John Anderson Generals 2/9/2017 Time: ~100 minutes Examiners: Klainerman (Chair), Yang, Gunning Special Topics: PDE and Differential Geometry The exam started late because it was snowing. Complex Analysis: [G]: Classify the behavior of holomorphic functions defined in the punctured disc. [J]: I talked about removable singularities, poles, and essential singularities. [G]: Give an example of a function with an essential singularity. [J]: e^((1 / z)) [G]: Talk about conformal mappings from the unit disc to a polygon. [J]: I talked about Schwarz Christoffel from the upper half plane to the polygon. [G]: Boundary behavior? [J]: Piecewise C^1 modulo the point at infinity [G]: Extensions? [J]: I talked about the Schwarz reflection principle. [G]: Favorite nontrivial analytic function? [J]: Gamma function [G]: Properties? [J]: I described the analytic continuation. [G]: If something satisfies this functional equation and is smooth, must it be the Gamma function? [J]: I did not know the answer, but I said that it reminded me of the Holmgren Uniqueness Theorem, so I said maybe you could try to cast this as a PDE and use Holmgren uniqueness. PDE: [K]: Use the Gamma function to talk about the \Chi^s distribution that is applicable to the wave equation [J]: I described the functional equation satisfied by \Chi^s which allows you to analytically continue [K]: How do you use this for the wave equation? [J]: Described [K]: How do you prove the forward fundamental solution is unique? [J]: I was not sure, but I said we could characterize that E_1 - E_2 = 0 in the distributional sense [K]: Uniqueness for the wave equation? [J]: Energy estimates [K]: How do you get energy estimates? [J]: Energy Momentum Tensor. [K]: Properties? [J]: I listed positivity for future directed causal vectors and divergence free properties of Q. [K]: Prove positivity. [J]: I started doing this wrong, so Klainerman helped me through it. [Y]: When is the energy momentum tensor 0? [J]: I didn't know. [K]: You know it has to be 0 outside of the future light cone by the positivity property. Differential Geometry: [Y]: How many quotients are there of S^2? [J]: I did not know, and I barely remembered that the projective space was an example. [Y]: Talk about Gauss Bonnet. [J]: I stated the Gauss Bonnet Formula and the Gauss Bonnet Theorem. [Y]: What about for a noncompact surface? [J]: Inequality instead of equality in Gauss Bonnet [Y]: Higher dimensions? [J]: Written down. [Y]: You can use this to classify quotients of S^n for even n. Yang walked me through this. [K]: Talk about the uniformization theorem. [J]: I gave the main ideas of the proof. Klainerman prodded me some and helped me with some additional points. [K]: Talk about linear elliptic equations on compact manifolds. [J]: Poincare inequality, existence, L^2 estimates. I talked about Ricci terms. [Y]: Bochner identity? [J]: Given [Y]: What is the intuition for the degeneracy of the estimates when Ricci can be very negative? [J]: I was not sure, but I was eventually led to the connection between Ricci and the isoperimetric (Sobolev) inequality Algebra: [G]: Sylow theorems [J]: Stated [G]: Usefulness? [J]: Classification of finite groups. [G]: Classification of finite Abelian groups? [J]: Stated [G]: Matrix normal forms? [J]: Rank is only invariant in general, then JNF, then normal operators and spectral theorem [G]: Example of operator that cannot be diagonalized? [J]: Gave example. Real Analysis: [K]: Favorite result in Fourier analysis? [J]: Calderón-Zygmund. [K]: Describe [J]: Stated. Klainerman preferred a different formulation, but I only vaguely remembered this, so we walked through this other formulation. The committee was extremely nice. Moreover, they did not mind if I did not know something (I knew aboslutely nothing about quotients of spheres). Also, I made mistakes at the board, but they were happy to help me, and I felt very comfortable working through things. They mostly cared about me knowing the main ideas to the results.