special topics:
Harmonic Analysis, Probability theory and Random processes
Committee:
Stein(c), Sinai, Katz
Tuesday May 9th 2000
I got there a little bit early. after a while Sinai came. but Katz was
not to be seen. we waited for some time and then stein went look for him.
he was not in his office. Sinai suggested that maybe he is in the
Institute.
stein called a couple of people so that they will find him and remind
him of today's exam and we started without him.
Sinai
----------------
sinai began with some probability: tell me what you have studied. which
theorems you find exciting or useful?
I named SLLN(both versions) and CLT(elementary, Lindeberg, Feller, UAN)
and ergodic theorems of Birkhoff and vonNeumann,
0-1 laws and Kolmogorov's three series thm,...
and added that I think I am well prepared to be asked about martingales.
specially about convergence thms and relation to harmonic analysis
which I was anticipating. and also some brownian motion stuff
and stoch. diff. eq.
but I ended up not being asked any question about most of them!
(you see, they are more interested in what you do NOT know than what you
know) so I first gave an elementary version of CLT and sketched a proof.
then sinai asked for ergodic theorem for Markov processes. instead I
stated the ergodic theorem for Markov chains with countable state space.
he said that this is not what he asked but it is OK.
so we went to random processes.
I defined Wiener measure, said why it existsand why a continuous version
exists (Kolmogorov's continuity thm)
can you give some properties?
(a.s. not BV on any interval. in fact a.s. nowhere differentiable)
can you relate it to random walk on a lattice?
- you mean symmetric random walk? -yes.
it is the weak limit.
so how big is approximately each step and the time to go each step?
I couldn't write it down immediately so he suggested maybe I can give an
idea of how far I expect the brownian traveler to be from origin?
what is the probability that he remains from some point of time always to
the left? -zero. -why? -I don't know. -think a little bit.
back to the order of distance from origin. so what can you say about that?
I suggested maybe we should calculate the fourier transform. it turned
out not to be such a good idea. then I suggested to use law of iterated
logarithm. -can you prove it? -no! -it is not what I wanted anyway!
I said that I expect the brownian traveler to be dense in dim 1,2 but go
to infinity in dim at least 3. this was also not what he wanted.
by that time Katz was there, so we moved to algebra.
-more probability later. Sinai said.
Katz
------------------------------------
so do you know anything about finitely generated abelian groups?
I gave the structure thm. -are any of those factors associated "naturally"
to your group? -I don't know. -what do you think? -I suspect yes. maybe
you could count elements of some specific order.
-you are wrong. -but at least the rank of torsion-free part! -well.
-now can you give any canonical form for matrices? -which for example?
-talk about Jordan form. -should the field satisfy some property?
-is there any relation between this and the last question?
-they are both special cases of the structure thm of finitely generated
modules over PID's. what are they in these two examples? I blanked for
the Jordan canonical form. so he suggested to move to galois theory.
give the fundamental thm of Galois theory. what does it mean "galois
extension"? OK. is galois correspondence a bijection? for all subfields
or for stable subfields? OK. what is the galois group of x^7-1 over Q?
what is "natural" in your answer? is the generator associated "naturally"?
now x^5-2.
-OK I am done.
then he fell asleep!
Stein
---------------------------------------------------
let's do some complex. I am looking for an entire function with
prescribed zeros of prescribed orders. can you give any?
under what conditions? is it unique? to what extent?
why do you need those scalings? you say it will not converge if I omit
them?
(Katz got up, made the comment that they actually do converge sometimes
and back to sleep) now go to the example you just wrote down. what is its
significiance? -it is a very famous function.
what is the relation to gamma function? to Sin function?
how do you define the gamma function? properties?
(I was not able to give a satisfactory answer why we need those factors,
but he did not mind)
Now some harmonic analysis. which theorem excites you. name one!
I did not know which theorem excites me more. -so tell us about what you
have studied. I named interpolations, general stuff about harmonic
functions, Riesz transforms and spherical harmonics, some multiplier thms
Hardy spaces (only a little)
so we began with interpolations. I stated and gave the idea of proof of
Riesz convexity thm. stated a generalization. then used it to prove
Hausdorff-Young inequality both about convlutions and fourier transform.
then stated Marcinkiewicz.
-do you know generalization to Hilbert-space valued functions?
-I have no idea.
-OK so talk about pseudo-differential operators.
-again no idea.
-so let's do some singular integrals. what is your favorite thm?
I stated the thm about Calderon-Zygmund kernels.
-what is it good for? -I can use it to prove boundedness properties say
for Hilbert transform.
-write explicitly how you use this to prove bddness of Hilbert transform.
then I realised that I had not written a correct statement.
stein suggested maybe I should change it to context of distributions.
so I did but still unable to fix it. so I went to state a simpler version
which I knew would give the result about Hilbert transform. he was not
interested. I could not find my mistake and we moved.
Sinai(2)
---------------------------
absolute continuity? Radon Nikodym? a continuous measure singular w.r.t.
lebesgue measure? (can you believe it? I blanked at this point!!)
with some help from stein I remembered.
it seemed that sinai did not want to ask anymore probability questions.
That was it. I left the office and after a while the door opened and Stein
said: congratulations. you passed! I was completely exhausted by this point.
took two hours and a half.
comments:
the committee was very very nice. they helped me a lot finding my way
through these questions. many many times I made mistakes but they did
not mind.
something else: it is vital to prepare many things in advance, but as in
my
case they might not question you on those. because as soon as they feel
you know something they will move to something else!
as you can see I had just one complex analysis question and just one real
analysis (more a formality) they were more interested in special topics.
so be prepared for the exam pretty well, but don't expect to answer any
question you will be asked. they themselves don't expect you to do so!