Jeff Vanderkam's generals, 5/18/95
Committee: Nelson (chair), Flach, Sarnak
Subjects: Functional Analysis, Analytic Number Theory.
Complex Analysis (mostly Sarnak)
State and prove the Riemann Mapping Theorem (I used the standard
normal family proof in Ahlfors). Do you know another proof? (No)
That got us started on solving the Dirichlet problem in regions
in C, I brought up Poisson's integral formula for solving it in
the disk. He got me talking about approximations of the
delta function and how one might solve Dirichlet using Hilbert-
space methods (using H1(region)+L2(boundary) as the space), but
then he realized he was getting pretty far afield since it was
supposed to be complex.
Real Analysis (mostly Nelson)
Suppose you have u' = f(u) in some region of R^n. What kind of
conditions on f are useful in knowing a solution exists? It
took me way too long and a lot of hints to realize that we
were dealing with u->\intf(u) as a contraction mapping that
would thus have a fixed point.
Some stuff about Lp spaces: what are conditions on a space that
force Lp to be contained in Lq when pf(x_o) a bounded function on K? Despite the
order these questions were asked, it took me a minute to
realize that one used the c.g.t. here.
Analytic Number Theory (Sarnak, of course):
Talk about the proof that all odds are the sum of three primes.
He didn't want much detail, but apparently someone claimed to
have solved Goldbach last week, and he wanted me to understand
why he was skeptical.
What is Dirichlet's theorem for primes in a progression? How do
you go about proving it? How do you show L(1,X) is non-zero?
What is the functional equation for the L's? What's that tau
function that you just wrote up? A little discussion of how
it's actually the discrete version of the gamma function, which
I didn't know (neither did Nelson, it seemed, from his reaction).
What's the big step in proving the prime number theorem? How do
you find the zero-free region?
And that was it, it took about 2 hours. It was a bit frustrating
at times because I knew I knew the stuff, I was just having
trouble with recall (in particular that contraction-mapping thing).
Sometimes it was a bit tough for me to figure what they were
getting at, but they weren't above dropping some (occasionally
huge) hints.