Terry Tao's generals
My examiners were Stein, Kleinermann and Rudnick. (Kleinermann came in
about five minutes late).
The first questions asked were about Harmonic analysis. They asked me what
I knew, and I said basically singular integrals and functional analysis -
no real harmonic theory at all. So they examined me on singular integrals
instead, essentially.
Give examples of singular integrals (Calderon-Zygmund). What theorems do
you know about them, how to prove boundnedness. (I stated a T(b) theorem,
which nobody seemed very acquainted with, and said with some brief
explanation that knowing boundedness on L^2 - L^2 was sufficient to prove
L^p-L^p for 1=5). What
other simple groups do you know. (The monster? But I backed down quickly,
saying I knew not how to define it.) Then the discussion meandered a bit
as the professors talked among themselves whether the group of order 168
(which I mentioned) was sporadic.
After this, they decided to pass me, though they said that my harmonic
analysis was far from satisfactory. :( They didn't ask any real or complex
analysis, but I guess from my handling of the special topics they decided
that wasn't necessary. Besides, we were almost getting snowed in.
The exam lasted 2 hours.