Wei Ho
May 16, 2005 2pm-4:40pm
Manjul Bhargava (chair), Ramin Takloo-Bighash, Edward Nelson
Algebraic Number Theory, Representation Theory
-----------------------------------------------------------------------
ADVICE/REFLECTIONS: (I'm putting this first, since those of you with
different special topics and examiners probably won't read this
transcript.)
Take the right attitude towards the exam and the requisite studying. You
will probably pass, so try to actually learn some math and enjoy studying
for it. Don't bother memorizing proofs! (They won't have the patience to
sit through a 5-minute proof of some basic fact anyways.) You should know
the main steps in most big theorems and have a sense of what results imply
what other results, but that's about it. They really are testing your
understanding of the material, so they'll ask lots of examples and some
random questions that are impossible to practice ahead of time. You won't
be able to answer most of your exam off the top of your head.
Stay calm during the exam. This was my biggest problem. I got very
frustrated for not knowing something or messing something up, and then
couldn't think at all. Of course, knowing that you're not supposed to
panic doesn't mean you won't, so if you think you're liable to do this,
take mock generals with older students to get used to not having the
answer. I think that they aren't really interested in a question once
it's clear you can solve it, so you will spend the majority of time trying
to work out problems that aren't easy for you. Also, if your committee
knows that you are pretty comfortable with a particular topic, they may
not even ask about it.
Picking a committee chair that might want to work with you helps. While
it is intimidating to think that your future advisor might not think so
well of you after your generals, it's more important that someone on the
committee is 'on your side' to some extent; he/she can temporarily steer
the conversation to things you know if things are going poorly. And make
sure you talk to everyone on your committee before your exam.
Just try to relax and get a lot of sleep before your exam :) Good luck!
-----------------------------------------------------------------------
I'm sure I've forgotten many things that happened (or rather, blocked them
out of my memory), but here's a version of my general exam. Most of the
time, they were happy to give hints when I needed them (though even
happier when I didn't need them!). I've included some of them, but there
were many many more.
Throughout, they were all very nice and they tried to create a relaxed
atmosphere. I was too nervous in the beginning to realize this, but the
second part of the exam, in particular, was more like a discussion than an
exam, and I actually didn't mind being there.
They asked me to pick the topic I wanted, so I thought I'd just get
analysis over with. (bad idea in my case!)
REAL ANALYSIS: (all Nelson)
N: Talk about absolutely continuous functions and functions of bounded
variation.
I defined the former and mentioned the fundamental theorem of calculus.
He asked for a Lebesgue version of the fundamental theorem that didn't
look like the usual calculus one, and I didn't really know what he wanted.
He asked me about absolutely continuous measures, and how they related to
the functions of the same name... I still wasn't sure where he was going,
so I threw out a lot of vaguely related things (Lebesgue decomposition,
Radon-Nikodym) but I definitely got the sense that this still wasn't what
he was looking for. Eventually we moved on, but by this point, I was
completely flustered, and that lasted for most of the real and complex
analysis sections...
N: If you have a sequence of continuous functions on [0,1] with integrals
converging to 0, do they have to converge to a continuous function? (no)
Give an example. Does the limit function have to be continuous somewhere?
(yes, use the Baire category theorem. I started explaining this in
detail, and he cut me off once it seemed clear I knew the answer.) Does a
pointwise-convergent sequence of L^1 functions on [0,1] have to have a
limit function that's Lebesgue integrable? (no) Give an example.
Somewhere in here I said something about nice things happening with
monotone or dominated convergence. He didn't seem to care though... There
was also a discussion at some point about Riemann integrability. He asked
me what made a function Riemann integrable. I first said you needed only
countable discontinuities, but he said I was missing something (the
obvious part!).
He suggested we move on to complex. I wasn't sure if the committee was
satisfied or not (I wasn't!), so I tried to indicate that we could do more
real, if anyone wanted...
COMPLEX ANALYSIS:
N: If a power series has radius of convergence 1, can it be continuous on
the entire unit circle? Can it be analytic?
I didn't process the question at first, and asked him to clarify, but I
think it just confused me more. At some point, we left a very awkward
discussion, but returned to it at the end, when I was able to think a
little better and answered it. As a warning, there was a bit of confusion
about 'Taylor's theorem' -- I was referring to the theorem in Ahlfors, but
he thought I meant the real-variable version...
N: Suppose you have a function analytic on the top half of the unit disk,
continuous on the real interval [-1,1]. Can you define an analytic
extension to the whole disk? How do you prove it?
T: Compute the Fourier transform of e^{-\pi x^2}. (I tried to just say,
well, it's basically the Gaussian, but he wanted me to compute it from the
definition directly. I didn't remember the computation, but fortunately
got it with help.)
This also led to a discussion of the Fourier transform, and why one would
want to normalize. I said 'inversion formula' immediately, but I think I
should have also said something about the isometry for L^2.
N: Consider all germs of meromorphic functions at 0. What kind of
structure does this have? (C(t), so a field) Is it algebraically closed?
Why not?
B: Talk about conformal mappings. This led to Schwarz's lemma. (At some
point Ramin told us about some generalization of Schwarz's lemma, but I
have to admit that I didn't really process any new information since I was
trying to prepare myself for the next question.)
At this point, they wanted to move on, with the possibility of returning
to real and/or complex. (I was hoping they would return, so I could
redeem myself.) About one (excruciating) hour had passed at this point.
(The rest of the exam flew by!)
ALGEBRA:
B: Classify groups of order... umm... how about 14. (I started to say
something about applying the Sylow theorems, but was immediately told to
do it an easier way.) Why do you know that there's a group of order 7?
(I said Lagrange's theorem, to some laughter... They told me the correct
name, which I have now forgotten!) Are all index-2 subgroups normal?
T: What's Cayley's theorem? (again, I didn't know which theorem the name
corresponded to, but fortunately they told me it had something to do with
permutation representations...) Give an example where you don't need to
use S_n for an order-n group. Give an example where you do. (I said
Klein four group, since that's the first thing that popped into my head,
though he was looking for the more obvious answer of Z/pZ)
B: What is an integral domain? Prove that all finite integral domains are
fields.
T: Let's do some representation theory (finite groups). Tell me about the
representations of D_4. (I essentially wrote down the character table,
which I had worked out pretty recently, and was periodically asked to slow
down and explain what I was doing.) What is the representation that
corresponds to the second line of your character table (a nontrivial 1-dim
rep)? Oh wait, that's too obvious. What is the 2-dimensional
representation? How can it be interpreted geometrically?
T: What's an induced representation? Write down the definition. I don't
like your definition; do you know another? (no...) Well, here's a better
one (in terms of functions). Now why are these equivalent?
T: Does induction preserve irreducibility of a representation? (no, e.g.
inducing a restricted rep gives a direct sum of the original rep.) What
can you say about the kernels of the irreducible representations?
(normal subgroups) How about kernels of direct sums of irreducibles?
What kind of functor is induction (e.g. left-exact or right-exact)?
B: We should do some Galois theory... What's a splitting field? Give an
example of something that's not. What's its Galois closure? Draw its
lattice of subfields. (Since I chose the easiest example of
Q(cuberoot(2)), this came back during number theory of course...)
ALGEBRAIC NUMBER THEORY:
B: Do you know the ring of integers of Q(cuberoot(2)) offhand? Given that
it's Z(cuberoot(2)), what's the discriminant? Which primes ramify?
Wait, what happens at infinity? (Oops, forgot about it) What happens to
the prime 5? What proportion of the other primes decompose like that?
What proportion of primes split completely? What's left?
T: (in the middle of the discussion above) What theorem are you using?
State Chebotarev. (I asked if they wanted me to define things like Artin
symbol, but they declined.)
T: What classical theorem... (I didn't even listen to the rest of the
question to know that he wanted Dirichlet's theorem on arithmetic
progressions. I offered to prove it via class field theory but they
weren't that interested in the details.)
B: What's the key step? Why doesn't L(1,\chi) vanish? (I gave the same
answer Manjul had given in his generals (!), that the product over all
\chi of L(1,\chi) is essentially the Dedekind zeta function. But Ramin
raised some objection about how that might only work for even characters,
so I also gave the class field theory reason.)
B: What's a Hilbert class field? What's the Hilbert class field of Q?
Why?
T: Here's a crazy question. What's Spec of the adeles? Wait, do you know
what the adeles are? (yes. But he didn't ask me to define them.) Ok,
then take 30 seconds to think about it and give me your best guess.
They were happy with number theory, even though I was surprised that I
wasn't asked to compute a class number, prove the units' theorem, or state
the main theorems of class field theory.
REPRESENTATION THEORY (Lie algebras):
B: Talk about the representations of sl_2, say over C first. (I
immediately said that the representations all looked like wedge^k V, for V
the standard representation. Oops! He asked a few very careful
questions, until I realized that I had meant Sym...) What's a root?
What's a weight?
B: What are the weights of Sym^k V? (I said something stupid again, that
the weights were k to -k with everything in between. Oops again!)
T: Can you decompose Sym^2 (Sym^2 V)?
B: What is the trivial representation in there?
T: Describe the Weyl unitary trick. How do you know there's a compact
group associated with your complex Lie group? (construct the split form
and take the invariants of the conjugate linear to obtain the compact
form) Take the standard representation of su_2. What happens when you
tensor with C to get sl_2? How about for other representations? Wait,
this question is too easy...
T: What does SU2 look like algebraically? What does it look like
geometrically? (S^3) Why? (I didn't know why, so he made me write it
down and actually do the computation.) The determinant condition doesn't
quite define a sphere, so what's wrong? (I chose a bad embedding)
B: What is a root system? Where does the 2(b,a)/(a,a) come from, in terms
of Lie algebras? (I commented that I always got the beta and alpha mixed
up when writing down the reflection formula, and they said that they did
too. I felt better.)
B: What are all the root systems of rank 2? (I started drawing Dynkin
diagrams, so...) What is a Dynkin diagram?
I drew the rank-2 Dynkin diagrams and root systems, and named the
corresponding Lie algebras. They were pleased that I knew that sp_4 was
isomorphic to so_5, which was apparently crucial in Ramin's Ph.D. thesis.
And they were amused that I kept forgetting to fill in the root system,
since I was drawing the simple roots off the top of my head. But in doing
this without working any details out, I mixed up the lengths for sp_4, so
I asked if it was ok if I worked out the roots on the side. They said
that I probably should've done that first anyways! Luckily I managed to
fix it...
T: How do you get the Lie algebra from the Lie group? For an infinite
dimensional representation, what could go wrong for a representation of
the group when going to the algebra? Like if it's not smooth? (I had a
lot of trouble with this, especially since it turned into a real analysis
question when we tried a concrete example.)
We agreed to go back to complex since they had also noticed how nervous I
was in the beginning, and we tied up some questions that I hadn't finished
earlier (and this time around, they seemed so obvious!). I still felt
unsatisfied with my analysis performance, but they asked me to leave. A
few minutes later, the door opened and they congratulated me!