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.