Eduardo Duenez' Generals Committee: P.Sarnak (Chair), D.Christodoulou, T.Hewett. Date: 10-21-97 Time: 9:15 (actually 9:30) Duration: Slightly over 2 hours. Special topics: Analytic Number Theory and Riemannian Geometry. Following Dan Grossman's advice, I arrive with this enormous plastic cup (the "Cool Stuff" one) filled with water, and when Sarnak sees it the first thing he says is "You must have a big bladder!". The exam was scheduled to be held in Fine 314 but (to Steve Miller's great disappointment!) Sarnak suggests that we move to someone's office, so we go to Christodoulou's. COMPLEX ANALYSIS (Sarnak and Christodoulou) S: Draw one doubly connected region in the plane. Now another. Can they be mapped conformally to one another? Prove that such a region can be mapped conformally to an annulus (if no component of the complement in the sphere is a single point...). Now what about triply connected regions? Can they be canonically mapped to a circle with circular holes punched? Cultural parenthesis in which Sarnak says that the Riemann Mapping Theorem isn't actually Riemann's. He asks what the key step in the proof is, so I say Montel's Theorem and he says Riemann actually used the existence of a function attaining the supremum without proving it. S: What is a Green function? Can you have a harmonic function on the sphere with just a logarithmic singularity at infinity? C: Say you have a Riemannian surface (i.e., Riemannian manifold of dimension 2). How can you connect it to complex analysis? (conformal coordinates give it a complex structure, so it becomes a Riemann surface.) S: State the theorem of uniformization for Riemann Surfaces. C: What would uniformization say in connection with the Riemannian surface above? (I thought this was very vague.) S: More specifically, say you have a Riemannian surface, diffeomorphic to the sphere, with a metric g. What does uniformization let you conclude? (we can map it conformally to the Riemann Sphere and pull back the sphere's metric to find a metric g', conformal to g, such that the curvature of g' is identically 1.) At this point our host (Christodoulou) leaves the office and doesn't return until I'm almost done with algebra, which takes maybe half an hour. ALGEBRA (Hewett... also Sarnak) H: Do you know the statement Weddenburn's theorem? (any finite division ring is a field.) Prove it. H: Now give an example of a division ring which isn't a field (Hamilton's quaternions). Define the quaternions. Why are there inverses? S: What is the group of unit quaternions topologically? What does it have to do with SO(3)? Prove a finite subgroup of the multiplicative group of units of a field is cyclic. Can a polynomial over a division ring have more roots than its degree? (there are eight fourth roots of unity in the quaternions.) H: Classify all groups of order eight. Now find the Galois group of x^4-2 over Q. REAL ANALYSIS (Sarnak and Christodoulou) C: Define the total variation of a real-valued function on a closed interval. What can be said about differentiability of such a function? S: Say that F defined on [a,b] satisfies F(x)-F(a) = int_a^x f(t)dt for an L^1 function f. Is F of bounded variation? What is its total variation? What is the relation between functions of bounded variation and monotone functions? C: If a function is of bounded variation on [a,b] is it necessarily equal to the integral of its derivative? (assuming the latter exists a.e.!) (I say Cantor's function is a counterexample, and they make me construct it). S: What is special about functions that DO equal the integral of their derivative? (absolutely continuous). State the Radon-Lesbesgue-Nykodym theorem (for complex measures on R, with respect to Lesbesgue measure). What does mutual singularity mean? Give an example of a measure that's mutually singular to Lebesgue measure (the one associated to the Cantor function). How about an easier example? (a point mass). He then asks if I have ever thought about the Fourier transform of the Cantor function. I say no and he says it is unfair to ask me that question if I have never thought about the matter, but that the answer is neat. He also asks me what the Hausdorff dimension of the Cantor set is. S: What kind of set is the following? {x in R | For infinitely many positive integers q there exists a rational number a/q such that abs(x-a/q) < exp(-q)} Sarnak starts to seem hurried, because he has a seminar at noon, so he asks Christodoulou to do geometry. RIEMANNIAN GEOMETRY (Christodoulou... and Sarnak) C: What is the exponential map? How do you know geodesics exist? Why is the exponential map defined on a neighborhood of the origin of Tp(M)? What are normal coordinates in a neighborhood of a point? What does the Taylor expansion of the (components of the) metric around the point p look like? (with respect to normal coordinates.) What does this say that happens to the lengths of vectors in T(TpM) upon exponentiation? S: State the Gauss-Bonnet theorem (first for two-dim Riemannian manifolds, the he asks if I know a generalization and I state it for arbitrary even-dimensional Riemannian manifolds). What is that form that you are integrating? (the Gauss-Bonnet form). Describe it explicitly (I write it down in terms of Cartan's curvature two-forms). What is the Euler Characteristic of a compact manifold? How can you express it in terms of Betti numbers? At this point Sarnak is more than eager to finish as quickly as possible, so we turn to ANALYTIC NUMBER THEORY (Sarnak, Sarnak, Sarnak!) Talk about the proof of the Three Prime Theorem. Where is N odd used? What is the key step in proving that the minor arcs contribute a smaller amount? (Vinogradov's estimates for the generating function S(N)). What is the really powerful idea behind Vinogradov's method? What happens if you try to adapt the proof to prove Goldbach's conjecture? Here Sarnak decides to stop and they ask me to go out, only to be summoned again shortly thereafter and have Sarnak shake my hand enthusiastically after telling me I pass. I pick up the laaarge cup whose contents I never touched, and leave after thanking them.