Ryan Peckner's Generals Committee: Sarnak (chair), Skinner, Nelson Special Topics: Algebraic number theory, Functional analysis May 12, 2011, 1:30 PM - 4:30 PM REAL ANALYSIS [N]: Suppose you have a sequence of functions on the unit interval converging in L^1. What can you say about pointwise convergence? Much to my humiliation I completely flubbed this one. He was even kind enough to suggest thinking about a subsequence, but I was too flustered to do anything but spew some nonsense about G_deltas. He moved on quickly. [N]: Consider the set of irrational numbers in the unit interval. What can you say about closed subsets thereof? I said that since the irrationals have measure 1, inner regularity of Lebesgue measure implies that there are closed subsets with measure arbitrarily close to 1. [N]: Consider the Fourier transform, except with exp(-itx^2) in the integral instead of the usual expression. What can you say about this function (which he amusingly dubbed the "Furious Transform")? I showed that it belongs to C_0(R) a la usual Riemann-Lebesgue, by showing that the "Furious transform" of a Schwartz function is Schwartz. This prompted Sarnak to have me replace x^2 in the exp with an arbitrary smooth function g(x) whose first derivative is bounded below near 0. I multiplied the integrand by g'(x)/g'(x) and integrated by parts. COMPLEX ANALYSIS [Sa]: State the Riemann mapping theorem. What was Riemann's original (flawed) proof? I talked vaguely about Green's functions, harmonic conjugates, etc. I hadn't studied them in detail but fortunately knew enough to squeak by. [Sa]: Suppose you have a closed semicircle in the plane, and an analytic function on it which is bounded by 1 on the circular arc and 2 on the straight line below. Can you give a better bound than 2 for points inside the region? This led to a discussion of the Hadamard three circle theorem. Ultimately I realized that I had to construct a harmonic function on the semicircle which is constant on the arc and on the line, but I had no idea how to do this. Sarnak told me that the answer is the angle subtended by the point and the radial segments. ALGEBRA [Sk]: How can you tell when two matrices over a field are conjugate? Routine discussion of rational canonical form, modules over a PID, etc. [Sa]: For what matrices B can you solve exp(A) = B? [Sa]: Tell me about representations of finite groups. I basically said everything I know (which isn't much): Maschke's theorem (with proof), Artin-Wedderburn, representations of D_8. ALGEBRAIC NUMBER THEORY [Sk]: Consider the field Q(sqrt(5)). What can you tell me about the behavior of 7 in the ring of integers? I reduced the minimal polynomial of (1 + sqrt(5))/2 (mod 7), figured out there were no zeros, hence 7 is inert. [Sa]: You got lucky because 7 is small. What if we take a much larger prime? I screwed up a bit in stating quadratic reciprocity and was corrected. Sarnak asked me to prove it using anything I wanted. I gave the cyclotomic fields proof, which Sarnak liked. [Sk]: You said the Galois group (of Q[zeta_q], q a prime) is (Z/qZ)*. What element of this group does the Frobenius of p define (p another prime)? [Sk]: A famous theorem of Dirichlet... He was of course asking about the one on arithmetic progressions. Sarnak then asked if I knew Tchebotarev density. I stated it and got about three-quarters of the way through the class field theory proof before they stopped me. [Sa]: Give a one-line proof that infinitely many primes split completely in an arbitrary extension of number fields. The Dedekind zeta function has a pole at 1. FUNCTIONAL ANALYSIS [N]: Suppose we have a bounded operator on a Banach space. What is its resolvent set? Why is it unbounded? Why isn't it the entire plane? [N]: What kind of continuity conditions should one impose on a one-parameter group of unitary operators? What else can you say about it? I said that strong and weak continuity are equivalent, and stated Stone's theorem. This led to a discussion of the spectral theorem for unbounded self-adjoint operators. [N]: What property do all unbounded self-adjoint operators have? (They're closed.) What does essentially self-adjoint mean? Give an example of a symmetric operator that isn't essentially self-adjoint. I gave id/dx acting on C_0^1(R_>0) inside L^2(R_>0). [N]: Take the Laplacian acting on L^2(R^n). What multiplication operator is it unitarily equivalent to? It took me an embarassing amount of time to realize I should use the Fourier transform to get (x_1)^2 + (x_2)^2 + ... + (x_n)^2. [Sa]: Let's talk about unitary representations of locally compact groups. I started talking about Pontryagin duality but was cut off. [Sa]: Take a compact Lie group. How do you exhibit an irreducible finite-dimensional representation? I said Peter-Weyl but he wasn't satisfied. [Sa]: No, I want you to show me an actual representation. I started with the regular representation of G on L^2(G). They agreed that it was completely reducible but wanted to know why the summands were finite-dimensional. I said that they're the eigenspaces of a compact operator, which I wrote down (convolution). They asked me what the name for such operators is in general. I said "integral operator" and they weren't amused. After thirty excruciating seconds of not knowing what they wanted I finally blurted out "Hilbert-Schmidt", much to their relief (and mine). I was sent into the hallway to await my fate. A minute later Sarnak came out and congratulated me. ----- Some words of advice: You will probably freak out on numerous occasions while preparing for this exam. That is natural. However, try to bear in mind that the vast majority of students who take generals pass them, and that the professors truly are on your side in this. No one wants or expects you to fail. Don't obsess over details of proofs of minor theorems and lemmas. Your committee won't expect you to know everything (not even close). That said, you should have a good understanding of the important elements of the proofs of the major theorems in your special topics. The big picture matters much more than technicalities. Finally, you will probably find yourself feeling much better during the exam than you had for several weeks (or months) preceding it. My committee was very friendly and helpful, and didn't seem to mind even when I made some really ghastly mistakes. Relax. You're going to be fine.