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.