Dano Kim 16 May 2002, 13:00 - 15:00 Committee: Pandharipande, Gunning, Ozsvath Special topics: Algebraic Geometry, Algebraic Topology Gunning let me choose the subject to start with and I said complex analysis. [G] What can you say about an analytic function in a nbd of the origin..? - stated a theorem (called as Casorati-Weierstrass ) and then proved it. And some more discussion of poles and essential sing. (It's funny that preparing for the general I studied pf of the little Picard to be able to say 'yes' when they ask if I know, but even the thm itself didn't occur to my mind at the necessary moment) [G] About Riemann mapping theorem, let's say, you have a triangle on the complex plane. What can you say? - described Schwarz-Christoffel formula for triangle. ( Roughly it is linear combination of arg(z - z_1 / z_2 - z) etc where z_1,z_2,... on the real line go to the vertices of the given polygon. ) Then Gunning went on to real analysis. [G] State the fundamental thm of calculus. Then he asked a question which I don't remember, but the answer was 'absolute continuity'. [G] Can you have a series of L_1 functions which converge but not ptwise converge. There was more discussion about dominated convergence thm, etc. Pandharipande asked about algebra. [P] Why does a n by n matrix satisfy its characteristic polynomial? (Jordan canonical forms) [P] How to prove the fundamental thm of algebra using Galois theory. - need to use Sylow thm at the last step [P] How many irreducible representations for S_5? ( = # of conjugacy classes of S_5 = 7) [P] Are there two nonisomorphic groups with the same representations? ( D_8 and the quaternion group, but he didn't pursue pf) Pandharipande asked me to choose which special topic comes first. I went with algebraic geometry and he continued. [P] Can P^1 map to a curve of higher genus ? [P] What about curve of genus g1 map to curve of g2 ? When can you have g1=g2? (use Riemann-Hurwitz) [P] Cohomology computation for the normal bundle of the rational normal curve in P^3. - started from the exact sequence of sheaves 0 -> T(P^1) -> T(P^3) -> N -> 0 to get the long exact seq. , but had trouble with T(P^3) and took long time, got many hints from both [P] and [O]. [P] Let's talk about a curve which is a complete intersection of P^n. Given the degrees, what's the genus? - repeated use of adjunction to get the answer. mentioned a hyperelliptic curve can't be ever complete intersection. [P] Let's think about surfaces. How many types of quadric surfaces in P^3 do you have? (nonsingular, cone, ... ) [P] Is the quadric cone rational? Y. What about a cubic hypersurface in P^3. - only mentioned that nonsingular one has 27 lines. He let me write down a specific one (x^3+y^3+z^3+w^3 = 0 ) and then apparently it had a projection to an elliptic curve, so not rational. [P] For a curve of genus g, what's the smallest d such that any line bundle of degree g >= d has some global section? - said the answer was g+1 and was going to give a pf , but he didn't pursue it. Now Ozsvath began with algebraic topology. He thought of a question which continues from the first one of AG; [O] Can you have a topological map from a (compact orientable) surface to another with higher genus? - Hint was to start from a very silly map (like map to 1 pt) , then I came up with a map from S^1 x S^1 to a surface of genus 2 by projecting to an appropriate generator of H_1. This map is obviously not null-homotopic, but I was lead to calculate the degree of the map which is zero! [O] What's your favorite theorem in AT ? - said Lefschetz fixed point theorem and he asked why. I replied " I don't know but it just occurred to my mind" and they all laughed. [O] Do you know any of its application? - suggested instead to verify the thm in an example and he let me go ahead. It was rotating a genus 3 surface (which is symmetric enough) around the central axis, so that we can actually compute the map btw H_0, H_1, H_2 and then the Lefschetz # = 0. Indeed the map doesn't have a fixed pt. [O] Now let's talk about the lens spaces. - gave the general definition and we focused on the 3-dim ones. [O] Compute the fundamental gp, homotopy gp, homology gp, cohomology ring. - standard computation and for the last, used Poincare duality [O] Think about connected sum of lens spaces. What's pi_1? Is it finite? N. What's the homology? [O] Think of its universal covering space. Is it compact? N. why? (from pi_1) Try describing it. - He let me begin with any covering space and I drew some pictures and used hand-waving, which was ok. By this time 2 hrs already have passed and I found that I didn't even touch the water bottles I brought in. They told me to leave for a while, and then came out to say I passed. Gunning said, "now go and write a thesis!"