sam lewallen's generals
committee: Szabo (chair), Conway, Gunning
date and time: May 14th, 2009. 1hr50m
special topics: Algebraic Topology, Compact Riemann Surfaces.
Here I give an approximate account, with some commentary. More commentary at the
end. This might not be too helpful for most people; see commentary. Later the
account does become more like a "question list."
Disclaimer: I've probably only remembered about half of the questions, in total.
C: "I've been doing generals for so long, I've gotten bored of asking so many of
the same questions. So last night, I thought up some new interesting ones to ask."
[I gulp; this is completely unexpected, and seems frightening.]
C: "Matrices come up all over mathematics. What we're going to do is think of as
many appearances of matrices in mathematics as we can, discuss the normal form in
each case, and the similarity (equivalence) relation."
[On the one hand this is quite basic; on the other, I hadn't prepared for it and
also have an innate fear of matrices. I was feeling not a small amount of
trepidation.]
SL: well, we should start with linear transformations from a vspace to itself,
obviously.
ZS+C immediately object: No!
And I remember that there are maps from one vector space to A DIFFERENT vector
space. I hadn't thought about this case in years!! Now what was the normal form?
I knew similarity was by pre- and post- composing with change of bases.
C: And so you can make the matrix look like...
I essentially had a total brain freeze, the most embarrassing part of my exam. I
had been thinking about Jordan normal form while studying, and so could not "free
my mind" to realize how simple the answer was. I finally remembered (after much
stalling) what I internalized so many years ago: the rigidity in the square-
matrix case comes from the fact that you're using the SAME basis in the domain
and range. If not, the only invariant is the rank. And so I write down the
answer.
C: Finally!
SL: Wow, sorry..
C: It's ok, I know it's hard to think at the beginning of these things.
This then continued for some time. We did square matrices, then quadratic forms,
(he asked if I could remember the name of the classification theorem (Sylvester's
law of inertia), but I couldn't, to my chagrin), orthogonal, unitary, symmetric,
and hermitian matrices. (Orthogonal was particularly bad, as I basically didn't
know the answer. You can have \pm 1 eigenvalues, or 2X2 "rotation" blocks, down
the diagonal. As I did this I realized that more generally, you get something
similar by applying Jordan form over the (non-algebraically closed) reals to a
general real square matrix; if the eigenvalue is imaginary, your Jordan "block"
will have 2X2 matrices "down the diagonal." I tried to make this connection out
loud, but was shot down).
[Finally we were done with that.]
G: Well, John, would you like to ask any real or complex analysis questions?
C: Why, yes.
[I don't remember this section too well. He asked:]
C: When does the integral of a sequence of continuous functions converge (to the
integral of their limit) [when they're converging uniformly].
C: When does the integral of a sequence of measurable functions converge (to the
integral of their limit) [when they're dominated]
C: When does the integral of the derivatives converge? or when do the derivatives
converge? I can't remember the precise question or precise answer.
Finally he asked for Taylor's theorem! And he wanted me to prove it, using a new
method he'd just discovered. He led me through it, tsk-tsk'ing along the way at
my inability to follow.
Next he wanted this for the complex case. I didn't know what he was talking
about, which I stated. But I also said that holomorphic functions have a power
series expansion. That was what he wanted.
C: Prove it.
I did, using Cauchy's integral formula. Along the way, when I was rewriting the
quotient, I got sloppy and Conway interrupted. Zoltan begged (first comment in 40
minutes), "let him finish it! He might get it." hilarious.
And that was the end of the general topics. Conway then got up to go--he had an
undergraduate thesis defense to attend.
Next, Zoltan did algebraic topology (actually, Zoltan did AT for a while, then
Gunning did CRS's, then Zoltan did some more AT questions).
[I can't remember this too well either. Questions asked include:]
ZS: What is an Eilenberg-Maclane space? How would you construct one in general
[start with sphere, attach disks].
ZS: Give examples [ RP^infinity, CP^infinity. I give definitions in terms of
S^infinity. I also mention K(Z_p, 1) and the "universal lens spaces," which he
likes].
ZS: pi_2 of S^1 wedge S^2? [I comment here that I'm using the fact that pi_n
(\alpha wedges of S^n)'s is \Z^\alpha, which is non-trivial, and not true if we
change the dimension of the spheres. He asks to prove this, I don't know how.
This leads immediately to the following question:]
ZS: Can you compute pi_3(S^2 wedge S^2) [I'm stuck, again because of a mental
block: I can't understand why pontrjagin-thom doesn't apply directly here to give
\Z. Of course, it's because S^2 wedge S^2 is not a manifold. Anyway, we finally
figure out together that it's \Z + \Z + \Z: two local linking numbers, one for
each S^2, then a global linking number between the two preimages. quite nice. We
then did some more homotopy, which I can't remember.]
ZS: pi_3(S^2), pi_4(S^3) [ more pontrjagin-thom]
ZS: Lefshetz fixed-point theorem [I give the proof for smooth orientable
manfiolds, using intersection theory in the product space].
[Finally, compact Riemann surfaces (again, these are just the q's I remember:]
G: What do you know about maps from Riemann surfaces to tori? [I start by saying
they're surjective or constant if g>1, but he corrects me and says he means
higher-dimensional tori. I say Jacobian, he nods, so I define it and give the
Abel-Jacobi theorem.]
G: Why is the embedding S --> J non-singular? [I write down Riemann-Roch]
G: How about other maps from a Riemann surface to a torus [I don't know. He walks
me through a proof that they all have to factor through the Jacobian. This is
quite nice].
Then we talked about the number of automorphisms [finitely many when g>1. Two
cases: hyperelliptic or not. If h.e., canonical factors through rational normal
curve followed by double cover, count branch points. Otherwise count Weierstrass
points using the Wronskian, etc.]
And the specific bound using Riemann-Hurwitz [I just described this vaguely].
They kick me out, talk for a minute, come out and congratulate me. Gunning says
to go and drink that night, which his advisor had told him after HIS generals. I
apologize to Zoltan for being an idiot, and he says don't worry -- these things
are an [un]necessary evil, and anyway, it's hard to follow JHC on your feet.
[[COMMENTS: Obviously my generals were a little weird, at least for the general
topics. Conway did ALL of them, and he asked almost nothing "from the standard
questions," though I'll be the first to say that in retrospect his questions were
rather trivial. At many times I was asked something I hadn't thought about, and I
had the choice to think carefully about it on my feet or sort of nod and sputter
random things while zoning out. I may regret it now, but I always chose the
latter. This led to some friction with Conway, who eventually complained "Sam,
this isn't Jeopardy, your answers shouldn't come in the form of questions" (I
guess I was just making random guesses and then asking if they were correct).
Indeed, Conway led me through most of his questions, at least the first part of
most of his questions, as if I were asleep. I was distracted by the fact that I
was getting none of the standard questions I'd studied so hard for: Galois
theory, group theory, any complex analysis or real analysis. (Still, the point
here is that, even if it's not ideal, it's at least possible to basically be a
complete idiot and not fail. This is important to remember).
Indeed, in the end, I can conclude that I really overstudied. My retrospective
advice is: don't go too crazy with the general topics. Know basic questions
"pretty well," know just OUTLINES of proofs, more or less just enough so you can
say specifically what part of the proofs you DON'T know (eg and then you do this.
and then you do that). On the one hand I feel like people who get "weird
questions" don't post their write-ups enough, so those write-ups that do appear
seem misleadingly uniform. (On the other hand, Zoltan and Gunning did ask many
questions they'd asked previously, as well as a few new ones. My experience w/
Conway may have been a special case). But the point is, as my experience shows,
you almost definitely won't fail the general topics portion as long as you do at
least SOME studying. On the other hand, from others I've talked to, it seems that
people fail one or two special topics because they've bitten off a big new topic
AND they haven't talked enough to their committee. I talked to Zoltan and
Gunning, and knew much of what they were going to ask. Do this with your special
topic examiners, and it seems that you will be fine. Try to find out before hand
how much they want you to know in order to pass you; if the topic seems too big,
you can choose a new topic before it's too late. Good luck!]]