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.