Peter Sepanski
Committee: Hsiang(head), Aizenman, McNeal
1/21/94
ANALYSIS (McNeal)
Let En be a sequence of measurable sets in [0,1] with m(En) --> 1. Does
there exist a subsequence whose intersections all have measure > 1/2?
State the inverse mapping theorem.
Say f:(1,infinity) --> C is continuous, and integral(1 to infinity) of
f(x)x^n dx = 0 for n>=2. What can you say about f?
Suppose f:unit disc --> C has the value 5 on the line x=y. If f is
holomorphic, what is f? How do you prove this? What if f is harmonic?
Give a nonconstant example.
Does there exist holomorphic f:unit disc --> C which has zeros at {n/(n+1):
n a positive integer}? Can such an f be bounded? Do you know Blaschke
condition?
Give all conformal equivalences of the open unit disc. How do you prove these
are the only such conformal equivalences?
Let U be a region. Let H denote the set of all analytic functions on U whose
L2 norms are finite. Is H a Banach space? Why?
(Hsiang) Set a(n) = 1/n + ... + 1/2n. Compute lim a(n) as n --> infinity.
FUNCTIONAL ANALYSIS (Aizenman)
Say f is in both L3(R) and L4(R). What other Lp spaces is f necessarily
in? What interpolation results do you know for linear operators?
Suppose A is a symmetric linear operator defined on all of L2(0,1). Can A
be unbounded?
Talk abount self-adjoint extensions of the Laplacian operator on L2(0,1)
whose intial domain is {f in C(0,1): support(f) lies in the open interval
(0,1)}
Give sequence in L2(0,1) that converges weakly to zero but not in norm.
What's your favorite result from functional analysis?
Let A be a bounded symmetric operator on a Hilbert space H. Fix h in H.
Consider the function z --> <((z-A)^(-1))h,h>. What can you say about the
set where this is defined? Prove that it takes the upper half-plane to
itself.
Do you know the variational method for finding bounds for eigenvalues?
(Also known as minimax.)
ALGEBRA (Hsiang)
symmetric forms and matrices: (He wanted an intuitive "geometric"
argument for how to diagonalize a symmetric matix: i.e. talk about
ellipsoids and principal axes.)
Talk about how Galois theory applies to trisecting an angle with compass
and straightedge.
Can Q have a non-separable extension? How about Z/p? Why not? Give an
example of a non-separable extension.
What is the structure of a finite field? Prove that the multiplicative group is
cyclic.
State fundamental theorem for finitely generated modules over a pid. How
does this lead to rational and Jordan forms?
ALGEBRAIC TOPOLOGY (Hsiang)
State theorem of VanKampen.
Calculate fundamental group of double torus.
Prove that a subgroup of a free group is free.
Compute pi3(S2). What is Hopf fibration?
Talk about differential structures on manifolds. How do you define the
tangent bundle? What do you know about fibre bundles?
Compute cohomology ring of CPn.
State Poincare duality. Do you know how to prove it?