Pavel Bachurin's generals (May 10, 2002)
Topics: Dynamical systems, Stochastic processes.
Commitee: Ya.G. Sinai (chair), J. Mather, H.Oh
(lasted about 1:40)

Algebra:

What is a Lie group? Examples. What is a simple Lie group?
What can you say about SU(2). How does SL2(R) appear in dynamical 
systems? Subgroups corresponding to geodesic and horocyclic flows.

Prove that A_n(n\geq 5) is simple. What is a separable extension.
Give a polynomial with S_3 as a Galois group. 

Real Analysis mixed with Stochastic proceses:

State Radon-Nykodim thm., give example of a meager set of full measure,
of a nonmeasurable set. What is a Banach space, L^p. Hilbert space.  
What is a conditional expectation? Why does it exist? When do two Wiener 
processes induce absolutely continuous (wrt to each other) measure on 
C[0,1]? Is it true, that for every sequence a_n of real numbers there 
exists a function whose nth derivative at zero is equal to a_n? Is the 
integral of a Markov process a Markov process itself?

Complex analysis:

Prove Riemann Mapping theorem. What is an entire function? Weierstrass 
factorization theorem. Why there are no doubly periodic functions of 
order one? Give an example of a Phragmen-Lindelof theorem.

Dynamical systems:

Poincare-Bendixson theory. What is a rotation number of a homeomorphism 
of a circle? When can a diffeomorphism be conjugated with rotation. What 
about smoothness of conjugacy? Do you know the Hermann Theorem? 

What is a hyperbolic fixed point? What are Anosov diffeomorphisms? What 
can you say about them (structure stability, periodic points are dense)? 
What is a homoclinic point? What can you say if you have an intersection
of stable and unstable manifolds? What do you know about family of 
unimodal maps (x\to ax(1-x))? Define topological and metric entropies.