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]: CalderÃ³n-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.