John Stogin's generals -- May 2013 Committee: Klainerman (chair), Warren, and Gunning Topics: Hyperbolic PDE and Riemannian Geometry Total time: 1hr 45min Note: For the questions I didn't answer right away, I tried to include the hints I received so that you can have a better idea of what is actually expected on the exam. Warren and I were waiting outside Klainerman's office when Gunning showed up and muttered something like, "let's pass him quickly and get out of here!". Soon after, Klainerman showed up and let us in to his office. ////////////////////////// I chose to start with Complex Analysis. Gunning: Classify singularities of a holomorphic function on the punctured disk. Provide examples. How does the principal part relate to these singularities? [I wasn't sure if having infinitely many terms in the principal part meant there was an essential singularity, so Klainerman suggested I redefine essential singularity to be precisely that.] What properties determine these singularities? Prove Casorati-Weierstrass. Gunning: State precisely the Riemann mapping theorem. Why does it not apply to the entire complex plane? Gunning: What is your favorite special meromorphic function? [gamma function] What are its properties? How do you extend it? Can you plot it along the real line? ////////////////////////// Then Complex Analysis turned into (elliptic) PDE theory Klainerman: What properties of holomorphic functions can you conclude from PDE theory? [Cauchy integral formula, maximum modulus principle, reflection principle, Morera's theorem] Continue discussing the maximum modulus principle. Does this generalize to other operators? What is the statement precisely? [I stated the strong maximum principle for elliptic operators.] Can you prove it? [I did, but forgot a detail or two which they provided.] Warren: Suppose you have a positive harmonic function on the ball of unit radius such that the function takes values less than or equal to 1 on the boundary. Can you bound the oscillation of this function on the ball of radius 1/2? [It didn't automatically occur to me that he was referring to Harnack's inequality, so I started proving some pointwise bound for the magnitude of the derivative instead. Then he reminded me the meaning of the word oscillation and I realized he was talking about Harnack. They asked me to sketch a proof.] Klainerman: Talk about estimates for solutions to Poisson's problem. [I mentioned the trivial estimate for the homogeneous H^2 norm and also the Schauder estimates. I think he wanted me to state a more general Sobolev inequality.] What if you have \triangle u = dfdg (in 2 dimensions) and f,g are in H^1? Is u in L^\infty? [umm...] Try working it out. [I bounded dfdg in L^1 without too much trouble. Then I assumed I could bound u in W^{2,1} by \triangle u in L^1 (which turns out to be impossible) and wondered whether W^{2,1} embeds in L^\infty. Since I didn't know, I was asked a few simple additional questions.] Does the inequality (embedding W^{2,1}\subset L^\infty) scale appropriately? [yes] Can you prove this for 1 dimension? [yes--fundamental theorem of calculus] Ok. So repeat the proof for 2 dimensions. [Ok. so the embedding is true.] Now, are you willing to bet that you can control u in W^{2,1} by \triangle u in L^1? [I said yes.] Actually you can't. [oops. ok] Klainerman: Ok. Same problem as before except what if you have instead *(df\wedge dg). Can you conclude something then? [I observed this was Wente's inequality from his lecture notes (so yes)] Can you prove it? [I said I wasn't familiar with the proof. A smile formed on Klainerman's face...] Ah--you should have come to class yesterday! [Everybody laughed. He didn't seem to really care that I couldn't prove it.] Ok. Let's move on to Algebra. ////////////////////////// Algebra Gunning: What is the structure theorem for finite abelian groups? How about for finitely generated abelian groups? Gunning: State the fundamental theorem of Galois theory. What if the extension is not Galois? [I gave the standard example Q adjoined cube-root 2 and said its automorphism group is trivial. That was sufficient.] Gunning: Groups of order 15? State the Sylow theorems. Gunning: What special forms can be used to show similarity between matrices? [I stated Jordan canonical form and then my mind completely blanked on what the second was called. (I knew this would happen at least once during the exam!) Gunning said the word "rational" quietly, and then I mentioned the rational canonical form.] Define them. Gunning: How can you quickly tell if a matrix is positive definite? [He wanted Sylvester's theorem, but I didn't know what that was.] Ok. I'm satisfied. Let's do some Geometry. ////////////////////////// Geometry Warren: With a lower bound on Ricci curvature, bound the volume of a ball by the volume of a ball with twice the radius. [I didn't know how to do this, so Klainerman suggested we do something more closely related to general relativity.] Warren: If sec<0 and dim=3, then show a stable hypersurface has negative curvature. [I wrote down Gauss's equation, and worked it out after much mumbling and a few hints. After I solved it, they asked for a simpler observation which had me stuck for quite a while despite their very straightforward hints: "how does /that/ expression [the determinant of sectional curvature] relate to /that/ expression [two terms in Gauss's equation] for a 2D hypersurface in an orthonormal frame?" (they're equal)] Klainerman: How about some general relativity? Tell us about the initial value problem for the vacuum. [I began by describing initial data.] What are the constraints? [I wrote down the constraints and then began talking about the need to choose a gauge. I asked if they wanted proof of propagation of the gauge and Klainerman said yes. In the proof, I mistakenly said div(ric)=0, so they asked me to compute what it actually is. I then clarified my statement (it is true in vacuum, but I couldn't assume that yet.) When I wrote down Bianchi's 2nd identity, Klainerman said that was enough.] Step outside and we'll decide. Just don't jump out of any windows or anything. ////////////////////////// After a minute or two, they stepped outside and congratulated me. Interestingly enough, there didn't seem to be any real analysis topics, and the "hyperbolic PDE" topic seemed a lot more like elliptic PDE. Advice: It seems there are generally two types of questions that are asked. Some are very straightforward (eg. define Jordan canonical form) and you should be able to answer most of these (Klainerman told me this a while ago). But obviously you don't have to know all of them (eg. I didn't know Sylvester's theorem). Then there are those which require you to do some more serious thought, but the committee will guide you through the parts where you get stuck. Good luck!