General Exam - Chi Li April 25, 2008, 12:30pm-2:00pm Commitee: Gang Tian(Chair), Kollar, Kohn Special Topics: Algebraic Geometry, Differential Geometry Algebra: If you have a degree 2 field extension K/F, is it Galois? If you have another degree 2 field extension E/K, is E Galois over F? (Not necessarily) Draw a picture of degree 4 mapping between Riemann surfaces which shows that the extension of function fields is not Galois. (If it's Galois then the automorphisms of the map act transitively on inverse image points of a point, so it's not Galois if there are three inverse image points brached at only one) Analysis: Explain different notions of convergence and relations between them. Try to find examples if one does not imply another. For example, find a sequence of functions which are L^p convergent but not uniformly convergent. Prove the Ascoli-Arzela theorem. Suppose you have a bounded harmonic function on R^n, what can you say? (constant) Prove it. (derive gradient estimates) How do you solve the Laplace equation in a ball, and in general domains? What's Green function? Algebraic Geometry: Calculate the genus of plane curve defined by y^3=x^6-1. (use Riemann-Hurwitz formula. It's singular at infinity but splits into 3 branches) Differential Geometry: What's Chern-Weil theory? (Chern-Weil homomorphism) Prove that the De Rham cohomology defined by invariant polynomials of curvature is independent of chosen connection. Calculate the Chern classes of CP^n? What 2 form represents the cohomology class of hyperplane?(Fubini-Study) The Committee are very nice. They didn't ask difficult problems, and when I got nervous and stuck sometimes, they gave me hint and I could go on quickly.