Seth Patinkin my generals - May 22 2001 Nelson (chair), Fefferman, Stalker duration: about 3 hours. In the following, take T to be the unit circle and U the open unit disk. algebra: Define the derived series. Define the commutator. State and prove two nontrivial theorems about derived series. Define simple, prime, and finitely-generated. State a theorem about the derived series involving finitely-generated groups. real analysis: Show that for a lebesgue-measurable set E, E-E contains an interval. Give some etymology of the convolution f*g, i.e., the relation with the distribution function of the sum of two random variables. Give an orthonormal basis for L^p(R). Exhibit some statements about the relation between C(T) and L^p(T) to prove the Riemann-Lebesgue lemma. Give some heuristic reasons why the Riemann-Lebesgue lemma should be true, i.e., give an explanation that "engineers would understand". Consider the powers of the continuous function f defined on [0,1]. State and prove a theorem about uniform converge of the power of f to 0 on [0,1]. Why are L^1, L^2, and L^infinity estimates usually the "easiest" estimates. Why is an additive group necessary for doing fourier analysis? How is fourier analysis on T different from fourier analysis on R^n? What is the difference between holomorphic and C^infinity? Which property is useful for proving global pde theorems? complex analysis: Prove Cauchy's theorem. What is Goursat's theorem? Use the del operator to reformulate the Cauchy-Riemann equations. State the generalized Cauchy-Riemann equations. How does the complex function sin(z) differ from the real function sin(x). On the real line, sin^2 + cos^2 = 1, but in the plane sin^2 + cos^2 is not bounded. How does one reconcile this notion, i.e., are the given definitions for sin(z) and cos(z) the right ones? Assume f has a complex derivative bounded by 1 on T. What can you say about the boundedness of f and its complex derivative on U, T? Also what can you say about the boundedness of f on T? Define what it means to be Riemann-integrable on T. Consider the powers of the complex function f. Make some statements about the convergence of these powers using Harnack's theorems. game theory: Define Nash equilibrium. Are Nash equilibria unique? Under what conditions can one have multiple Nash equilibria? State some open problems concerning Nash equilibria for 3-player games. What is the Kalai-Smorodinsky line; are there alternatives to Nash equilibria? What are Kuhn's main contributions to the theory of sequential games? State a theorem about non-sequential games. What are the properties of the utility function? Why is convexity important? Characterize the Nash bargaining solution. Explain why your work on (i) classifying 3-player games and (ii) classifying algorithms for finding Nash equilibria in n-player games is important. singular integrals: Define the Calderon-Zygmund class. Define singular integral; can there be singularities besides 0 and infinity? Define the Hilbert transform. Why is traditional "definition" of the Hilbert transform ill-defined. Define cancellation. What is the role of smooth cutoff functions? State a theorem about the value of the Hilbert transform on characteristic functions of intervals. State a multiplier theorem. How does one usually prove almost-everywhere convergence theorems? State a theorem relating the Riesz transform and partial derivatives.