The following is, to the best of my memory (not very good), the questions I was asked on my qualifying exam. I do not remember who asked what.
Lie Algebras and Representation Theory
- Define sl2. Describe its representations. Tensor the dim 3 irreducible and the dim 4 irreducible, and decompose it. How do you know all representations decompose (show this directly).
- Can you classify all representations of the 1 dimensional abelian Lie algebra? What about two dimensional?
- What is the Harish-Chandra theorem? What does it say when the Lie algebra is abelian?
- What is your favorite non A_n type lie algebra? Draw its root system. Describe its representations.
- Take the whitney umbrella x^2 = zy^2. What are its singularities? Sketch a picture. Compute the blow up. Describe the blow-up in terms of the picture. Is the blow-up smooth? If not where is it singular?
- What is an elliptic curve? Write down an explicit example and show it is genus 1. Write down a map to P^1 explicitly. State Riemann-Hurwitz. What does it tell you about this map?
- Compute fundamental group, homology, cohomology, cup product for torus.
- Compute homology and cohomology for the banana sphere (S^2 with two points identified). Its cohomology and homology satisfy the conclusion of Poincare duality, but does it really apply? What is the map given by Poincare duality?