Rolfe Schmidt
Here is a brief account of my exam.
Committee: Stein (chair), McNeal, Shimura
At the beginning, they asked me to leave the room, so
that they could make their "plan of attack," and they
came up with a pretty good one. Everyone was very friendly
and the whole thing was very relaxed, even when I was
displaying my complete ignorance of the most basic ideas.
I'll write things down in the order they happened.
ALGEBRA (Shimura)
This is very sketchy, I didn't know what was going on,
and I didn't even understand what the questions were
at some points.
State the Chinese Remainder Theorem, in any form you
like.
I didn't know, so I was told to write down all of the
ingredients for the theorem, and then they tried to
get me to figure out what hypotheses I needed for
the theorem to work. I did terribly, but we didn't really
move on. From there we started to talk about polynomial
rings in one and several variables.
If a|f and b|f does ab|f? what assumptions do you need?
what is special about these rings that allows you to
do this? Can you give an example of a UFD that is not a
PID? I gave polynomials in several variables, of course.
What else is special about this ring? (They wanted
Noetherian.)
What are the symmetric polynomials? If you were teaching
a class what theorem would you put on the board right now?
Then Stein suggested to move on to Real (whew!)
Algebra went so poorly that I was sure I had failed, so
I was pretty relaxed, thinking about all of the time I
would have before I went through this again. I was asked
nothing about the standard topics that I had prepared for.
REAL ANALYSIS
The analysis questions were asked by Stein and McNeal.
I am not including any of the standard problems I was asked
that are already recorded in other peoples exams. I am only writing
what I think will really help people.
What is a measurable function?
What is a measurable set?
Given a function on R^n, how can you test to see if it is
integrable? (They wanted me to cut off the function appropritely
and say that the integrals of these cut off functions were
bounded.)
Why are there more Lebesgue measurable sets than Borel?
What is L^1?
How do you get a hold of the limit function when showing completeness?
How do you construct a non measurable set?
OK, enough of the pathologies.
HARMONIC ANALYSIS
What is your favorite definition of the Fourier Transform?
State some theorems about it.
Why do you care about the Fourier Transform? (I said PDE's)
State a specific result.
(I talked about constant coefficient elliptic PDE and regularity.)
How is this a statement about the Fourier Transform?
(I said I was using a multiplier theorem.)
State the multiplier theorem you're using.
Could you explain why this works in L^2?
What is the simplest example of an interesting multiplier
you know? (Hilbert Transform.)
Talk about the Hilbert Transform.
What is the Complex Analytic interpretation of this.
What can you say about the L^p boundedness of the Hilbert Transform?
Can you give an elementary proof of this?
How do you use the Fourier Transform to solve PDE?
COMPLEX ANALYSIS
These were all standard questions that other people have already written
down. There was nothing tricky.
SEVERAL COMPLEX VARIABLES
What is a domain of holomorphy?
Can you characterize these?
What is pseudoconvexity? Say the boundary is smooth.
You said pseudoconvexity was equivalent to being a domain of
holomorphy, is there an easy direction to this proof?
(I said I needed to introduce more machinery, and I talked about
psh exhaustion functions and convexity with respect to a family
of functions, then showed that domains of holomorphy were pseudo-
convex.)
What is the holomorphic hull of B(0,1)\B(0,1/2)?
How do you prove Hartogs extension theorem?
Why doesn't this work in 1 dimension?
Do you know about the Cousin problems?
When can you solve Cousin I?
(I gave a sketch of the proof.)
What about Cousin II?
Do you know Oka's principle?
What do you know about the dbar Neumann problem?
That about does it, it wasn't too hard.
I hope that this is some help to you.
Good luck,
Rolfe