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).