Chris Yang Xiu's generals
Committee: Zoltan Szabo(Chair), Peter Ozsvath, Gustav Holzegel Topics:
Algebraic Topology, Differential Topology
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I was allowed to pick which topic to start, and I said Algebraic
topology
Algebraic Topology: S: Pick a matrix in SL_2(Z). It induces a map from
torus to torus. Take T\times I, glue the two ends by this map. What
are its homology groups and homotopy groups? Can you give it a CW
structure? What is the definition of a CW complex? What's its
universal cover? (R^3) We didn't get to all the homology groups.
O: What's the cup product structure for RP^2 # RP^2? I got the answer
with the simplicial structure. Given a connected sum of two manifolds,
do you have any general methods of computing its cup product
structure? I struggled a little and Prof. Ozsvath told me to do
S^1\times S^3 # S^2\times S^3 as a start. (Answer is mapping the sum
to each component.) In the meantime, I was asked to state universal
coefficient theorem and apply it to some manifold.
O: What's your favourite theorem in Algebraic topology? I said Kunneth
formula.
O: Let's look at H^1 of a manifold. What are the elements? (Maps from
\pi_1 to Z) How would you characterize homotopy classes of maps from a
manifold to S^1. I said H^1, but I didn't know how to prove it. Then
Prof. Ozsvath led me through the argument using H^1_DR.
Real and Complex analysis
H: What is a Banach space? Take C[0,1]. Give it a norm to make it a
Banach space. (sup norm) Why?
H: If I have a uniformly bounded family of functions in C[0,1] and I
want any sequence to have a converging subsequence, what conditions do
I want? I said equicontinuous, by Arzela-Ascoli theorem. Do you know
the proof? I mentioned diagonalizing argument. Why do we need
equicontinuous? I gave an example when the statement fails without
it. Would derivatives being uniformly bounded work? Yes. Why? How
about for holomorphic functions? "Uniform boundedness gives
equicontinuous." How to prove it? I sketched the proof.
H: Can you map the complex plane into the disk? No. If a holomorphic
function is dominated by log|z|, what can you say about it? Still a
constant. Why? I did the same argument as the proof of Liouville's
theorem. How about dominated by |z|? Divide it z and we still have an
entire function.
Algebra
O: What are the subgroups Z^2? How about F_2? Free. How many
generators can you have? Any number up to countably many. Can you find
one with 3 generators? I said that I had to use covering space to do
it. Then Prof. Szabo said: "So are we asking more algebraic topology
questions?" Prof. Oszvath replied: "I don't know what you are talking
about." Obviously, I wasn't in the mood of laughing. I drew a covering
space of S^1vS^1 and was asked to write down generators. Then I was
asked to find subgroups 4 generators and countably many generators. Is
the subgroup with 4 generators you find normal? Why? The one I drew
was not a norm covering. What is a normal covering? Why isn't it
normal? Can you find a normal one?
O: What is a projective module? I said I couldn't remember projective
and injective and flat which one is which one. Just give one of the
definitions. Then I remembered flatness corresponds to tensor
product. Do you know an example of flat module? Q. What else? Free
modules. Now what's the definition of projective and injective? I
wrote down the two short exact sequences of Hom. I will tell you Z is
projective. Figure out which definition is which.
Differential topology
S: Tell me everything you know about Chern classes. I was asked to
state the signature theorem.
S: Talk about the cobordism groups. Worked out generators of 2
dimension using stiefel whitney numbers. How about oriented ones?
Generators in 4 dimension?
O: A complex submanifold of CP^2 corresponding to 3 times a generator
in H_2. What is it? I struggled big time here, with lots of hints, I
showed it's a torus.
There were possibly more questions that I don't remember in this
section.
General advices: Focus more on concrete examples than on general
theory.
Don't worry about messing things up. It's almost bound to happen when
you are on the spot and under pressure. My committee was so nice and
they didn't get mad even when I messed up easy things.
My general certainly didn't go through as smoothly as it would seem
from this transcript. I got lots of hints and nudges.