Examiners: Rodnianski (chair), Hanselman, Yang
Topics: Riemannian Geometry and PDE
May 22, 2020
Time: 1:00pm to 3:30pm
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Algebra
[H] State Sylow’s theorems.
[H] Pick one part and prove it.
[H] Classify groups of order 21.
[H] What do you know about rational canonical form. How does the
existence of such a canonical form follow from general theory?
[H] State the fundamental theorem of f.g. modules over PID’s, and
outline the proof.
[H] Describe the fundamental theorem of Galois theory, and apply it to x^3 -2.
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Complex Analysis
[Y] Suppose you have a holomorphic function on the punctured disk -
describe possible singularities.
[Y] Is there a similar theorem for classifying singularities of
harmonic functions?
—> I say something about the relation of blow up rates of harmonic
function to that of the green’s function, turns out it’s very wrong.
But it opens a discussion about Green’s functions.
[R] What’s the green function for the laplacian in d=2 and higher
dimensions? Derive it explicitly.
[Y] State the Riemann mapping theorem.
[Y] Can you describe a conformal map from a rectangle to the disk?
[Y] Suppose you have a holomorphic function with norm bounded by a
polynomial power of |z|^k. What can you say?
Proof? Where do derivative estimates come from?
[R] For a complex polynomial in z, what can you say (geometrically)
about the relation of its roots to the roots of its derivative
polynomial?
—> I blank for a while, and R recommends starting with the quadratic
case. The root of the derivative is the midpoint of the polynomial
roots, and we start talking convexity. Turns out the roots of the
derivative lie in the convex hull of the roots of the polynomial.
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Real Analysis
[R] State Fatou’s lemma + show an example where it’s sharp.
—> Example of bumps escaping to infinity.
[R] What do you know about monotonic functions on R?
—>Mention differentiability a.e., types of discontinuities they can
have, that they’re in BV
[R] Talk about BV. Why is it interesting?
[R] You mentioned uses of BV in higher dimensions. How do you define
BV on bounded domains of R^n? How would I relate this definition to
the one involving variation in 1D?
—> I tried recalling the general definition of variation for functions
I had read on Wikipedia, but thankfully we didn’t dwell on this.
[R] People always ask about the Fourier transform - what about
Laplace? Define the Laplace transform. If you were to define a one
sided Fourier transform analogously, what important property of the
Fourier transform would fail? (Inversion) Do you have inversion for
the Laplace transform? (I didn’t know the inversion formula).
[R] On what spaces do you have decay at infinity for the Fourier
transform? Prove Riemann-Lebesgue.
[R] Explicit example of a function in Fourier transform decaying like
1/x at infinity?
—> First tried to pass off a singular function of |x|, but wasn’t what
R wanted. Then remembered that the Fourier transform of the
characteristic function of the interval is the sinc function.
[R] Suppose all moments of a continuous function on [a,b] are zero.
What can you say about it?
—> Zero, by density of polynomials in the space of continuous functions.
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Geometry
[Y]What’s the second fundamental form for a surface in R^3?
[Y]What are the compatibility equations?
—> Wrote down the formulas in Do Carmo, was told to specialize to the
case of surfaces.
[Y]What does the Codazzi equation imply for the derivatives of the
second fundamental form?
—> Y pointed out that this implies totally umbilical surfaces have
constant curvature (*)
[Y] What do you know about conformal maps on R^n, n \geq 3? (there are
only a few types). Can you see a relation with the above question, and
how (*) might suggest a proof?
—> Get to ‘conformal maps map spheres to spheres and planes to
planes’, but we drop the question.
[Y]State the local version of Gauss Bonnet.
[Y]What happens when you sum these terms over a compact manifold?
[Y]Higher dimensional version?
—> Stated Chern-Gauss-Bonnet.
[Y] Version for non-compact surfaces?
[Y] Applications of Gauss-Bonnet?
—> Mentioned how it relates to special cases of uniformization, e.g.
how it’s a key step in showing metrics on the 2d torus are conformally
flat.
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PDE
[R] Were you at the colloquium this week? (I shuddered a little) The
speaker mentioned that the nonlinear heat equation u_t - \delta u +
u|u|^(p-1) = 0 can’t have any singularities. How would you prove this?
—> Start with L^infinity bounds for u. R tells me to think of the
maximum principle, and has me prove a maximum principle for the
nonlinear equation.
—> From there, guided to write the solution using Duhamel’s formula,
and consider the regularity of the heat kernel.
—> Notice I can take strictly less than one derivative of the Duhamel
expression, and the expression remains integrable, so by Young's
inequality we get bounds on derivatives. From there we can use
interpolation/product estimates to bootstrap to higher regularity.
[R] Write down the energy for the wave equation. What’s true about this?
—> Energy conservation.
[R] What about a local energy? Do you still have conservation?
—> Define a local energy for an open set, talk about monotonicity of
energy on the domain of dependence of the set.
[R] Suppose I give you smooth initial data, and a solution u(x,t) to
the linear wave equation. How do you know the energy of u(x,t) in the
ball B(x,T-t) goes to zero as t -> T?
—> I mention that one can use non-concentration estimates, but told
that’s overkill for the linear problem. I list all the facts I know
about the linear wave equation, but they don’t seem to lead anywhere.
—> Guided to consider the regularity of the solution. Linear wave
equation has all higher order energies conserved, so the derivatives
are in L^infinity by Sobolev embedding. Thus energy has to go to zero
as the domain’s volume shrinks.
[R] I claim some type of data is dispersed at t=0, but converges to a
point at t=1. Describe such data.
—> [much confusion] R points out he’s asking about the fundamental
solution, not about smooth initial data, so doesn’t contradict the
previous problem.
—> Mention that one can flow out the fundamental solution to time 1,
then reverse. In particular, the data is localized on the sphere of
radius 1.
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They entered a zoom breakout room for a few minutes, and then they
said I passed!
The actual exam experience went by a lot quicker than I’d anticipated,
and I was fortunate that many of the early questions were straight
from previous generals. The transcript definitely doesn’t show how
often I asked for clarification, or how nervous I was throughout (esp.
when R was asking questions). Not that this advice is applicable to
all committees/all people, but I found questions more comfortable when
I was honest about what I didn’t know, and thought out loud. I think
it was clear to everyone when I was using language/theorems I wasn’t
totally comfortable with. Best of luck to everyone going forward!