Stephen Miller's Generals
May 17, 1994

Committee: Browder(chair), Aizenman, Solovej
Topics: Algebraic Topology, Functional Analysis

My test was at 10 am.  I had spent most of my preparation thinking of
excuses for failure and getting used being awake and thinking at 10 am. 
So I of course went to bed at 11 pm the night before and didn't fall
asleep for 3 hours.  Then I woke up at 7 am.  So I was essentially asleep
during the test.  The whole thing seemed like a dream.  My committee was
very nice and made me feel very relaxed during the test, except during the
functional analysis section, in which I actually was asleep.

Real Analysis (Aizenman, Solovej)

- Does every measurable subset of the real line have density points?

- Prove the Riemann-Lebesgue Lemma.

- If a sequence f_n --> f, must their integrals converge?  Give conditions
  and counterexamples (I mentioned 1/n on R, but they weren't amused).

- For which (real) values of s is the integral (on R) of sin(x)/x^s finite?

- Can you find an open dense subset of [0,1] with measure 1/2?  
  (I suggested taking the complement of certain Cantor sets, but they
   wanted the "Fat Rationals.")

 - Are the rationals a G_delta?

Complex Analysis (Aizenman, Solovej)

- What is the Cauchy Integral Formula?

- Given the values of a meromorphic function on a curve, can you tell me
  the multiplicities of the zeroes and poles inside?

- Under what conditions is the point-wise limit of analytic functions
  analytic?

Algebra (Browder)

- How many fields are there of order p^k?  Why?

Functional Analysis (Aizenman, Solovej)

We came to this section twice because I screwed up royally the first time.
It was all essentially one big question.  Like I said, I was asleep so I
can't recreate that effect here.  Or maybe I already have.

Algebraic Topology (Browder)

- Tell me about maps from S^6 into other spheres and manifolds.

- State Poincare and Alexander duality.  

- What does the degree of a map from S^6 to a manifold tell you?

- Prove that RP^2 cannot be embedded in R^3.

- What is the Jordan Separation condition?

Each of the last three questions led to much discussion.


Comments - At the end of the algebraic topology section, Aizenman mentioned
to Browder that he was not satisfied that he had seen my full knowledge of
functional analysis, so we came back to it.  He and Solovej asked a few
questions about compact operators, to which I gave vague answers.  They
let me off of the hook.  We probably spent a half-hour on the first three
parts and an hour each on the second (Browder and Solovej commented that
it was getting boring).  Browder guided me through the problems mentioned.  I
never really answered any of the problems in functional analysis, so I'm
not quite sure why they let me pass.  In summary, this was an extremely
nice committee.  I felt much more comfortable once the test started than I
had in months.  Of course, if I was awake it would've been a lot nicer.