Eric's generals
4/29/2015, 1:30-4:30pm
My special topics were PDE and Riemannian geometry.
Examiners
[AC] Alice Chang (chair)
[PC] Peter Constantin
[ZP] Zsolt Patakfalvi
Algebra
[ZP] What is the structure theorem for finitely generated modules?
[ZP] Give examples of modules over a polynomial ring K[x] over a field K. What are all the submodules of K[x] over K[x]?
[ZP] How does the module K[x]/(x(x-1)) over K decompose explicitly in the way stated by the structure theorem?
[ZP] How many finite fields of a certain order are there and why?
[ZP] State the Sylow theorems.
Complex Analysis
[PC] State and prove Rouche's theorem.
[PC] Under what conditions does a sequence of pointwise converging holomorphic functions converge uniformly?
[PC] What is the residue of a function and how is it computed?
[PC] Given two infinite sequences of points in C intended to correspond to poles and zeros, can you write down a function that has exactly these poles and zeros?
[AC] Describe a conformal map from the complex plane with the segments (-\infty,-1] and [1,\infty) removed to the unit disc.
[PC] What is the maximum modulus principle? What about Phragmen-Lindelof?
Real Analysis
[PC] What is a sequence in L^2[0,1] which converges weakly but not strongly?
[PC] What conditions can guarantee the precompactness of bounded sets in L^p[0,1], for 1\leq p<\infty?
[PC] What is the dual space of L^p[0,1]?
[PC] State the Radon-Nikodym theorem.
[AC] When do second derivatives of a function commute in R^n? What about on a Riemannian manifold? What is f_{ijk}-f_{jik} on a manifold?
PDE
[AC] What are some nice properties of harmonic functions? (eg. mean value, analyticity, Harnack, maximum principle)
[AC] Prove Liouville's theorem. Prove the Harnack inequality (for positive harmonic functions).
[AC] Give the gradient estimate for harmonic functions on manifolds with Ricci curvature bounded from below.
[PC] What is the decay for solutions of the heat equation on R^n with L^2 initial data?
[PC] What is the variational problem associated with the equation \Delta u+u^3=f? Show a minimizing sequence for the variational problem is bounded below and converges to a weak solution of the PDE. How do you obtain higher regularity of the solution?
[PC] What can you say about the growth of solutions to u_t+=u (a in R^n fixed) given some intial data?
*[AC] State the ABP estimate for elliptic equations and sketch the proof. What does the constant in the inequality depend on?
Riemannian geometry
[AC] How does the length of the hypotenuse of a right triangle compare to the lengths of the other two sides in hyperbolic space and why? Also show this by direct computation in the Poincare ball model.
[AC] What is the Bochner formula? What is Calderon's theorem (W^{2,p} estimates)?
[AC] What is Ricci curvature?
[AC] State the volume comparison theorem. What hypotheses on curvature are needed? How is it proved?
*[AC] Talk a little about optimal transport, Brenier's theorem, and applications.
*These were among some extra material in addition to regular PDE and geometry that Alice asked me to know for the exam.
The committee was friendly and I had help from hints and suggestions during the exam. I've listed the main questions I recall - other more minor questions would also come up while I was responding to the ones above.