Andy Manion
Zoltan Szabo (chair), Paul Yang, Nick Katz
Special Topics:  Algebraic Topology, Differential Geometry
April 20, 2011
Length of exam: 2 hours

Note: Even though I wrote most of this right after my exam, there were still things that
I didn't remember clearly- the exam went by very quickly. I hope this is a more or less
accurate summary of the questions that were asked, but no guarantees. Also, it's possible
that some of the questions were asked in a different order. 

Algebraic Topology:

sz: Talk about the classification of closed surfaces. How do you know they are different?
K: How would Riemann have produced a surface of genus g? How about genus 1?
(wrote down polynomial for elliptic curve)
K: Say you have the same equation but with more factors on the x side. What's the genus?
How many points is it branched over?
Sz: Consider CP^3 and S^2 x S^4. Are there invariants of these that are the same? Different?
Sz: Write down S^1 wedge S^2. Talk about its homology, cohomology, homotopy groups, etc.
Sz: What's \pi_3(S^2 wedge S^2)?
Sz: What's \pi_4(S^3)? (Did the spectral sequences argument)
Sz: Can you do it like you did for S^2 wedge S^2? (eventually relies on \pi_1(SO(3)))
Sz: Do you know other homotopy groups of SO(3)?
K: Anything in general about \pi_k(SO(n)) for various k,n?
K: How about for a general Lie group?


Differential Geometry:

Y: Do you know Gauss-Bonnet? (Stated and proved)
Y: How about for non-compact surfaces? (didn't require proof)
Y: Give an example where strict inequality holds in the non-compact version
Y: Gauss-Bonnet for four dimensions? (Talked about Pfaffian for general even
dimensions)

Then we took a break. The first half took about an hour.


Complex analysis:

K: Rouche's theorem
K: What's an explicit conformal transformation from the upper half plane to the 
disk? (z to (z-i)/(z+i), which is called the Cayley map)
K: Can you have a double cover of the sphere branched over 7 points? How about 6?
K: Are all genus-g Riemann surfaces hyperelliptic? (I fumble for a while)
K: How many genus-g surfaces are there? How many hyperelliptic ones? (Eventually I write
down 3g-3 for the general case although I got the formula wrong at first. They lead
me through a rough calculation of how many hyperelliptic surfaces there are: if
you have y^2 = (x-a_1)...(x-a_k) then you can put any 3 of the a_i's in standard
position and you're left with 2g-1 parameters)
(Somewhere along the way, Zoltan asked just for the special case of genus 2
(all are hyperelliptic) and genus 3 (not all are hyperelliptic))
K: Say you have a meromorphic 1-form on the plane. When can it be written as df
for some meromorphic f?
K: Same question for a compact Riemann surface. (Mentioned genus = dim of space of
holo 1-forms)
K: What does the Hodge decomposition have to do with this?


Algebra:

K: Fundamental theorem for finite Abelian groups? Is this canonical, i.e. can you have
automorphisms which don't preserve the decomposition? (didn't ask for proof of theorem though)
K: Galois theory in 30 seconds.
K: Finite fields in 30 seconds. Can you have a degree 7 irreducible polynomial over F_p?
How about a degree 14 irreducible polynomial? (Yes in both cases)
K: I'm going to ask a question that might not be kosher. Consider SO(n). Does there
exist a Zariski open set which is also an open set of R^m for some m?
Sz: That's algebraic geometry.
K: Some people would call that topology.
Sz (laughs): I wouldn't.
K: OK fine, I'll just ask your opinion then. (I say I think there isn't)
K: Your opinion is wrong. (It turns out that the answer involves the Cayley transform,
which is related to the Cayley map above. If you define a map o(n) --> SO(n) by sending
A to (I-A)(I+A)^{-1}, you get a bijection between the vector space o(n) and the set of
special orthogonal matrices which don't have -1 as an eigenvalue.)



Real analysis:

K: Say I have a C^k function on the circle. What can I say about the decay of its Fourier
coefficients?
K: Say I have a continuous function on the circle. Does its Fourier series converge?
K: Talk about the Fourier transform.

At this time Zoltan had to leave so the exam ended.

Thoughts: The exam ended up being less scary than I had anticipated. My committee was
generous, and they gave me lots of help when I got stuck. Of course, this is also what
I had heard from many older students about the exam, and the preparation for the exam
was still stressful. That stress is probably unavoidable, but here are a few things I
learned that might help:

-I scheduled my exam to be very early in the year. I think this was a good decision; it's
nice to be done with the exam. Having a few extra weeks might seem safer, but by the time
late April arrives, you'll probably be more than prepared. A lot of the difficulty for me
was staying as fresh as possible on a large amount of material, and extra time just makes
this task harder.

-I started with the topic I knew best (algebraic topology). Some examiners might prefer
a general topic first, but if you get the choice, I'd recommend starting off on a good
note. It'll make you more confident for the rest of the exam if you pick your favorite
topic (special or general) as the first one.

-When I answered questions, I tried to approach it as if I were giving an informal
presentation. I wrote on the board when possible and volunteered arguments or even complete
proofs when I knew them. The more time you spend talking about things you know, the more 
confident you'll be. I also tried to be as honest as possible when I didn't know
something, and my committee didn't seem to be upset when I said I didn't know a particular
proof or result. Sometimes they would walk me through a proof, and other times they would
just move on. 

-The week before the exam, I found it very helpful to rehearse answers to some of the most
commonly asked questions, e.g. proofs of Gauss-Bonnet (for differential geometry), Riemann
Mapping Theorem, fundamental theorem of calculus for Lebesgue integrals, Sylow theorems,
computation of \pi_4(S^3) (for algebraic topology), etc. Chances are you'll be asked some
subset of your committee's favorite generals questions, which you can deduce from past 
exams, and if you prepare beforehand these will be a breeze. My committee let me give
nearly all of the proof of Gauss-Bonnet, and I was very glad I had practiced it. It's
especially helpful to practice at a board (the board in the first-year office works
great). Mock generals were also very useful.

Good luck!