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.