Vadim's generals
Last for slightly more then 2 hours.
My special topics were Dynamical systems and Stocastic prosseses.
My commite was Y.Sinai (chair), J.Mather, J. de Jong
We start with algebra:
What is a field? (J)
Structure of finite fields? Isomorphisms. (S)
Prove the fundamental theorem of algebra? (J)
(I proved it using Lioville thm.) J. asked to give another
proof. (I gave a proof using Cauchy formula).
Find all prime numbers in {\bf Z}[i].
Find all normal subgroups of A_4.
Real analysis.
State Radon-Nikodim thm. (S) (Now questions related to
Radon-Nikodim thm from Dynamical systems and Stocastic prosses).
Give an example of absolute continuity
in the theory of hyperbolic diffeomorphisms.
Problem (S) Take two diffusion prosesses in the line.
dx_i(t)=a_i(x)db(t)+ f_i(x) dt, i=1,2
When they are absolutely continious one to the other
(Answer if a_1(x)=a_2(x). Remark use law if iterated logarithm.)
What is Fourirer transform? (S)
State Plancherel thm, the inversion formula (M).
State Riemann mapping thm. (I mentioned also uniformiz. thm)
Give an example of residual set (countable intersection of
open dense sets) which has measure zero.
Give an example of nowhere dense set of arbitrary large
measure.
At this point we switch to special topics.
What is Anosov diffeomorphism and Anosov flow?
Are they stable?(M) Why?
I gave an idea why they are stable.
Are they chaotic?
Describe baker's transform.(M) What chaos it has.(M)
What thm from dynamical systems was proved by algebraic
geometrists?(S) I stated Artin-Mazur thm.
Why did they study periodic orbits?(S) I said that they
introduce dynamical zeta function.
Moreover, I said that zeta function is rational for
Axiom A system.
de Jong got excited by that fact.
Do I know any easy proof of that?(M)
I said that I have to introduce markov partitions
subshifts of finite type, and so on.
Mather said that there is an easy proof using Lefschetz
fixed point formula. I stated the Lefschetz fixed point formula and
said that that formula sums periodic points with different signs.
Then Sinai and de Jong got interested and asked Mather
to explain why it is easy. He said that there is a simple reason
why it is true, but he don't remember what.
A few questions from hyperbolic geometry.
What is upper half plane? Describe a metric of constant
negative cuvature. What group act on that metric space
isometrically. Describe any two dimensional
surface with a metric of constant negative curvature.
What can you say about geodesic flows on surfaces
of constant negative curvature? (ergodic, mixing,
exponential decay of correlations)
When you have an Axiom A diffeomorphism what can you say
about invariant foliations (How smooth they are,
how about and what is absolute continuity)?