CheeWhye Chin
May 15 (Friday), 1998, 2pm -- 4pm
Committee: de Jong (chair), Trotter, Pappas
Special Topics: Algebraic Number Theory & Algebraic Geometry
COMPLEX ANALYSIS:
T: Consider a meromorphic function whose only poles are at 1 and 2i.
What can you say about its power series expansion about 0?
(Converges to the function on the open disk of radius 1 about 0)
T: Suppose I allow negative powers of z. What can you say now?
T: Write down a formula for the coefficients of the expansion.
(I wrote down Cauchy's integral formula.)
OK. Now this formula makes sense also for negative values of n.
So where does the corresponding series converge?
T: (Moving on to a different question.) What can you say about the
zeros of an entire function?
(It can have at most countably many, and if it does have infinitely
many, then they must go to infinity.)
dJ:Why must the sequence of zeros go to infinity?
(Otherwise they would cluster in the plane, and the function is
then a constant zero function.)
dJ:OK. Just checking!
T: So, suppose I give you a sequence of points converging to infinity.
How would you construct an entire function with those points as
zeros?
(Use Weierstrass factors. I tried to define them, but got myself
totally confused instead...)
T: [Seeing that I've embarassed myself...] How about doing it for the
Gamma function?
REAL ANALYSIS:
dJ:Let's work on the real line. What is the measure of a set?
(??? Not every set is measurable...)
dJ:Suppose I just want to know how big a set is...
(I defined outer measure.)
dJ:OK. When is a set measurable then?
(I stated the criterion.)
dJ:Name some measurable sets.
T: What are the operations that you can do to measurable sets to get
measurable sets?
(They form a sigma algebra, so countably many set operations are
always OK.)
dJ:What is the integral of a function?
T: Suppose f is a function in L^1(R). Define g(x) as the integral
over the real line of the function cos(x*t)f(t) with respect to t.
What can you say about g?
(more-or-less the real part of the Fourier transform of f.)
T: Why is the integral finite for all x?
(Because cos(x*t)f(t) is in L^1.)
T: Why is it in L^1?
(Because cos(x*t) is in L^infty.)
T: Now prove that g is continuous.
(I tried to fend off the question by stating that Fourier transform
sends L^1 into C_0, but he didn't accept that. After some pretty
explicit hints, I mentioned dominated convergence theorem and he
was satisfied.)
dJ:Give an example of a singular measure. (with respect to Lesbegue)
(I gave a Dirac measure.)
: Look at the distribution function. What can you say?
(It has a point of discontinuity.)
T: Can you give a singular measure that has a continuous distribution
function?
(Take the Devil's staircase; they laughed at my nomenclature of the
Cantor function. I was then asked some crazy questions that I
don't even remember, and certainly couldn't do on the spot. After
that...)
T: Define absolute continuity of a measure. (with respect to another)
ALGEBRA:
dJ:Ah ha! I know how to get from analysis to algebra. Prove the
fundamental theorem of algebra.
(I talked through the standard Liouville's theorem proof.)
P: Can you prove it using Galois theory?
(I talked through the Galois theory proof as well, stating
precisely what facts are needed about the real numbers which have
to be established using analysis.)
P: Let me ask you a bit about representation theory. Suppose you have
a finite p-group, and you have a representation of this group on a
finite dimensional vector space over a finite field of
characteristic p. What can you say about it?
(Yucks! OK... what else can I say? I said I knew nothing about
these things; he wasn't happy.)
P: Write down a nontrivial example of such a representation.
(I gave a two-dimensional representation of Z/pZ over F_p.)
P: OK. What are the invariants of this representation?
(Well, it has a one-dimensional subspace which is pointwise fixed.)
P: Good. Can you generalized this somehow?
(It took me a while before I realized that the crux of the problem
wasn't anywhere close to representation theory! So I ended up
saying that every irreducible representation in such a case is
one-dimensional, and talked through the proof by mentioning the
orbit formula and its consequence for a p-group action.)
dJ:What is the one-dimensional representation?
(I thought I heard "What is a one-dimensional representation", so I
said it is given by a character.)
dJ:No, no, that's not what I want. Hmm... What kind of character is
it for this irreducible representation?
(Oh... the trivial one.)
ALGEBRAIC NUMBER THEORY:
P: Look at Q(cuberoot2). What is its ring of integers?
(A naive guess is Z[1, cuberoot2, cuberoot2^2].)
P: How would you prove that?
(Embarrassingly, I blanked at this one.)
P: Look. I'm sure you can figure out how to prove it. Think for a
while.
(I mentioned something about the discriminant of this candidate
ring as compared to the discriminant of the whole ring of
integers.)
dJ:OK. What's the big deal about the discriminant?
(The primes dividing it are ramified.)
dJ:If you localize the ring of integers at a prime, what do you get?
(A (noetherian) integrally closed local ring of dimension one, aka
a DVR.)
dJ:How would you know it is integrally closed?
(Maximal ideal is generated by one element.)
dJ:SO???
(Oh boy! Such explicit hints! I finally got what he was driving
at, and talked through how one could, in principle, show that the
candidate ring is actually the whole ring of integers of
Q(cuberoot2), but I didn't really carry out the computations for
the example on the board.)
dJ:OK. Now localize this candidate ring at 2. What is the maximal
ideal?
(I wrote down the two obvious generators.)
dJ:So, is it generated by one element?
(Yes! Phew! That was close!)
P: Can you tell me, roughly, what proportion of primes in Q remain as
primes in Q(cuberoot2)? Like 10%, 50%, or what?
(I said my gut feeling is 1/3 of the primes.)
P: And how would you prove that?
(I tried to avoid really doing the work by saying that one has to
look at how the Frobenius of the prime in Z act on the primes in
the Galois closure; again he wasn't happy, so I went ahead to do
the computations. I first drew the subfield lattice in the Galois
closure...)
dJ:Why is Q(cuberoot2) not Galois over Q?
(cuberoot2 has two other conjugates not in the field.)
dJ:Why are the other two conjugates not in the field?
(Take a real embedding of Q(cuberoot2); the other two conjugates
are not real.)
dJ:OK. Just checking!
(As I was trying to determine which conjugacy class of the group to
pick...)
dJ:What are the conjugacy classes of a symmetric group?
(Given by the cycle type.)
dJ:How would you determine the number of primes in Q(cuberoot2) over a
rational prime p?
(Factor the polynomial X^3-2 modulo p; I ignored all the ramified
primes right from the start.)
P: So, what's your answer for the proportion of inert primes?
(It took me a while longer before I convinced myself that the two
element conjugacy class of the 3-cycles was the one I wanted, and
I went on to talk about why that is the correct one, by looking at
the factorization of X^3-2 modulo p.)
T: You have reduced the polynomial modulo p. What else could you have
done?
(???)
T: For instance, would you have reduced it modulo p^2, p^3, etc?
(??? What was he asking for???)
dJ:Well, you know this. You could have factor the polynomial over Q_p.
(Oh!!)
dJ:[Sensing that the magic word hasn't been mentioned so far...]
So, you are using Tchebotarev, right? Can you prove it?
(Yes, assuming some results from class field theory.)
P: Why don't you do it?
(I started to erase the board...)
dJ:No, no. It's OK.
P: Let him do it.
(I offered to just sketch the proof, much to the relief of
everyone. I wrote the statement of the theorem on the board.)
dJ:Say, what exactly are you assuming from class field theory?
(Since I had to just sketch the proof, I said I might as well take
it for granted that the theorem holds for abelian extensions; he
seemed pleased. Then I went ahead to talk about the proof for
finite Galois extensions.)
T: Do you know of a PID that is not Euclidean?
(Yes, Z[(1+sqrt(-19))/2]. I confessed that I merely knew this as a
fact, but couldn't remember how one would actually prove that it
is not Euclidean.)
dJ:How would you prove that it is a PID then?
(This is the ring of integers of Q(sqrt(-19)); use Minkowski's
bound, reduce the problem to checking that a finite number of
ideals are principal ideals; he was quite happy even though I
didn't really do the work.)
ALGEBRAIC GEOMETRY:
dJ:OK. Let's do algebraic geometry. What is a curve?
dJ:What is an elliptic curve?
(Curve of genus one with a rational point. I forgot to mention
nonsingular, but he didn't notice it.)
dJ:What do you mean by a rational point?
(I defined that, and [I think] also said that it is the identity of
the group.)
dJ:Write down a cubic plane curve.
(Y^2*Z = X^3 - X*Z^2)
dJ:Where is the identity?
([0:1:0])
dJ:Define the group law for this curve.
(I defined it using divisors.)
dJ:How do you define it geometrically?
(I drew the real points of the curve in the usual affine part and
described the group law I wrote earlier in terms of the picture,
adding auxiliary lines along the way.)
dJ:Your curve is nonsingular. How do you define the group law for a
nodal cubic plane curve?
(Use Cartier divisors.)
dJ:Er... that's not what I want. How do you do it geometrically?
(I didn't know, so...)
dJ:That's OK. I mean, you could take a family of nonsingular curves
that degenerate to the nodal one... Let's go on.
P: Speaking of families... how would you define a family of curves?
(I said you need a base scheme to parametrize the family, but the
key requirement is flatness.)
P: Why do you want flatness?
(Numerical invariants like genus, Hilbert polynomials etc don't
change in a flat family, so it makes sense to call that a family.)
P: Now, suppose you have a not-necessarily flat family over a base
scheme. How would you impose conditions to make it a flat family?
(I didn't remember the local criterion for flatness, so I just
declared that I did't know about such stuff; he didn't pursue
either.)
dJ:Here's a hard one for you. Take a quadric and a cubic hypersurface
in P^3 and intersect them. Suppose the resulting curve is as nice
as you want. What is its genus?
(I admitted that I had not studied that, but said that if the
dimensions were one lower, then I can say something in that
flavor.)
dJ:What can you say?
(I mentioned the adjunction formula.)
dJ:State the adjunction formula.
dJ:Define genus.
(Dimension of global sections of canonical sheaf.)
dJ:What is the canonical sheaf?
(Top wedge product of the sheaf of differentials.)
P: Which is???
(Huh??)
P: You're on a curve, right?
(Oh, I forgot! Then just take the sheaf of differentials.)
COMMENTS:
It appears that the committee was happier seeing that I can tell
exactly what I know and what I don't, than seeing me prove anything at
all. They seem to be testing that the student knows:
0) how to use a result when the situation demands it (e.g. fraction of
inert primes --> Tchebotarev)
1) how to define things (e.g. what is a curve?)
2) how to state a key theorem precisely (e.g. Tchebotarev) and
every concept related to the theorem (e.g. Dirichlet density,
Frobenius element, etc)
3) the main ideas of the proof (e.g. give a 1-minute sketch);
the relative difficulty of the steps involved (e.g. use class field
theory here; do a computation there).
I was lucky to get a committee that showered me with hints to many
questions; often the hints were embarrassingly explicit! But it took
me a while before I realized that some of the questions were actually
hints in disguise!