The following is a fictional account of my generals. I say "fictional"
simply because I don't remember the exact sequence of events. I remember
some version of the following questions were asked, and that many were
clumped together in a long string of questions meant to explore a larger,
further reaching question.
Nick Katz was my session chair, along with Robert Gunning and Jiu-Kang Yu.
Therefore Dr. Katz led most of the session, though Dr. Gunning and Dr. Yu
did conduct the exam from time to time. My declared topics were Algebraic
Geometry and Algebraic Number Theory.
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These questions are pretty much in the order that they occurred, but one
must recall that this is a "fictional" account.
- State Riemann-Roch for curves and Riemann Surfaces.
- Define the arithmetic genus and geometric genus of a variety.
- What can you say about global sections of \Gamma(X,\omega)? Sections of
H^1(X,O)?
- Serre Duality?
- What is the genus of y^2 = x^11 - 1? How about y^2 = f(x) with f(x) of
odd degree and distinct roots?
To my own disappointment I could not give the general genus formula for
such a curve. Together with Katz I walked through the Hurwitz computation
of the genus, and in particular we focused on understanding the
ramification over infinity. The resulting discussion touched on several
topics in commutative algebra, including local fields, complete rings, etc.
I had to do a lot of thinking on my feet here, but all of the committee
members were very nice in dealing with this.
- What is the Galois Group of the degree 11 extension over K(y) given by
the equation x^11-(1+y^2)?
- Talk about intersection theory for a surface. We discussed intersection
pairings, including a statement of B'ezout's Theorem. Katz pointed out
that everything I had done would work on a surface as well as in P^2 and
hence that maybe I knew more surface theory than I thought (I claimed that
I knew none).
- Why does an elliptic curve have a cubic form in P^2?
- What is the invariant differential of an elliptic curve?
- State Abel's Theorem for an elliptic curve.
- What about higher dimensional varieties?
- What does Hasse's Theorem say about the number of points on an elliptic
curve over a finite field.
- How does this relate to the Weil Zeta Function of a projective variety?
- What does this zeta function look like for an elliptic curve?
- What does it look like for P^n?
Here is where Katz showed me that the previous two zeta functions have an
identical denominator. Hence one might expect the same sort of thing in a
far more general situation.
- Where does the coefficient a_p come from in the elliptic curve's Zeta
function? Here I spoke about Galois Representations, the trace of the
Frobenius, etc. This is stuff I have a personal affinity to and may not
be relevant to the average algebraic geometer.
- Suppose that you have a large finite set of equations over a finite field
and the number of points on the resulting variety over F_{q^n} for all n.
Further suppose that you know that the resulting variety is a hypersurface.
How might one determine the genus of this hypersurface? (Hint: Look at the
numerator of some zeta function.)
- State a theorem of Dirichlet. I chose the theorem of primes in
arithmetic progression, which lead to the following questions.
- How does this relate to the Cebotarov Density Theorem?
- Suppose you are given a finite Galois extension K/Q by f(x)\in Z[x] such
that deg(f)=n and Gal(K/Q)=S_n. What can one say about the roots?
Note, this is a converse to a more typical Galois question: what are
sufficient conditions for a polynomial over Q(x) to give rise to a Galois
field extension with Galois group S_n. There one discusses having precisely
two complex (conjugate) roots, while the remaining are real, provided that
n is prime. It took a moment of reflection to understand that we were
discussing the converse of the expected question and that I didn't (and
really still don't) understand the full answer. One might state that these
conditions are also necessary, but my work didn't directly answer this
question.
- How does one use CDT to determine the average number of roots of f(x)
modulo a prime p (downstairs).
This questions covered a fair amount of time in my exam and I find the
result fascinating. In essence one wants to understand how the
factorization of f(x) (mod p) relates to the conjugacy class of the
Frobenius element's conjugacy class in Gal(K/Q)=S_n. The key point is to
understand the relationship between the respective lengths of the
irreducible factors and the cycle lengths in the disjoint cycle
decomposition of a Frobenius element over p (they are identical!).
- What are the inertia and decomposition groups? Are these quotient groups
or subgroups? What about the Galois group of a resulting finite field
extension?
- Classify all quadratic extensions of C((T)).
I was caught off guard with this question as I had focused too narrowly on
number fields. However, when I explained that my intuition lay with number
fields the committee helped me to find an answer to a related question in
local field theory. Once I was able to answer the question there we
returned to the original C((T)) question.
This too was another exploration into commutative algebra. This time we
discussed tools such as Hensel's Lemma, including a discussion as to when
it holds. We also discussed when one can answer questions for different
fields K and K((T)), and Katz mentioned that Puiseux series were a central
point (i.e. Puiseux had a lot to do with the first answers to this
question). If one explores the question it will result that a very nice
answer appears which quickly allows one to answer the same question for all
extensions of a finite degree n. Additionally we briefly touched on
ramification (including wild and tame ramification for local fields).
- State and prove Liouville's Theorem.
- State Cauchy's Theorem. Katz explicitly said he didn't want the integral
formula (as I had already stated it at this point) but a more naive
theorem. I needed help here and was told guided to a simpler statement
than my original answer of Morera's Theorem (which came up when we all were
trying to determine if Cauchy was the right person to attribute this
theorem too).
- Define locally exact (residues vanish).
- How does this relate to Stokes Theorem, Cauchy's Integral Theorem and
Green's Theorem?
- Cauchy's integral formula: suppose our contour is |z|=1. What can I say
about the resulting function inside the contour?
- Outside the contour?
- Is the resulting function analytic?
- Is that bothersome?
Here Gunning pointed out that it was preferable to use, say, f(x) under
the integral sign of CIF and F(x) on the lhs of the equation. If f(x) is
already analytic, then one can abusively use the same letter for
both (as Ahlfors does).
- State the Riemann Mapping Theorem.
- Discuss conformal maps (because I had trouble recalling the statement of
the RMT).
- Give a conformal map of the upper half plane to the unit disk.
- What is the Poisson kernel? I responded that I didn't know. Note, this
is a topic I made a conscious decision to skip in my studies. It should be
noted that it seemed ok with the committee that I didn't know anything
about this. It certainly would have been nice to answer something, but one
should recall that the committee's job is to find out what you *don't* know
as well as what you do.
- State something pathological about measure theory.
- Consider continuous functions on a given real (bounded) interval. Define
a metric via the sup norm.
- What is the completed space of functions?
- Now, via the L^1-norm (d(f,g)=\int |f-g|dx) what is the completed space?
- Talk about relative containment of these spaces.
- Can you give a series which converges in L^1 but not uniformly?
- What is Jordan Normal Form for a matrix?
- How does this relate to the classification theorem of finitely generated
abelian groups?
- Is A^t similar to A?
I'm sure that there are other questions which I forgot. I apologize for
any omitted questions, misattributions, etc. The entire exam lasted three
hours. In the end I must admit that I enjoyed the exam. It was stressful
to prepare for, but it was stressed I placed upon myself, not the committee
members. Each of them was very nice and helpful throughout the course of
the exam. It's worth noting that it was ok to be able to answer some of
the questions, and hence that one shouldn't expect to be able to answer
every question that I did.