Yakov Shlapentokh-Rothman's Generals
Special Topics: PDEs and Riemannian Geometry

May 9, 2011

Chair: Rodnianski (R)
Yang (Y)
Sorensen (S)

Algebra:

(S) What are the Gaussian integers?

(S) What are the primes in the Gaussian integers?

I couldn't say anything intelligent about this at first and was slowly
guided to conclude that for a prime (not 2) to split in Z[i], it must
be 1 mod 4.

(S) What is the symmetry group of a tetrahedron?

(S) What is the symmetry group of a cube?

(S) What is the symmetry group of an icosahedron?

These questions went terribly, for example I have no idea what an
icosahedron looks like. Eventually we moved on to

(S) How many ways can you color the tetrahedron with C colors if we
identify symmetric colorings?

I noted that we needed to count the orbits of the S_4 action.

I wrote down Burnside's lemma and after some difficulty counting
elements in conjugacy classes, determined the answer.

(S) When do conjugacy classes in S_n split in A_n?

(S) What is a perfect field and why is this important?

(S) Give an example of a non-perfect field.

(S) Describe the Galois theory of cyclotomic extensions.

(S) What is the maximal real field in a cyclotomic extension Q(z_n)/Q?
I had no idea how start here. It was suggested that I consider the
field Q(z_n + z_n^{-1}).

After some hints, I realized I could produce a degree 2 polynomial
which z_n + z_n^{-1} satisfies.

(S) What are the radical ideals in Z?

(S) Describe the nullstellensatz and sketch a proof.

Complex Analysis: This really was my "complex analysis" section!

(Y) State and prove the maximum principle for holomorphic functions.

(Y) Does this also hold for harmonic functions?

I stated the weak and strong maximum principles and proved the weak
maximum principle.

(Y) Does this hold for a wider class of operators?  I explained how
the previous statements generalize to elliptic operators.

(Y) State and prove the Hopf Lemma.

Real Analysis:

(R) What are approximations to the identity?

I talked about convolutions with rescaled bump functions and gave half
of a proof of the relevant properties via approximation by smooth
functions of compact support and the fact that the support of the
rescaled bump functions was shrinking.

Rodnianski didn't like my proof and suggested a different proof using
Fubini.

(R) Can you have approximations to the identity that do not have
	compact support?  I blanked on this. Eventually the Poisson
	kernel and Gaussian were suggested to me.

Geometry:

(Y) What is Gaussian curvature?

I gave a definition in terms of the Gauss map.

(R) What about a more intrinsic definition.

I described how one gets a Riemannian connection from a metric and
gave the definition of sectional curvature.

(Y) Given a metric and second fundamental form, when can you find a
surface in R^3 with these?

I briefly explained the fundamental theorem of surfaces.

(Y) Do you know generalizations of this?

I vaguely remembered reading something about this and claimed that you
needed one more compatibility condition. This seemed to satisfy Yang.

(Y) What about embedding 3-mflds into R^4?

I had nothing to say about this and Yang told me something about
locally conformally flat beging a relevant property...

(Y) Does there exists a negative Ricci metric on S^4?

I recalled reading somewhere that there were no topological
obstructions to negative Ricci curvature. So...yes? This was correct
as it turns out.

(Y) Does there exist a negative einstein metric on S^4?

I had no clue how to proceed. After a minute Yang told me this was an
open problem.

(R) What are normal coordinates?

(R) What is special about exponential coordinates?

I really wasn't sure what he was after. I blabbered out everything I
knew about exponential coordinates while managing not to satisfy
Rodnianski.

Later in the exam, after some prodding, I realized he wanted me to
Taylor expand the metric in exponential coordinates and note that it
was euclidean to first order and that curvature dictated the second
order terms.

(R) and (Y) During the above stream of consciousness we got
sidetracked, and I was asked to produce a theorem involving the
injectivity radius.
 
Theorems related to the sphere theorem and positive curvature were
rejected. After a painfully long time I was led to conclude that
volume bounds from below, diameter bounds above, and curvature bounds
from above were enough to control the injectivity radius.

(R) What do you know about the exponential map and negative curvature?

I said that the exponential map was a covering map. I briefly
explained that this followed from expressing the derivative of the
exponential map in terms of Jacobi fields and then using the Jacobi
equation to show that no conjugate points occur. Rodnianski was not
interested in the details.

(R) Does the sphere admit a metric of negative curvature?

PDEs:

(R) What can you say about the long term behavior to solutions to the
homogeneous wave equations with smooth functions of compact support.

I mentioned conservation of energy, L_{infty} - (1+t)^{-(n-1)/2} L_1
bounds, Klainerman-Sobolev estimates for the derivatives, and Huygen's
principle.

(R) How do you know Huygen's principle holds?

I said that in theory one can write down a solution to the wave
equation using spherical means and then it is apparent. Luckily I was
not asked to carry this out.

(R) Draw a picture for Huygen's principle. This took me way too long
to do satisfactorily...

(R) How would you define the wave equation on the sphere?

(R) Do you have decay?

I showed how you can solve the wave equation via separation of
variables. From this it is clear that there is not decay.

(R) Do you know the eigenvalues of the Laplacian on the sphere?

I didn't.

(R) What about the wave equation on a compact manifold?

I wrote the solution via an expansion into eigenfunctions of the
Laplacian.

(R) What can you say about long term behavior?

With the formula from the previous question I showed that L^2 norms of
arbitrary time derivates and powers of the Laplacian were controlled.

Then via elliptic estimates we can control all derivatives in L^2.

(R) What about L_{infty} bounds?

Since it is compact we can cover with finitely many coordinate charts
and sobolev on each chart.

(R) Does this remind you of anything we talked about earlier?

Given volumes bounds below, diameter bounds above, and curvature
control we can estimate how many coordinate chart are needed.

(R) What about the heat equation on a compact manifold?

I wrote the solution down in terms of eigenfunctions and mentioned
that the solution will decay to a harmonic function, i.e. a
constant. The decay rate is governed by the first non-zero eigenvalue.

And that was it.