Adrian Banner May 7, 1998, 9:30am - 11:30am Committee: Fefferman (c), Sarnak, Ram Special topics: Singular integrals, representation theory of Lie groups While waiting for Sarnak to arrive, I was asked by Fefferman in which order I'd like to do the topics. I was then given a bridge question (that is, bridge the card game) to solve; I was quite bemused by this but it broke the ice somewhat. Shortly afterwards, we began. All questions were asked by Sarnak unless otherwise noted. This is not to say that the other committee members did not contribute to the general discussion. COMPLEX ANALYSIS: State the Riemann Mapping theorem. Why can't you map the whole plane to the disk? Talk about multiply connected domains. How many degrees of freedom in choosing non-conformally-equivalent annuli with slits missing. Hadamard 3-circles theorem, generalize to annuli with slits missing. REAL ANALYSIS: (F) What can you say about convex functions? (When I mentioned Jensen's inequality, Sarnak said that it was on a Danish (?) stamp, whereupon the committee wondered whether Euler was still on the Swiss 10 Franc note. They decided that he wasn't.) Specifically, can a convex function have countably many points of non-differentiability? Could these points have an accumulation point? Could there be uncountably many points of non-differentiabilty? Sarnak took over at this point and asked about monotone functions, functions of bounded variation, absolutely continuous functions. Specifically, a function of bounded variation which is not absolutely continuous. More real later, said Sarnak, but this didn't seem to happen. SINGULAR INTEGRALS: Suppose x is in l^2 (square summable sequences). What can you say about $\sum_{m \neq n} (x_m x_n)/(n - m)$ ? (Hmm.) View it as a singular integral? (Of course, it's related to a discrete Hilbert transform.) Properties of (normal) Hilbert transform. Prove L^2 boundedness, state weak 1-1 and L^p boundedness - how would one go about proving these (no proofs required)? Now replace the (n - m) term in the denominator of the above sum with its absolute value. Do you still get l^p boundedness? How does the Hilbert transform generalise to higher dimensions? (having mentioned Riesz transforms) What are the kernels and multipliers associated to Riesz transforms? It seems as if every singular integral operator we talk about is bounded on L^2 and weak-type 1-1, hence bounded on L^p for 12 by duality). Are there multipliers which are only multipliers for L^2 and no other p? (Yes, Fefferman's result about the characteristic function of the unit ball in R^n, n at least 2.) What about the characteristic function of the unit ball in R^1? Does Fefferman's theorem apply? State the Lebesgue differentiation theorem. What are the main tools used in the proof? (maximal function) Define the maximal function. What are the main results concerning it (statements only)? (At this point, only about 45 minutes had elapsed.) ALGEBRA: (R) Questions for quite some time trying to get me to state the Hilbert basis theorem. Since I didn't really know it, it's not surprising that it took so long for me to work out what was being asked. When asked to think of two big theorems of Hilbert, I mentioned the Nullstellensatz and was asked to state it. What is the nilradical of the ideal I? What is special about elements in the nilradical? (I said nilpotent, general derision from the committee, whoops - I mean, I-potent! That is, some power of each element is in I, of course.) Sarnak said that he'd better ask something basic (since I am supposed to be an analyst, after all! - his words). So: prove that the multiplicative group of a finite field is cyclic. I used the structure theorem for abelian groups along the way, saying that this was perhaps a little weighty, but Sarnak said that it was exactly what he wanted to hear. How does one prove the structure theorem? (modules over PID) What is the module and what is the PID in this case? (R) Talk about conjugacy classes in S_n (symmetric group). The whole committee spent about 5 minutes trying to get me to state that the number of conjugacy classes in a general finite group is equal to the number of irreducible representations, but I couldn't see what they were driving at, looked confused and mentioned things I could think of until I finally stumbled upon it, much to the relief of the committee. REPRESENTATION THEORY OF LIE GROUPS: Sarnak asked what I had studied; I mentioned compact Lie groups and SL(n,R). So: Why are irreducible representations of a compact Lie group finite dimensional? Connection to Peter-Weyl theorem and compact operators (they wanted me to say Fredholm but I couldn't put a name to the theorem - my functional analysis was a little rusty!). State the Weyl character theorem and describe many of the ideas and terms involved (maximal torus, Weyl group, roots, weights, Weyl chambers, integral lattice etc.). During this, Ram asked about the algebraic interpretation of this and I somehow stumbled onto the fact that he was talking about the Artin-Wedderburn theorem about semisimple artinian rings. I stated the theorem. (R) Consider elements in root spaces. What subgroup do these elements generate? (I said that I'd have to think about it, but I guessed that it would be unipotent elements, which is correct.) Talk about the Weyl theory applied to SU(n). In particular, apply the dimension formula in the special case of SU(2). What is Weyl's unitarian trick? (I got this wrong because I genuinely thought it was just averaging over the group. In fact it is the reduction of the study of representations on, say, GL(n,C) to their restrictions to a compact subgroup such as SU(n) and using properties of holomorphic functions to extend them back to GL(n,C).) What is the unitary dual of SL(2,R)? Why does the picture look like this? For what representations is the matrix coefficient (gv,v) in L^2(SL(2,R))? A few other comments on the diagram, a tiny mention of property T and then it was over. I never did answer that bridge question though! GENERAL COMMENTS: The committee was very friendly. They were generally interested in the main ideas and not the minutiae of proofs. I was given considerable freedom to talk about what I knew, especially in the last topic. The questions were sometimes a little vague but I was steered in the right direction on a few occasions. I guessed a few answers, saying "My intuition is that..." and was fortunately correct most of the time. Also, the exam was more like a discussion than a series of unrelated questions.