Andras Juhasz's general exam
5/10/2005, 10-11:30 am
Committee: Browder, Gunning, Szabo (chair)
Special topics: Algebraic Topology, Differential Topology
ALGEBRAIC TOPOLOGY (Browder):
What is your favorite homotopy group ? I said pi_3(S^2), but Browder
said let's do something more complicated, say pi_4(S^3). I offered
several proofs, and I had to work out the one using spectral sequences.
Freudenthal suspension theorem. Write down a long exact sequence
containing the suspension map (embed S^n into the loop space of S^{n+1}).
What about pi_6(S^4) ? I sketched the proof using the
Pontrjagin-Thom construction, but I didn't remember every detail.
DIFFERENTIAL TOPOLOGY (Szabo, Browder):
State the h-cobordism theorem. Outline the proof. Where did you use that
the cobordism is simply connected ? Why do you need to use
gradient-like vector fields ?
How do you derive the Generalized Poincare theorem from h-cobordisms ?
Why do we only get a homoeomorphism with S^n, not a diffeo. ? What
happens in dimension 4 ? I mentioned Freedman's theorem.
Characteristic classes: What are Stiefel-Whitney classes ? What are the
S-W classes of RP^n ? Prove it.
What is the signature of a manifold ? Does the intersection form have to
be nondegenerate ?
State Hirzebruch signature theorem in dimensions 4 and 8 (I forgot one
of the constants).
How would you compute the coefficients in the formula ? I said that the
oriented cobordism ring over the rationals is generated by complex proj.
spaces of even dimensions.
What do you know about exotic S^7's ? Group structure induced by
connected sum.
I had to outline Milnor's constrution of an exotic 7-sphere. How do you
obtain rank 4 vector bundles over S^4 ? (You can add homotopy classes in
pi_4(BSO(3)).)
REAL ANALYSIS (Gunning):
Does a convergent sequence in L^1[0,1] have to converge pointwise a.e. ?
Give a counterexample. Prove that there is a subsequence that converges
pointwise a.e. Fundamental theorem of calculus. Why doesn't it hold for
functions of bounded variation ? I mentioned the Cantor function, then
they asked me to construct it.
COMPLEX ANALYSIS (Gunning):
How can an analytic function fail to be conformal (globally and
locally)? State the Riemann mapping theorem. Why cannot the domain be
the whole plane ? I used Liouville's theorem, then they asked me how
would I prove it.
Then I had to give an analytic function that is not elementary. I chose
the gamma function. I wrote down the functional equation and said it has
poles at the negative integers. Then Gunning asked me to write down an
explicit formula, but I didn't remember exactly. I just said that we can
construct an integral formula by applying Mittag-Leffler to the
logarithmic derivative.
ALGEBRA (Gunning):
How many groups are there of order 8 ? How do you distinguish between
the abelian ones ? How would you prove there are no other groups of
order 8 ?
COMMENTS: I was asked to choose the first topic, so I chose Algebraic
Topology. After this I could pick the next subject, which was
Differential Topology. I highly recommend that if you are offered the
choice, go with the one you know the most, so that you give a good first
impression, and also less time remains for the topics you prefer less.
Talk about things that you know, many times you can lead the
conversation in a favorable direction. The atmosphere was really
relaxed, and everyone was very nice during the exam. If you don't
remember something, they will give clues. And you can get away with not
knowing some things. It's a good idea to talk to your examiners before
the exam about what to learn. And check out their favorite questions
on these pages.
Good luck !