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.