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!"