Will Cavendish
Dave Gabai (chair), Paul Yang, Nick Katz
Special Topics: Riemannian Geometry, Algebraic Topology
April 29, 2009
Length of exam: 2.5 hours
Complex Analysis :
(Y) What is a holomorphic function? What are its basic properties? Prove the Cauchy integral formula.
(K) What if the derivative is not continuous? How does the proof work in this case?
(Y) What can you say about the real part of a holomorphic function? When is a harmonic function the real
part of a harmonic function? Why does the harmonic conjugate exists locally and how do you construct it?
(Y) Describe how to solve the Dirichlet problem on the disk. On an arbitrary domain?
(Y) What is the uniformization theorem for Riemann surfaces and how is it proved?
(Y) Prove the Riemann mapping theorem.
(Y) What about convergence on the boundary of the domain?
(K) Explicitly write down the map from the disk to the upper half plane. What 19th century mathematician’s
name is associated to this map? (Cayley, it turns out)
(K) How do you know this map does what you say it does?
Real Analysis:
(Y) What do you know about Fourier series?
(K) What can you say about decay of Fourier coefficients? How do you prove this?
(K) What can you say about the Fourier transform?
(K) What is Schwartz space and is its relevance to Fourier analysis?
(Y) Why does smoothness of a function imply decay of its Fourier transform?
(Y) Suppose you have a function on the plane whose gradient is in L^2, what can you say about its decay?
(G) What is the measure of the Cantor set?
(G) What can you say about Cantor sets in general? Does a Cantor set always have measure zero?
(K) What happens if you take a middle 5ths set instead of a middle 3rds set?
Algebra:
(K) State the fundamental theorem of finitely generated abelian groups. What about this is natural?
[I had seen this question in another transcript but still didn’t give a good answer. What Katz wanted to
hear is that the torsion part is canonically a subgroup, while the free part is a canonically a quotient]
(K) What theorem is this a consequence of? Can you prove it?
(K) In ten seconds or less, describe Galois theory.
[This part of the test took about 20 minutes]
(K) What is the Galois group of Q(\sqrt(2),\sqrt(3)) over Q?
(K) What is the Galois group of Q(\sqrt(n_1),\sqrt(n_2),...,\sqrt(n_m)) over Q(\sqrt(n_1)+...+\sqrt(n_m))?
(K) What can you say about subgroups of a free group? How is this proved?
Riemannian Geometry:
(Y) What is the fundamental theorem of hypersurfaces?
(Y) How do you prove that a surface with negative curvature bounded away from zero can’t be embedded in R^3?
(Y) Prove the Gauss Bonnet theorem.
(Y) Is there an analogue of the Gauss-Bonnet theorem for non-compact surfaces? What happens if you rescale a
paraboloid in R^3?
(K) What about generalizations of Gauss-Bonnet? Can you say anything for odd dimensional manifolds? Do
Chern-Simons forms mean anything to you?
Algebraic Topology:
(G) Exhibit a surface of Euler characteristic -1.
(G) How many distinct topological types of curves are there on a surface of genus 6?
(G) What can you prove about the topology of a simply connected 3-manifold?
(G) Let M be a manifold which fibers over the circle. Let S be a surface in the homology class of the fiber,
and let M' be the infinite cyclic cover of M. Does S lift to M'?
(K) Let V be a vector field on a closed manifold M and let X be its zero set. Assume X is a smooth manifold.
What can you say about how the topology of X relates to the topology of M?