Preliminary Oral Examination, 11:30am, 19 October 2006
General Topics: Real, Complex, ODE, Probability, Functional
Committee: Steve Childress, Jeff Cheeger, Marco Avellaneda

Questions were asked by:
[SC] - Steve Childress
[JC] - Jeff Cheeger
[MA] - Marco Avellaneda

[SC] What is a function of a complex variable? What is an analytic function? Prove Cauchy's theorem.
- I showed on a rectangle
[MA] Suppose an analytic function grows like a polynomial at infinity. What is it?
- I proved it was polynomial with Cauchy's Integral Formula.
[SC] Evaluate integral of sin(x^2) on positive x axis.
- I changed variables and then screwed up for a bit. They gave me a hint for the contour and I made the estimates.
[MA] Prove the Riemann-Lebesgue lemma.
- I approximated by C1 and integrated by parts.
[MA] How do you know C infinity is dense in L1? How about C infinity with compact support?
- Used approximate identity argument, multiplied by characteristic function of [-n,n], then smoothed out the ends.
[JC] What is Vitali's covering lemma?
- I forgot so made some stumbling attempts.
[JC] Is L infinity the dual of a space? How about L1? l1?
- Yes, no (proof with Krein-Milman and weak star compactness), c0.
[SC] What is c0?
- sequences converging to 0
[JC] What is the dual of Lp, 1 < p < infinity?
- Lq
[MA] Prove Holder.
[MA] Describe the pendulum.
- I wrote x'' = sinx and then fixed it to -sinx
[SC] What does x correspond to? Describe movement around x=0.
- I linearized and looked at eigenvalues, etc.
[MA] How do you find the flow lines everywhere else?
- I found hamiltonian and drew potential diagram
[SC] Relate these to the physical system.
[JC] What is weak convergence? Weak star? Do you know criteria in a Banach space for weak star convergence?
- I talked about weak-star compactness of the unit ball in a dual space.
[JC] Do you know conditions in Lp for a weakly convergent sequence to converge strongly?
- Radon's theorem. If norm fn converges to norm f.
[MA] What is the relation of weak star convergence to convergence of probability measures? What is tightness?
- Look at them as a subset of the dual of C(X), X compact.
[MA] Give an example to show that tightness is necessary.
- Delta masses escaping to infinity.
[MA] What is a martingale? If Xn, Fn is a martingale sequence is it true that E[Xn+1|X1,...,Xn] = Xn?
- I said I don't know but probably not since sigma(X1,...,Xn) may in general be a proper sub-sigma algebra of Fn.
[MA] What is Stirling's formula?
[SC] What is the Poincare-Bendixson theorem?
[JC] What is the entropy of a probability measure?


Special Topics: Percolation and Random Cluster Measures
Committee: Charles Newman, Dan Stein, Jinho Baik

Questions were asked by:
[CM] - Charles Newman
[DS] - Dan Stein
[B] - Jinho Baik

[CM] What is percolation? Prove there exists a phase transition
- I showed Peierls argument for d=2 and general argument for other d.
[DS] Describe dynamic renormalization arguments. Why are they useful?
- I indicated how one would prove theta(pc) for the halfspace is 0.
[CM] Define Ising model, Potts model, and their hamiltonians. What is the relation to the random cluster model?
- I showed coupling for RC/Potts.
[CM] Is there a relation between quantities in RC to some in Potts?
[CM] Prove phase transition exists in the Ising model with N.N. interaction using this coupling.
- I used FK comparison inequalities and infinite volume coupling.
[CM] Talk about uniqueness of random cluster measures.