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.