Special Topics: Algebraic number theory, Algebraic geometry
Committee: Chris Skinner (chair), Janos Kollar, Sophie Chen
Chen: State Fubini's theorem. [I screwed up a bit and stated one of the
conclusions as a hypothesis instead. Kollar asked me for a counterexample, and
I fixed the statement].
C: Use Fubini's theorem to compute the Gaussian integral.
C: When does the series of 1/n^p (for fixed real p) converge? What does this
imply if we replace p by a complex number?
S: What else can you say about the Riemann zeta function? [Meromorphic
continuation, functional equation, etc.]. Can you modify it to be contained in
some special class of holomorphic functions? [I had no idea what he was looking
for. He said something about the Hadamard factorization theorem, which I didn't
know].
C: Can you give a conformal mapping of the upper half-plane onto the unit disc?
[I didn't remember this, so I wrote down (az+b)/(cz+d) and then stared at it
blankly for a while. Eventually they prodded me to decide where I wanted some
particular points to go to compute the right coefficients].
C/S: How do you define the residue of a function at a pole? What's it good for?
[Residue theorem, to compute integrals]. What other consequences are there?
[Argument principle]. Then they gave me an integral to compute via the residue
theorem. I fumbled around a bit but eventually showed how I would set things
up, and they said I didn't need to actually do the computation.
S: Earlier you gave a conformal mapping of the upper half-plane onto the unit
disc; in what generality can you do something like this? [Riemann mapping
theorem]. Can you prove it? [Yes; I sketched the proof from Ahlfors.] If the
region has a "nice" boundary does the conformal mapping extend to take the
boundary to the boundary of a disc. [I said it sounded plausible but I didn't
know. He said it was true].
K: In your proof of the Riemann mapping theorem you mentioned Schwarz's lemma.
Can you state it? [I did but got confused over whether it needed a hypothesis
of "injective".] Well, then do the proof and see if you need that. [I stared
at the blackboard for a while. Eventually Kollar suggested I look at f(z)/z and
I figured out what to do.]
K: What can you say about the ring of functions that are holomorphic in
some neighborhood of the origin? What about multiple variables? [I said it
should be a UFD and guessed that you could prove this from using that the
formal power series ring is a UFD. He asked if I could prove that and I
said I didn't really remember how to do so for more than one variable. I
said something about the Weierstrass preparation theorem, but couldn't
remember the actual statement of it.] What about the ring of entire functions
on C? [I got that it was a domain, and Kollar eventually gave me enough hints
to produce an ideal that wasn't finitely generated and also wasn't contained
in any ideal generated by a linear function z - z_0.]
S: Which is your favorite proof of the fundamental theorem of algebra?
[Liouville's theorem]. How would you prove it for a group of bright
undergraduates? [I asked what I was supposed to assume the undergraduates knew,
and he said real analysis. I said I didn't know any real-analytic proofs of it
offhand and we moved on].
S: When are two matrices similar? [Rational canonical form]. How do you prove
this? [He wanted me to say the fundamental theorem of finitely generated
modules over a PID.]
S: State the fundamental theorem of Galois theory.
S: If we think of the Galois group of a polynomial as contained in S_n (acting
on the set of roots), when is it contained in A_n? [Use the discriminant]. Can
you compute the discriminant of a polynomial from the coefficients? ["Depends
on the polynomial". They prodded me on how I would do it in general but I had
no idea.]
Skinner then asked some questions about what the splitting of a polynomial over
a finite field says about elements of the Galois group of that polynomial over
Q, and also about what it says about prime factorization in the extension
corresponding to that polynomial. I fumbled over these questions pretty badly.
Eventually he said that he would come back to them later, but he never did.
K: Compute the genus of y^3 = x^6-1. [I normalized and used Riemann-Hurwitz.]
K: If X is a projective scheme over the rationals Q, K a finite extension of
Q, and X' the base change of X to K, can you relate the cohomology of the
structure sheaf of X' to the cohomology of the structure sheaf of X? [No idea.]
What if we just look at H^0? [I still had no idea.] How about for H^0 and if we
let X = Spec Q[x]/(x^2+1) and K a field containing Q[i]? [I worked it out in
that case. They eventually got me to write it in the correct form so that
I could guess that what he originally wanted me to say was that the cohomology
of X' was the cohomology of X tensored up to K. Then Kollar asked what
statement from Hartshorne would prove this. I said "flat base change" but that
I didn't know much about it. Kollar asked what linear algebra fact this
related to, and I figured out he was looking for me to say that if you have
a vector space and extend scalars to a larger field, it doesn't change the
dimension.]
S: How would you explain class field theory to someone who wasn't a
mathematician? ["Uh..."] Okay, maybe to a mathematician who isn't a number
theorist. State any version of the main theorem of class field theory. [I wrote
the local one]. What does the inertia subgroup of the Galois group correspond
to? [It took a bunch of hints for me to get to "units of the ring of
integers".] What is local class field theory good for? [I said "global class
field theory".] State the main theorem of global class field theory. What is it
good for? What does it let you say about L-functions? [I said a bit about
equidistribution results like the Chebotarev density theorem]. Do you know
about Artin L-series? [Not really.]
At this point they agreed that they were done asking questions, sent me
outside for a minute, and then told me that I passed. The exam took about 1
hour and 50 minutes.