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 p<q?  (Finite measure) What 
about the other way around?  
What are some inequalities with Lp spaces?  (C-S, Holder).
Can a function be continuous only at the rationals?  Can you name
some theorems that are similar to this?  What are some ways of 
saying that a set is "small"? (cardinality, measure, Baire
category).  

Algebra (Flach at first, but Sarnak mostly at the end)
What is a bilinear form on a vector space?  When are two forms
equivalent?  What is an ortho. matrix?  What's special about
them (fix some inner product)?
Field Theory:  If you have a finite field, what are possible 
orders?  What can you say about its multiplicative group?  Say
you have a field extension F:K with only finitely many 
intermediate fields--show F is a simple extension.  If you have
a cubic polynomial, how do you go about finding its Galois 
group?
Can you name some simple groups?  (I said Z/p, Sarnak, who had
been really anal up to that point, admitted that he had asked
for that)  I mentioned An and the matrix group w/ 168 elements,
which let him segue into matrix groups, and in particular: 
Representation theory (all Sarnak):  What does it mean for a
representation to be irreducible?  What kind of matrices can
be in a representation (now that I think about it, it's kind of
obvious that you need A^n=I, which is pretty restrictive, but I
was really blanking at this point)?  What is Schur's lemma?

It was 11:10, we took a ten-minute break.  After the break things
got much nicer.

Functional Analysis (mostly Nelson, occasionally Sarnak):
If A is an operator on an (infinite-dimensional) Hilbert space
which is both unitary and self-adjoint, what does it look like?
(definition of those terms, spectrum, spectral theorem to get
that H=H_1 + H_-1 where A=I on H1 and A=-I on H_-1)  Can you 
state the spectral theorem? (Which one?  Turns out Sarnak wanted
the multiplication operator form for bounded operators, I stated
three different forms before I got the right one).  I made a 
comment that Nelson's name was cited about five times in that
chapter of Reed&Simon, and Sarnak said "That's why I'm the one
asking this."  What kinds of measures can crop up with this?  Can
you give an example of a singular measure which is purely 
continuous?  (Cantor measure).
State the closed graph theorem.  Consider H=L2(\Omega), a region
in R^n, let K be a closed subspace of H which has all continuous
functions.  Is f->f(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.