Denis Kosygin
stochastic processes and dynamical systems
Sinai (chair), Mather, Hewitt
1 hr. 50 min.
Algebra. ( Hewitt and Sinai)
What is the Wedderburn's theorem? How does the group ring generated by
Z_5 over Q look like? What if we take the noncyclic group of order 4
instead of Z_5? The quaternionic group H instead of Z_5?
What is the Lie group? Define its unitary representation. What is the
Peter-Weyl theorem? What is the Lie algebra? The Jacobi identity?
What is the adjoint representation of a Lie algebra?
What is the commutator of two vector fields on a manifold?
Describe all the rational conjugacy classes of (3,3)-matrices with
rational entries which satisfy the equation A^4-A^3-A+1=0. Give a
representative in each class.
Real analysis (Mather and Sinai)
What are Baire categories? Present an example of a set of the first
category of a full measure. Give examples of properties of dynamical
systems which occur on sets of the second category in the appropriate
spaces of all dynamical systems.
Tell anything about Fourier series. Prove that the Fourier series of a
smooth function converges to it everywhere.
Complex analysis (Mather and Sinai)
How would you prove that a C^1 function which is complex
differentiable has actually infinitely many derivatives?
What is the Fragmen-Lindel\"of Principle?
Given a complex function f on the boundary of the unit circle can you
tell when it can be analytically extended inside. If f is real on the
boundary when it can be represented as |g(z)|^2 where g is analyic in
the unit disk?
Dynamical systems (Mather and Sinai)
What is a hyperbolic fixed point of a diffeomorfism. What can you say
about the structure of its small neighbourhood? How would you prove
the existence of the stable and unstable unvariant manifolds?
What is the Anosov system? What is the axiom A system?
What is a nonwandering point? (the last question was Hewitt's)
What can you tell about normal forms? What are Siegel and Poincare cases?
Stochastic processes (Sinai)
What is the probabilistic way to solve the Dirichlet problem? Take the
circle centered at the origin of the radius 2 and remove the segment
I=[-1,1] . Consider the Dirichlet problem with the boundary conditions
1 on the circumference and 0 on I. The Kac formula you wrote gives
the solution of this problem. On the other hand the only harmonic function
which has constant values on the boundary of a circle is a constant
function. Explain why there is no contradiction with your prevoius formula.
(The answer is of course that the solution of this Dirichlet problem is not
harmonic on I).