Examiners: D. Christodoulou, W. Browder, A. Ram
Topics: Riemannian Geometry, Alg. Topology
Monday May 4, 1998 10:00 AM-12:40 PM
The questions actually alternated between subjects. Here I have
listed them by subject.
Analysis
Give an expression for the first Dirichlet eigenvalue of a domain
in R^n.
Give a pointwise upper bound for a harmonic function in a domain
of R^n if its gradient squared integrates to 1, depending only on
the distance from the boundary.
Is the derivative of a BV function in L^1?
Algebra
Eigenvalues of a symmetric matrix.
Galois group of x^2-2?
Why is x^2-2 irreducible?
What does factorization over Q[x] say about factorization over Z[x]?
State the structure theorem for abelian groups.
What is a semi-simple algebra?
State the structure theorem for semi-simple algebras.
What is a matrix algebra?
State Schur's lemma.
Do you know an example of a local ring? Another one?
What about completions?
Define Lie Algebra? Where do they come from? What is the Lie bracket
of a right invariant and a left invariant vector
field on a Lie group?
Complex Analysis
What is the Riemann surface of sqrt(z), over the disc and over the
Riemann sphere? Give an example of a map from the torus to the Riemann
sphere branched over 4 points.
Topology
Exhibit the solid torus as a cover of the solid ball, branched
over two intervals. State Brauer's Fixed Point theorem. What happens
when you replace the closed disc with the open disc? What can you say
about involutions of the open disc? What about of the surface of genus
5? Talk about homotopy
groups of spheres. How do you compute pi_n(S^n)? Use spectral sequences
to compute the first stable higher homotopy group.
Geometry
Characterize the value where the function "distance from a hypersurface"
in Euclidean space stops giving a diffeomorphism of a tubular neighborhood
with
the normal bundle. Give a complete proof of the volume comparison theorem,
i.e. given a manifold with Ric(M)>=0, prove that the volume of a ball
of radius R is less than in Euclidean space.