Committee: Fernando Codá Marques [M], Peter Constantin [C], Gabriele Di Cerbo [D] April 28, 2021 1:30-4:30pm Riemannian Geometry [M] Talk about curvature. I wrote the definition. [M] What is the connection appearing in the formula? I defined a connection and talked about the Levi-Civita connection. [M] What does the space of connections look like? I showed that the difference of two connections is a (1,1) tensor. [M] Talk about a curvature theorem. I wrote down Synge-Weinstein and the proof in even dimensions. [M] Use Synge-Weinstein to give examples of manifolds not admitting positive sectional curvature. I talked about RP^n \times RP^m. [M] Does T^n admit positive sectional curvature? I said that Synge-Weinstein takes care of even dimensions. After a digression about the second variation of area, he said there is a more elementary argument. I said that positive sectional curvature implies finite fundamental group. [M] What theorem are you using to say positive sectional curvature implies finite fundamental group? I wrote down Bonnet-Myers and said the fundamental group is finite because the universal cover is compact. [M] Can you prove the fundamental group is finite using Bonnet-Myers without appealing to covering space theory? I said that if the fundamental group is infinite, then there is a sequence of minimizing geodesics in distinct homotopy classes. Eventually I realized that this sequence has uniformly bounded length by Bonnet-Myers, so there is a uniformly convergent subsequence. Real Analysis [C] What is your favorite integral convergence theorem? I wrote down the monotone convergence theorem. [C] Why do the functions need to be nonnegative in the monotone convergence theorem? I said you could take negative constant functions converging to 0. [C] Bound the L^p norm of the convolution f*g. In what spaces should f and g live? I wrote down the L^{\infty} bound using Holder. He guided me through a derivation of the p < \infty bound. [C] Give the idea of the proof of Fourier inversion. I wrote down the statement and said you introduce a Gaussian and use Fubini. Algebra [D] State the Sylow theorems. I wrote them down. [D] What is your favorite application of the Sylow theorems? I said you can classify groups, but my favorite application is the proof of the fundamental theorem of algebra using Galois theory. [D] Classify groups of order 30. I wrote down the possible numbers of Sylow 2, 3, and 5 subgroups. He walked me through the case analysis thereafter. [D] Prove the fundamental theorem of algebra using Galois theory. I wrote down the proof. [D] If F/K is Galois and E is intermediate, why is F/E Galois? After he suggested using the normality/separability characterization instead of the Galois group characterization, I wrote the proof. [D] If F/E is Galois and E/K is Galois, is F/K Galois? No. I gave the example Q(2^{1/4}) over Q(2^{1/2}) over Q. [D] Give an example of a non-separable extension/is every quadratic extension separable? I wrote down (x-a)^2 = x^2 – 2ax + a^2, and he pointed out I was done (characteristic 2 versus characteristics not 2). Complex Analysis [M] Define complex differentiability. I wrote down the definition and the Cauchy-Riemann equations. [M] What is the geometric meaning of complex differentiability? I said it is related to conformality. He asked me to write down the precise relation. [M] Talk about the convergence of holomorphic functions. I said the uniform limit of holomorphic functions is holomorphic. I proved it using Morera’s theorem. [M] When can you extract a convergent subsequence from a set of holomorphic functions? I said the functions should be uniformly bounded. I proved it using the Cauchy integral formula. [M] What can you say about the Dirichlet energies of a sequence of holomorphic functions converging uniformly on compact subsets? He suggested I use Fatou, which gives lower semi-continuity. He asked if there is a counterexample to continuity. He guided me to think about Blaschke factors on the unit disk. [C] In one word, what would you do to prove the fundamental theorem of algebra using complex analysis? I said Liouville. PDE [C] Consider the equation u_t + v(t,x)u_x = 0 with given initial data. What norm is preserved? He suggested I write down the characteristics. I wrote them and said that the solution is constant along characteristics. Hence, the L^{\infty} norm is preserved. [C] Talk about Burger’s equation: u_t + uu_x = 0. I said that shocks can form from intersecting characteristics. He asked me to draw the picture. [C] Consider the equation u_t + (K*u)u_x = 0 for K = e^{-x^2}. Is there global existence? If the initial data is compactly supported, does u remain compactly supported? He asked what I could say if K*u is bounded. I said you get a bound on the speed of the characteristics, so it remains compactly supported. Then he asked how to bound K*u (i.e. see the real analysis section), and what approach I would use to show existence. He said he was looking for the word “approximation” or “iteration”. [C] Consider the equation -\Delta u + u + u^5 = f on R^3. How would you solve it? Are such solutions unique? I wrote down the energy and eventually showed it is lower semicontinuous. He reminded me that uniqueness follows from convexity of the energy. [C] Consider the equation u_t - \Delta u = f, u(0,x) = 0 on R^n. How would you solve it? Can you control the L^p norms? What about the L^p norms of the spatial derivatives? I wrote down the convolution with the fundamental solution. He reminded me that I could use the convolution inequality from before. I wrote down the derivative and thought about the L^1 norm of the new kernel. Disclaimer: Much hemming and hawing was omitted from the transcript.