Mikhail Smirnov
Commitee: J.J.Kohn (chairman), P.Sarnak, M.Bridson. (2 hours 40 min)
ALGEBRA.
What groups have nontrivial isomorphisms.
Formulate structure theorem for abelian groups.
What is Galois theory. Formulate its main theorem.
What is Galois group of the cyclic (i.e. $x^n-1$) extention of $Q$
.(It is isomorphic to the multiplicative group of residues of the
numbers relatively prime with n).
Why $(x^p-1)/(x-1)$ is irreducible over $Q$.How to prove it.
What is Eisenstein criterion.
What is the structure of finite fields. What automorphisms do they have.
How to prove it.
REAL ANALYSIS
What is the Fourier transform. What is teh image of $L_1(R)$ under
the
Fourier transform. (Uniformly continous on $R$ functions which tends
to zero at infinity.) Will all such functions belong to the image of
Fourier transform of $L_1$. No. Why. It is possible to use Banach inverse
operator theorem to prove this. Explicit examle is in the book of
Katznelson.
How Fourier transform acts in $L^2$. What is Plancherel theorem.
How to prove it.
What space is conjugate to $L^p[a, b]$. When $L^p$ is contained in $L^q$.
Holder inequality.
What is Fourier transform of the compactly supported function.
Paley-Wiener theorem.
COMPLEX ANALYSIS.
What is the order and type of entire function.
Proof of the Riemann mapping theorem.
DIFFERENTIAL GEOMETRY
What do you know about differential geometry. Connections, curvature
tensor,
Christoffel symbols, connection arising from Riemannian metric.
What is the Theorema Egregium (Gauss-Bonnet formula). How to prove it.
Calculate explicitely the curvature of Lobachevskii plane(model in
the circle, curvature will be -4). Can torus have a metric of
nonnegative gaussian curvature.
Global Gauss-Bonnet theorem. What can be said about the surfaces
which covers the surface of the genus g.
What are geodesics. Their equations. Variational equations. Why they
exist.
How long geodesic curve be extended. Jacobi fields and conjugate
points.
What can be said about geodesics on the surfaces of the positive and
negative curvature. On the manifolds of negative sectional curvature.
I mentioned the existence of closed geodesics on the closed surface
diffeomorphic to the 2 dimensional sphere.
The existense of three nonselfintersecting closed geodesics follows
from Lyusternik-Schnirelman theory.
Then Sarnak mentioned recent result on the existense of infinitely
many closed geodesics on the closed surface diffeomorphic to the 2
dimensional sphere. But fortunately I knew that they may have
self-intersections).
Then there were some questions on global differential geometry
mainly from Bridson.
SEVERAL COMPLEX VARIABLES.
What is the Hartogs' extention theorem.
What is the domain of holomorphy. Examples of domains which are not
the domains of holomorphy. How to extend a holomorphic function from
the neighborhood of the sphere.
What is the Levi form, pseudoconvexity, PSH function. Idea of the
proof of Levi's theorem. How to extend function through the point
which is not pseudoconvex. Cohomologies of the domains of holomorphy.