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.