Qualifying Exam Syllabus 

Joel Kamnitzer

December 17, 2002


Major Topic: Lie Groups

Lie groups, Lie algebras, exponential map, nilpotent Lie algebras

representation theory for compact groups, complete reducibility, 
characters, Schur's lemma, Peter-Weyl theorem, character tables for finite 
groups, representations of tori

maximal tori, root decomposition for compact Lie groups

root systems, positive systems, simple roots, Dynkin diagrams, Weyl groups, 
Weyl chambers, root and weight lattices

complexification of compact Lie groups, Borel subgroups

classification of semisimple compact and complex Lie groups with given
Lie algebra

Bruhat decomposition, topology of  G/B , Bruhat order

classification of irreps by dominant weights, Borel-Weil theorem

Weyl character formula, Kostant multiplicity formula, Steinberg's
formula

SL_n  theory: construction of Schur functors, weight multiplicities 
using Gelfand-Cetlin patterns and Young tableaux, representations of  S_n, 
Schur-Weyl duality


Major Topic: Symplectic Geometry

symplectic vector spaces, vector bundles, and manifolds

Lagrangian, isotropic, coisotropic, and symplectic subspaces, subbundles 
and submanifolds

symplectic structure on cotangent bundle, Lagrangian 
submanifolds of the cotangent bundle, conormal bundles, 
symplectomorphisms, the method of generating functions 


symplectic structure on coadjoint orbits

Moser's trick, Darboux's theorem, Weinstein's Lagrangian tubular
neighbourhood theorem

compatible almost complex structures, K\"ahler manifolds,
integrability for almost complex structures, K\"ahler structure on complex 
projective space

Poisson bracket and Poisson manifolds, Poisson structure on the dual of a 
Lie algebra

Hamiltonian vector fields, momentum maps: existence and uniqueness, 
Noether's principle

Marsden-Weinstein symplectic reduction, reduction in stages

Atiyah/Guillemin-Sternberg convexity theorem, Schur-Horn theorem

Duistermaat-Heckman measure, Duistermaat-Heckman theorems

Delzant polytopes, classification of toric symplectic manifolds, Delzant 
construction for toric symplectic manifolds

Minor Topic: Algebraic Topology

singular, simplicial, and cellular homology and cohomology

long exact sequence of a pair, Mayer-Vietoris sequence

cup and cap products

manifolds, Poincar\'e and Lefschetz duality 

homotopy groups, Whitehead's theorem

fibre bundles, fibrations, the long exact sequence of a fibration

principal bundles, associated bundles, classifying spaces