Student: Jared Anderson
Examiners: MacPherson (chair), Hsiang, Stalker
Topics: Algebraic Topology, Representation Theory
Duration: about 2 hours 20 minutes
Date: October 21, 1997 (10 a.m.)
Real and Complex Analysis
S: Define "meromorphic function".
Give an example of a nonrational meromorphic function.
What are the poles and residues of 1/sin(z)?
S: Is the indefinite integral of a function in L2(R) continuous?
Prove it.
S: Compute the Fourier transform of 1/(1+x^2).
Algebra
H: Explain how the structure theorem for finitely generated modules over
a PID applies to a linear operator on a finite dimensional vector space.
H: Explain geometrically how you diagonalize a quadratic form.
H: Are all extensions of characteristic 0 fields separable?
Of finite fields? Prove it.
Give an example of a field extension that's not separable.
M: What's the Galois group of x^2+1 over Q?
What's the integral closure of Z in Q(i)?
M: Give an example of a PID with a unique prime ideal.
M: Classify real division algebras.
Representation Theory
M: Find the representations of the dihedral group of order 10.
H: Is there something in common between the proof of the Sylow theorems
and the proof of the theorem about maximal tori in a compact Lie group?
H: State Artin's theorem.
H: What is the orbit space of the action of SU(n) on itself by conjugation?
Algebraic Topology
H: Describe CP^n as a CW-complex.
Discuss the Hopf fibration.
Use spectral sequences to compute the cohomology ring of CP^n.
H: How does the intersection of homology cycles in a manifold relate to
the cap product?
About the exam:
There was initially some nerve-racking merriment because Hsiang
forgot all about the exam and had to be fetched. So we got underway about
15 minutes late. I was extremely nervous at the start and blundered about
embarrassingly over the simplest of questions, but my examiners did not
seem to mind too much.
All of my examiners were very nice throughout. Although serious,
they seemed to be relaxed and wanted me to do well. The questions
written above are mostly terse paraphrases of what was actually asked.
When I had trouble with a question, they were helpful, or simply skipped
it if I hadn't studied that area.
When it was over, MacPherson started to say "Perhaps you could leave
the room so we can discuss..." when Hsiang, friendly and efficient, got
the point across by jerking his thumb back towards the door, hitch-hiker
style.