Michael Greenblatt's generals
Committee: Okikiolu, Washnitzer, Sinai (chair)
Topics: Harmonic Analysis, Ergodic Theory
Examination time: 2 hours
Algebra:
Given a skew symmetric matrix S, show that U = (S + I)(S - I)^-1 is unitary.
Then find an expression for S in terms of U.
What is a Noetherian ring? If I is an ideal in a Noetherian ring with a unit,
what is the intersection of I^n over positive integers n. (This is a non-
trivial question, and they were satisfied for me to show it's 0 for various
examples e.g. polynomial rings)
Do you know what a group representation is?
Do you know what the trace of a group representation is?
I was then asked some questions about representations of the unitary group,
but when I said I hadn't a clue,they said that this was expected, and moved on.
All algebra questions were asked by Washnitzer
Real Analysis:
Define absolute continuity. State the Radon Nikodym theorem.
Do you know what the Haar measure of a compact abelian group is? Any ideas
on how to prove it exists? Do you know the Pontryagin duality theorem?
(This tied in with ergodic theory, and I suspect that's why I was asked this.)
Given a smooth f:[0,1] -> R, describe how the integral from 0 to 1 of
exp t(f(x)) behaves as t goes to infinity.
These questions were asked by Sinai.
Complex Analysis:
What does "simply connected" mean? State and prove the Riemann Mapping Theorem.
You used the existence of square roots of holomorphic functions with no zeros
on a simply connected domain. Why can you do this?
I was then given several quarter-circles with various holes chopped out, and
values assigned to different pieces of the boundary. I was asked if there
existed harmonic funcions on these domains with these boundary values.
I was asked to integrate 1/(1 + x^4) from -infinity to infinity.
These questions were asked by Okikiolu and Sinai.
Ergodic Theory:
Do you know any ergodic theorems? (I gave the L^2 ergodic theorem)
Do you know what the Lebesgue spectrum of a measure-preserving transformation
is? (I didn't, so he defined it for me) Show that the limit function in the
above ergodic theorem has L^2 norm = the mass at 1 of the spectral measure
corresponding to f and T.(I was helped out even more than usual on this one.)
What do you know about irrational rotations of the circle?
Is the map on T^2 given by (x,y) -> (x + a, x + y) mixing?
What do you know about invariant measures of diffeomorphisms of S^1?
Do you know about Liouville's theorem on invariant measures on flows induced
by vector fields? (I stated the only thing I knew) Try to prove it.
In addition I was asked a few other "do you know" questions the answers to
which were no. I don't remember exactly what they were.
These questions were asked by Sinai, with occasional input from Washnitzer.
Harmonic Analysis:
Given f in L^1(R^n), what is the maximal function of f. How do you know it's
measurable. From which L^p to L^q is f -> M(f) bounded? Prove it.
What is the Hilbert transform H(f). I was asked several questions relating the
smoothness of H(f) to that of f. I was then shown a graph of a function f con-
sisting of two positive compactly supported chunks, and was asked to describe
how H(f) behaves at endpoints of the two intervals. Next, I was asked to
state a boundedness theorem for the Hilbert transform, and to sketch out a
proof.
What do you know about Riesz potentials?
How is the Fourier transform of |x|^ -a, 0< a < n defined? Do you know a
formula for it? How is it proved? (They didn't press me on this last point.)
They then asked me to solve simple PDE's i.e. heat equation, wave equation in
1 + 1 dimensions using Fourier analysis.
These questions were asked by Okikiolu, with occasional input from Washnitzer.
General comments: I was fortunate to have a nice committee. They would help
me through anything I got stuck on for more than 30-60 seconds. The atmosphere
was extremely relaxed, chiefly due to Washnitzer's tendency to tell stories
and interrupt other questioners with related remarks/stories. Even though I
was asked to do things that were over my head at times, they seemed to expect
me to get bogged down from time to time and need to be led past some obstacle.