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.