Examiners: Rodnianski (chair), Katz, Yang
Topics: Riemannian Geometry and PDE
May 17, 2018
Time: 195 minutes (including 15-minute tea break).
They spent a minute talking about the furniture in Rodnianski's office, then decided to start with general topics.
Algebra:
K: What can you say about Galois theory? What does Gal(K/F) mean? What does it mean for an extension to be Galois? Can you give a non-Galois extension of Q? (I said Q(2^{1/3}). Can you give a non-Galois extension of Z/pZ (No, Frobenius automorphism).
K: What is the Jordan canonical form? What does this have to do with classification of abelian groups? (He was happy as soon as I said the words "Finitely generated module over a PID")
Complex analysis:
K: What does it mean for a function to be holomorphic?
I foolishly said that it's the same definition as real differentiability, which caused a digression about real-differentiable non-holomorphic functions on R^2.
Y: Can you write down the Cauchy-Riemann equations?
K: What do you need to say about a function that satisfies these to show that it's holomorphic?
I said C^1 is enough, mumbled stuff about elliptic regularity and tempered distributions.
R: So what can you say about the functions u and v (they're holomorphic)? What regularity do you need on a holomorphic function? What holomorphic functions are tempered distributions (I wasn't sure)?
Y: So assuming u and v are C^1, can you prove u+iv is analytic?
I eventually got an outline of the proof.
Y: Classify isolated singularities.
Y: What about for functions harmonic except at a point? What happens if they grow slowly near that point?
K: Who was Green?
Real analysis:
R: what is BV? Why do we care about it? Can you prove that a BV function has a derivative? Just do it for continuous BV functions.
I fumbled around with trying to prove that increasing functions are almost everywhere differentiable until Rodnianski said that it was close enough
R: does the fundamental theorem of calculus hold for BV (no, Cantor's function)? For what functions does it hold?
K: What is the history of the concepts of continuous and differentiable functions (I mumbled something about Weierstrass's example, and not the modern idea of a function being relatively new)?
Y: Can you state the Sobolev inequalities? What is this inequality called when p=1?
I didn't know. Eventually they had me write the Sobolev inequality for p=1 and the isoperimetric inequality as I knew it and asked me how they relate.
R: What about Sobolev inequalties on manifolds? What about Sobolev spaces with non-integer numbers of derivatives (I started talking about Littlewood-Paley and using Bernstein-type inequalities, all he wanted was interpolation). On a bounded domain in R^n, what else can you say (he wanted compactness of the embedding)?
I thought we had already finished complex analysis, but:
K: speaking of compactness, what does this have to do with the Riemann mapping theorem (I sketched a proof)? When was it proven?
Riemannian Geometry:
Y: What is the mean curvature? What is the second fundamental form? What type is it? What can you say about constant mean curvature surfaces? How would you find a constant mean curvature surface with a given boundary (I didn't know, this question would haunt the rest of my exam)? What variational problem does constant mean curvature correspond to (I didn't know, eventually they told me it's maximizing volume)?
Y: Write down the constant mean curvature equation. What type of equation is it (meaning elliptic quasilinear)?
Y: What examples of constant mean curvature surfaces do you know? How do you use these to find one with a presribed boundary?
I think this is where we took a tea break.
Y: State Gauss-Bonnet. Just give the version for triangles.
K: What does the right hand side mean?
Then they had me define curvature, covariant differentiation, etc.
K: So take a torus in R^3. What does this formula mean for it?
R: But how can you have different curvature at different points? It's a torus. You can just rotate it to get one point to another
K: So what does this formula mean for a sphere with g handles? Why do we care, since I already know the genus (I had no good answer to that one)?
Y: How can you see the Guass-Bonnet formula for the torus in R^3 (Gauss map, pick a convenient direction that has two points)?
K asked some question about degrees of vector fields and how they relate to Euler characteristic.
They also asked me some stuff about DeRham cohomology which I didn't know.
PDE:
R: Do you want me to ask a question about fluids?
I didn't have strong opinions either way.
R: Write down the incompressible Euler equation. If you have a very deep ocean, say it's irrotational, and u is initially 0, what can you say about it?
I did the energy estimate, then he wanted me to include the boundary terms at the bottom and the top.
Then they asked me some more stuff about solving elliptic equations like the constant mean curvature equation.
R: How do you solve quasilinear or semilinear hyperbolic equations? Why doesn't iterating that work for elliptic equations?
The answer he apparently wanted was that you can do things locally in time, but that doesn't make sense for elliptic equations.
R: What quasilinear hyperbolic equations do you know?
I said Einstein equation.
R: Yes, but what sort of nonlinearity does it have? While the constant mean curvature equation has Du^2D^2u.
I said that you have a minimal surface/constant mean curvature equation in Lorentz space. Then he told me that compressible Euler is hyperbolic and has a similar type nonlinearity.
They sent me out, then congratulated me a few minutes later.
I probably forgot some stuff. A lot of the time I wasn't sure what answer they wanted and would say various things until I said the right keyword or until they wer happy. Apparently Katz likes historical digressions.