My committee: Peter Sarnak (chair), Charles Fefferman, Peter Ozsvath.
My subjects: Algebraic Topology, Representation theory.
Me: Alessandra Pantano. (May, 23 2001)
COMPLEX
Since I probably looked a bit scared, they let me choose which subject I
wanted to start with. I picked complex.
The first question was (of course!) about the Riemann mapping theorem (F):
its proof and its extension to the boundary. Sarnak asked me about the
number of degrees of freedom that you have in choosing the conformal
equivalence. This brought us to talk about the Moebius transformations of
the disc (the group is SL(2,R) / {+I,-I}). Then we discussed very quickly
the case of polygons (I mentioned the existence of the Schwartz
Christoffel formula, although I couldn't remember the actual formula) and
we moved to Montel's theorem.
Next question (F & S): take a sequence of holomorphic functions converging
uniformly to some function f. Is f holomorphic? (yes, for instance you
can prove it using Morera's theorem).
What happens in the real case? (the uniform limit of C-infinity functions
need not be differentiable. Counterexample: any continuous function on {0,1}
can be approximated by polynomials in the uniform norm. Use
Stone-Weierstrass.).
Then Sarnak asked me to talk about "some" theorems about entire functions.
I chose small Picard, and I proved it. I mentioned that the existence of a
conformal covering map from the disc to the plane minus two points can
also be justified using the Riemann uniformization theorem. [This result is
pretty neat: up to conformal equivalences, there are only three simply
connected Riemann surfaces (the sphere, the plane, the disc.) Now ask
yourself of which Riemann surfaces are these the conformal universal covering.
The sphere only covers the sphere itself. The plane covers the plane, the
plane minus one point and the torus. The disc covers anything else].
Then I gave an explicit construction of the covering using the Schwartz
Principle and this opened the doors to TOPOLOGY.
I was asked to compute the group of transformations of the covering I had
just constructed (O). [This question seemed hard to me, but it was not,
and they gave me some hints. Just use the fact that the group of Deck
transformations of the universal covering is isomorphic to the fundamental
group of the space that is covered, which in this case is simply Z*Z].
At this point Sarnak asked the others if they wanted to go on with Riemann
surfaces or to move to move to real analysis. They said it was enough with
complex, so we moved to real.
REAL ANALYSIS
Now it was Fefferman's turn. He asked me to write down an integral (which
turned out to be the Gamma function) and we played for a long time with it,
basically using many times the theorems of passing to the limit and
differentiation under the sign of integral. Actually, I had not studied the
integral definition of the Gamma function, but they were very kind and they
helped me to work things out. Same thing for the Laplace transform.
ALGEBRA
The algebra session was very short: just three questions, and all about
finite fields. No more than 5 minutes, I would say.
1. What's the field with 25 elements?
[it's the splitting field of x^25 - x over Z / 5Z ].
2. What's the structure of its multiplicative group?
[it's cyclic, like for any finite group (with proof)]
3. What's the Galois group of the extension F_p^n , F_p ?
[it's cyclic, generated by the Frobenius authomorphism].
We took a little break before passing to the special topics.
ALGEBRAIC TOPOLOGY
Algebraic topology came first. [Most questions were asked by Ozsvath].
We first talked about the nice connection between graphs and free groups:
the fundamental group of a graph is free, and given a free group, there's
a graph whose fundamental group is the given free group. This is used to
prove that a subgroup H of a free group G is still free. Main idea: G is
the fundamental group of some graph T, so its subgroup H is the fundamental
group of some covering space of T, which is still a graph. Hence H is free].
The following question was then to determine a formula for the index of such
a subgroup H. I actually had to struggle a bit, but eventually found a
formula that uses the Euler characteristic of the corresponding covering
graph.
Next matter: describe SU(2) and SO(3) topologically. I gave a topological
proof of the fact that SU(2) is a double covering of SO(3). [SU(2) is simply
the sphere S^3, while SO(3) is RP^3, which is basically S^3 with the
antipolar identifications]. Then I computed homology, cohomology (with
coefficients in Z and Z / 2Z), fundamental group and higher homotopy groups
of SU(2) and SO(3). [Use the theory of covering spaces once again].
We were naturally brought to talk about higher homotopy groups of spheres
and I mentioned the Hopf fibration.
That was basically all with Algebraic Topology, so we moved to
Representation Theory.
REPRESENTATION THEORY
Sarnak asked me very general questions in Representation theory:
1. Tell me all you know about representation theory of finite groups.
2. Tell me all you know about representation theory of compact groups.
It was very nice because they just let me explain the whole theory without
further questions.
We worked out specific examples (D_4 and SU(2)) and then we moved to
SL(2,R), my favorite group.
First question: are all the irreducible representations unitary?
[No, the non trivial finite dimensional representations are not
unitrizable. Prove it, for instance, by showing that the matrix associated
to the corresponding representation of the Lie algebra is not skew-Hermitian].
Then a more open question: talk about the unitary dual. I described the
principal, complementary, discrete... series, specifying the unitary,
irreducible, inequivalent representations in each series. It seemed to me
that the thing they mostly cared about was the final picture of the dual.
Two more words about the classification of unitary irreducible
representations in terms of matrix coefficients and spherical functions, and
the exam was over. Finally!
GENERAL COMMENTS
The exam was, I guess, shorter than 2 and a half hours. At the end, they
seemed to be happy and I was even happier.
I believe I was lucky to have such a kind committee: they were very friendly
and often smiling. This helped me so very much to be calm and think properly.
But above all I should thank all my friends who gave me constant help during
the preparation of the exam.