Whom: Heather Macbeth
Who: Alice Chang (chair), Gang Tian, Alireza Salehi Golsefidy
What: Differential geometry, elliptic partial differential equations
When: 3 May 2011, 11 am - 3.40 pm
My committee gathers gradually from 11 am. Once everyone has arrived,
they send me out for a couple of minutes so they can plot strategy.
**Algebra**
[SG] What can you say about a linear transformation which preserves a
nondegenerate alternating form? [This mildly nonstandard question is
asked because my examiners know I like symplectic geometry. They are
after the fact that if k is an eigenvalue then so is 1/k.
Unfortunately, I am in no mental state for mildly nonstandard
questions. After half an hour of patient prodding, I've done the
dimension-two case, been asked (for motivation) about the proof of the
Diagonalization Theorem, and shown that if k is an eigenvalue of such
a linear transformation then the transformation factors through to the
map [multiplication-by-1/k] on the quotient by the symplectic
complement of a k-eigenvector. My committee then decide they are sick
of the problem (so am I! Although by this point we are closer to the
solution than I realize) and we move on.]
[SG] What are the possible images of the unit circle under a linear
transformation of R^2?
[SG] What is the character table? What field do its entries lie in?
What can you say about an element x of a group G, such that every
character of G is real on x?
[SG] What are the irreducible representations of finite abelian
groups?
[SG] Given a normal subgroup H of a finite group G, we lift all the
representations of G/H to representations of G. Show that the
intersection of the kernels of all these representations is precisely
H. What can you say when H is the commutator subgroup of G? [On this
last part I have no idea.]
**Complex analysis**
[C] What does it mean for a function on R^2 to be real-analytic? Why
is a holomorphic function real-analytic?
[T] What is Schwarz' lemma? Prove it. What are the complex
automorphisms of the disc?
[C] What are the possible images of a circle under a Moebius
transformation? Exhibit a conformal map from the upper half-plane to
the disc.
[T] Can you have a holomorphic function from the plane to the
plane-minus-the-origin? To the plane-minus-two-points? Prove the
result [Liouville] you just used.
[SG] Prove the Fundamental Theorem of Algebra. Prove the Argument
Principle.
**Real analysis**
[C] Talk about the dual of L^p. Why is the (L^p)-dual norm of an
element of L^{p/p-1}, the same as its L^{p/p-1} norm? [I give a proof
using the fact that (f, f^{p-1}) gives equality in the (p, p/(p-1))
Hoelder's inequality.] What pairs of functions give equality in
Hoelder's inequality? [With a hint from Alireza, I do the L^2 case.
We move on before I can think much about the general case.]
[C] What is the dual of L^1? Why isn't L^1 the dual of L^\infty?
What is the dual of L^\infty?
[T] A sequence of continuous functions converge pointwise. Is their
limit continuous? What is a condition you can impose to make the
limit continuous? [Me: uniform continuity.] What is a condition you
can impose on a sequence to ensure the existence of a uniformly
continuous subsequence?
[T] What is the measure of a countable set? Can an uncountable set
have zero measure? [Me: Cantor set.] [SG] Do you know a variation on
the Cantor set that has positive measure? [I don't, so they tell me
and have me prove it.]
[SG] Show that if A is measurable then A + A contains an interval.
[SG] Suppose given a group-homomorphism from the reals to the reals.
[I get excited, because I know the axiom-of-choice construction of
lots of such group-homomorphisms.] Show that if it is measurable then
it is linear. [I have no idea.]
Next we do the "equidistribution of fractional parts of n\alpha"
problem that Alireza had previously asked on Guangbo's generals: given
an irrational number a, and a continuous function f on the circle, to
show that as N tends to infinity the average of f on the set {e^{2\pi
a}, e^{2\pi 2a}, e^{2\pi 3a}, ... e^{2\pi Na}} tends to the average of
f on the circle. The hint is that this is easily proven for f(z)=z^k.
[SG] What is the Stone-Weierstrass theorem?
Tian has a doctor's appointment, so we break for lunch.
**Elliptic partial differential equations**
[C] Tell us some properties of harmonic functions.
[C] State Harnack's inequality. Show that the constants blow up as
the smaller open set grows to fill the bigger open set. [I give an
example where the function has a singularity on the boundary of the
bigger open set.] [T] Give an example continuous up to the boundary
of the bigger open set.
[T] Suppose given a harmonic function function on the disc, continuous
up to the boundary. Suppose that on some open subset of the boundary,
it and its normal derivative vanish. Show that the function vanishes
everywhere. [While tossing out ideas in trying to prove this, I
mention Hopf's lemma.] What does Hopf's lemma say? Can you prove it?
[We give up on the boundary-vanishing function question.] Okay, what
if a harmonic function vanishes on an open set?
[T] Talk about C^\alpha estimates for the Laplacian. What is the idea
of the proof of the inequality you just wrote down? [C] Could such an
estimate hold for \alpha = 1? [I don't know this counterexample, but
offer one for \alpha = 0 instead.] How do you extend these estimates
to general linear second-order elliptic operators? [I talk vaguely
about scaling, weighted Hoelder norms, and the need for C^\alpha
coefficients.]
[C] What is a weak solution of a linear second-order elliptic
equation? When must a weak solution be C^\alpha? [Embarrassingly, I
prove this for C^{large k} coefficients, using H^k regularity
estimates for weak solutions and Morrey's inequality, before being
reminded of the de Giorgi-Nash theorem. I then state de Giorgi-Nash
and state a Harnack-type inequality that can be used to prove it; they
seem happy.]
[C] Let's talk about general conditions for a function to be C^\alpha.
What Sobolev spaces contain only Hoelder-continuous functions? [I
state Morrey's inequality.] How do you prove it? [I state an
intermediate lemma, which says that L^2 functions with a certain
family of bounds on their variation are locally C^\alpha. Alice asks
a technical question about this lemma, and also has me show how to
deduce Morrey's inequality from it using Poincare's inequality.]
[T] What is the Green's function for a ball?
**Differential geometry**
[T] What is the Splitting Theorem? [I state it, then blank on the
proof. After a while, I mention the estimate for the Laplacian of a
distance function. We move on.] Okay, what can you say about
manifolds of positive Ricci curvature bounded away from zero? [I
state and prove Myers' theorem.] What is the Ricci curvature of a
bi-invariant metric on a compact Lie group? What conditions will
ensure that this metric has positive (as opposed to nonnegative) Ricci
curvature?
[T] Show that a compact manifold with nontrivial fundamental group has
a closed geodesic. Do you know about anything about closed geodesics
on compact simply-connected manifolds?
[T] You used the fact that geodesics are locally minimizing. How is
this proved?
I go into the hallway, and after a couple of minutes they come out
smiling.