2 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.