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)?