Jade Vinson's General Exams
8:30am, May 5, 1998
Topics: Harmonic Analysis of Wavelets, Ergodic Theory
Committee: Ingrid Daubechies (chair), Yakov Sinai, Thomas Hewett
H and then S arrive first. Since D is not yet present, they decide to
conduct the exam in 214 instead of the scheduled location of D's office.
I will record the questions asked on the exam as accurately as I can,
however mistakes are very likely in the ordering of questions, who aked
what, and the specific wording. I am also likely to forget several
questions which went by quickly -- either I knew the answer immediately
or didn't recognize any of the words in the question.
ALGEBRA:
H: Which matrices are diagonalizable? It took me a few minutes and
several hints before I remembered that these are the normal matrices -- I
then sketched a way to prove it. This is one of two questions which I
answered especially poorly.
H: Discuss some canonical forms for matrices. I discussed Rational form
and remarked that this could be proven using the structure theorem, which
I was asked to state. Then Jordan form is possible over C.
H: What if two diagonalizable matrices commute? I said that they could
be simultaneously diagonalized.
S: What do you know of Toeplitz operators? I said I knew what a T matrix
was and drew one on board. I gave hand-waving argument why the F.T. of
the cross diagonal gives approximately the eigenvalues. He then asked me
what a T. operator should be. I wrote something down and was correct.
H: Let's do Galois theory. What finite groups can be realized as Galois
groups. I said all of them could and sketched a proof. Which numbers
are constructible? I said t is constructible if [Q(t):Q] is a power of
two (incorrect). He asked me to sketch the proof. One direction was
easy, then I noticed the mistake going the other way. He told me the
correct statement and I completed the proof. Which regular n-gons are
constructible? I answered and sketched a proof. I knew this stuff cold
because I anticipated the questions [see e.g. Dan Grossman's generals,
but note that Fermat primes and not Mersenne primes are appropriate].
H asked some questions which led me to remark "This sounds a lot like
representation theory." He confirmed this and we discussed basic stuff
for a while including why representations can be decomposed into
irreducible representations. After a few hints I sketched why this is
possible and H points out where the proof breaks down in characteristic
P. I learn later that day that this is Maschke's theorem (spelling?).
H: If we asked you to prove FTA, how would you do it. I immediately said
Liouville's theorem and thought "Here comes the complex analysis." But
it never came: I WAS NEVER ASKED A SINGLE COMPLEX ANALYSIS QUESTION.
S: What do you know about Lie groups? I said not much, but ask anyway.
He did, and there were several words I had never heard before. S decided
it was probably time to move on to analysis, perhaps asking some
questions that relate to the special topics.
REAL ANALYSIS:
D: Why is Lebesgue integration so much better than Riemann integration?
I started to list some reasons: there are more Lebesgue integral
functions; the convergence theorems MCT,LDCT, and Fatou might not all
work with Riemann integration; the idea that sets of measure zero don't
bother Lebesgue. WRT the first point, I was asked to produce a fcn which
is LI but not RI. The first try didn't work, but was easily fixed. WRT
the last point, H asked a question which prompted me to say something
about completeness of Lp spaces.
S asked some sort of ergodic-theory question which was solved on the
second try using Fourier series, but I forgot the question.
S: How would you estimate the behaviour of
int[-infty,infty,(x^2n)*e^(-n^4)] as n->infty. I knew precisely how to
do this and outlined the steps. The exact calculation was not requested.
ERGODIC THEORY:
S: What is a measure preserving transformation? No problem. Then come
lots of other basic definition type questions. Also no problem.
S: What are goedesic flows? I made it clear that I did not know the
formal definition, but said that it is a flow on the tangent bundle of a
manifold and drew pictures. S asks when this is ergodic, and I say that
negative curvature is sufficient.
S: How do you get a unitary operator from a MPT? No problem.
S: What are Hamiltonian systems? I remark that I can only do this in
R^2n (ie not on manifolds) and sketch definition. He then asks me what
measures are invariant. I say Lebesgue. He says this is infinite. I
say restrict to surface of constant energy and outline how to get
invariant measure which is a.c. wrt surface area. S asks what this is
called, I can't remember until prompted to say "microcanonical distribution."
S: What do you know about MPT's with pure point spectrum? I say what
this means and then say that such systems are isomorphic to rotations of
a compact abelian group, and sketch how to prove this, making it clear
that I cannot remember how to prove the lemmas involved.
S: Can you solve f(x+t)-f(x)=h(x) for h given and h and f L2 fcns on
torus when t is irrational? I immediately remarked that a necessary
condition is that h have zero integral. It took me three different
attempts and a hint before finding a sufficient condition. A very fun
problem -- I won't spoil it for the reader.
S: Suppose f is a smooth doubly periodic function. How does they average
value on a circle with fixed center grow as the radius tends to
infinity? I say what the answer has to be, and am led through a proof.
I am still a bit skeptical of the proof and intend to work though it
carefully.
WAVELETS:
D: Why do wavelets form an unconditional basis of L^p? This is the other
question I answered very poorly. I got that deer-in-the-headlights look
and tried to classify L^p fcns in terms of wavelet coefficients, wrote
down the wavelet analogue of Littlewod-Paley characterization but could
not see how to relate this to the question. D reminded me that this was
not answering the question but let me struggle for several minutes (it
seemed like a lot longer) before offering a hint. After being prompted
to write down the correct definition of unconditionality I knew what to
prove, and then after another hint remembered how to prove it.
D: Why do wavelet series of L^2 fcns converge a.e.? She said the
corresponding result for Fourier series (Carleson's theorem) was very
difficult and was only recently proven. I offered to outline Fefferman's
proof of C's theorem but D declined. I proceeded to prove result for
compactly supported orthonormal wavelets. She was very particular about
each stage of the estimations.
The exam lasted just under two hours, and was actually kind of fun once
it got started. There were a few awkward silences, such as when I got
stuck or when they were thinking of a problem or deciding who gets to ask
next. Except for the two questions I answered very poorly, I was
satisfied with my performance. My general approach was to be honest and
let them know if I know a topic very well, know the main ideas but not
the details, or know very little. My committee seemed to prefer the
middle category.