office was being used (he was on leave at the time). When I came in, only Dr Yang was
there. We began to wonder if Dr Fefferman had made it to Princeton alright. Just when we
began to worry, they both walked in. Dr Fefferman asked which order I would like to take
my topics in. I said Complex, Algebra, Real, PDE, and Geometry. And then we began.
F: Paul, would you like to start with complex?
Y: Sure
Y: Do you know what the Cauchy-Riemann Equations are? And how they relate to Holomorphic
functions?
Me: At this point I was so nervous, I momentarily forgot what Cauchy Riemann equations
even were. They sounded familiar, and I know I had heard the phrase before. It took me 5
seconds to recover from this brain freeze, during which I bought myself time by repeating
"Cauchy-Riemann equations?" again and again. Once I recovered, I wrote them on the board.
Y: What is the geometric meaning of these equations?
Me: I mumbled something about f'(z) != 0 implying the mapping preserves angles (i.e. is
conformal)
Y: Assuming suitable regularity of u and v (f = u + iv), what can you say about their
analyticity?
Maybe it was the nervousness, but I did not understand what was being asked of me, and
proceeded to go off on a tangent. I started showing them why f'(z) != 0 implied f was angle
preserving at z.
...When I had finished
F: So, I think you just provided an explanation for the question that was asked two
questions before.
Me: Oh, of course...I'm sorry, what was the question again?
Y: Assuming suitable regularity of u and v (f = u + iv), what can you say about their
analyticity?
Me: Well we could show that C-R implies analyticity and then get a power series expansion
from the Cauchy Integral formula.
Y: Isn't there an easier way? What type of functions are u,v?
Me: Harmonic. So we can say that u and v are analytic
Y: How would you prove that?
By this time the discussion had morphed into PDE.
Y: What can you say about maximum principles?
Me: We have a weak Max principle for harmonic functions
Y: Can you prove it?
Me: Yes, and offered a standard proof
Y: What about a harmonic function defined on all of R^n? have you heard of Liouville's
theorem?
Me: Yes. I explained in words how one would prove such a statement.
They seemed to be happy with this verbal proof.
Y: Ok, how about a standard question? State, and tell me how you would prove the Riemann
mapping theorem
Me: I stated it, and gave them the main ingredients in the proof, and where they were
used. Montel's Theorem, Hurwitz's Theorem, Existence of branch of log, and Schwartz Lemma
all came up. Then I pieced together a proof. When I came to the final step of showing the
map was onto, Fefferman stopped me
F: I think that's enough for complex analysis.
F: What was your next topic?
Me: Algebra
F: Oh, yes right
And he motioned for Dr Oblomkov to start.
O: Do you know what a normal subgroup is?
O: Prove a subgroup of index two is normal
O: Given a finite field, how many elements does it have?
I always forget how to prove this one, but Dr. Oblomkov was very nice and game me lots of hints
O: Would you like anymore Algebra questions?
Me: Sure, I don't mind
O: How about some linear algebra? Suppose we are given two matrices that commute, what can we
say about them?
Me: I said one fixes the eigenspaces of the other. So, if they are diagonalizable, they can be
diagonalized in the same basis
O: What if a Matrix commutes with a Jordan block?
Me: I fumbled with the definition of Jordan block, and with help, finally wrote it down
O: Lets assume that we have zero's on the diagonal as well, what can we say if a matrix commutes?
Me: I wrote down what conditions the matrix would need to satisfy
O: So we can see that the matrix must be of the form P(J), where P is a polynomial, and J is the
Jordan block.
O: Ok, I'm happy with Algebra
F: Ok, my turn! Let's do some Real
Strangely enough this turned into a complex question also..
F: Let f:[0,1] --> R. For Im(z) > 0, define G(z) = Int_0^1( f(x)/(z-x) dx). What can you say
about G?
Me: Holomorphic?
F: (laughing) I wasn't expecting this to turn into a complex analysis question, but oh well
F: So why is it holomorphic?
Me: Dominated Convegence Thm! Then I wrote out the arguments, and somewhere along the way I stopped
writing lim_(h-->0)
F: You meant to write limits there, correct?
Me: Oh yea, sorry.
F: What is the dominating function?
F: What is the largest domain to which this function can naturally be extended to?
Me: C - [0,1]
F: Can the function G be extended to all of C
Me: Hmmm, I'm not sure
F: What is the behavior of G at infinity?
Me: It decays to zero.
F: So what can we say?
Me: Oh I get it. If such an extension existed, then the function would be bounded and entire. Which
would imply G was constant by Liouville's Thm. So no such extension can exist
F: Ok, I'm happy with Real, let's move on to PDE
Then the committee tried to figure out who should do PDE. It was decided that since Dr Yang would
cover Geometry, Dr Fefferman should cover PDE.
F: State and prove your favorite regularity theorem for Elliptic PDE?
Me: I said Nash-de Giorgi Iteration, which provided Local boundedness. Following which we can prove
C^alpha estimates
Me: I showed them how to set up the iteration, but this took a while. It seemed like 20 minutes, but
I really have no idea how long anything took. Somehow I miraculously setup the iteration correctly.
Me: Would you like me to show you how the iteration works?
F: No, I think that'll be fine
F: Paul, would you like to start on Geometry?
Y: State and prove the Gauss-Bonnet Theorem.
Me: I stated the version for triangles, and with some help from the committee finally figured out
whether I should add or subtract the exterior angles.
O: Sorry to interject, but what does geodesic curvature mean?
After explaining what it meant, Dr Yang had decided that I didn't need to prove the GBT.
Y: Here's a nice application of GBT theorem. Take the upper half plane, and any solution to Laplace's
equation with nuemman boundary conditions du/dn = 0 except for a "thin" subset of the boundary.
Me: What does thin mean?
Y: Let's say of Hausdorff dimension less than 1/4
Me: What's Hausdorff dimension again?
I think at this point we just assumed our "thin" set consisted of isolated points. But I really can't
remember.
Y: Write a metric for the upper half plane e^2u (dx^2 + dy^2). What can you say?
I drew a blank
Y: Well actually we can show that this manifold is flat.
Me: I started thinking about what would be the easiest way to compute the curvature. But, he stopped me
Y: Never mind that, lets assume we can do that. So the manifold is flat, which means we can isometrically
immerse it into R^2. Now let's suppose two things. Firstly, that this manifold is complete. Secondly,
that the geodesic curvature is an L1 function on the boundary of this manifold. What can you say?
Me: I mentioned that the immersed manifold in R^2 must blow up infinity at the isolated points, but then
couldn't say anything else
Y: Here, let me show you!
He came up to the board and showed me that there must only be a finite number of isolated points.
Because, GBT shows us that the integral of the geodesic curvature must be 2pi between each two isolated
points. And since the total integral is finite, so must the number of isolated points.
This commenced my generals.
They asked me to leave the room for a few minutes, and then came out and congratulated me. My committee
was very nice, and always happy to give me a hint. It was scary when I got stuck, but they helped me
through it.