Qualifying Examination Syllabus for Nicholas Proudfoot
Committee:  Robin Hartshorne (Chair), Allen Knutson, 
Robion Kirby, Robert Littlejohn (Physics).
Date and Time:  29 August 2001, 10:00am.

Major Topic:  Manifold Topology
Fundamental group and covering spaces.
Higher homotopy groups, and the long exact sequence of a fiber bundle.
Homology and cohomology groups:  Simplicial, singular, and DeRham theories.
Computational tools: 
Mayer-Vietoris, exact sequence of a Borsuk pair, Poincare duality.
Characteristic classes:  Axioms and obstruction theoretic construction.
The Thom isomorphism and the pushforward in cohomology.
Pontrjagin-Thom construction.  Application:  pi_{n+i}(S^n) for i=1,2,3.
Cobordism and surgery.  Cobordism invariance of the signature of a 4k-manifold.

Advanced Topic:  Symplectic Geometry
Symplectic manifolds.  
Examples:  C^n, cotangent bundles, coadjoint orbits, toric manifolds.
Moser's theorem and Darboux's theorem (equivariant version).
Complex geometry:  Almost complex structures, Kahler and almost Kahler metrics.
Moment maps.  Examples:  C^n, coadjoint orbits, toric manifolds.
Obstruction to finding a moment map, and example of a nonhamiltonian action.
Marsden-Weinstein symplectic reduction.  
Examples:  toric manifolds, polygon spaces.
GIT quotients and statement of equivalence 
with symplectic quotients for reductive group actions.
Symplectic cuts.
Atiyah-Gullemin-Sternberg convexity.  Application:  Horn's problem.
Statement of nonabelian convexity.  
Example:  U(n-1) acting on a U(n) coadjoint orbit.
Duistermaat-Heckman theorem.  Proof of circle case; applications.
Equivariant cohomology.  Borel and DeRham models.
Statements of Kirwan's theorem and Chang-Skjelbred theorem.  
Example:  cohomology of Flag(C^n).

Minor Topic:  Algebraic Geometry
Affine varieties.  
Hilbert's Nullstellensatz and 
the correspondence between varieties and prime ideals.
Projective varieties.  
Examples:  the Segre and Veronese embeddings, the smooth quadric in P^3.
Morphisms of varieties; regular and rational functions.
Singularities and blow-ups.  Examples:  nodes and ordinary cusps.
Schemes:  Spec and Proj, general definition and
 elementary properties, morphisms of schemes.
Sheaves of modules:  (Quasi-)coherent sheaves, 
tensor operations, module of global sections.
Divisors:  Weil divisors, Cartier divisors, and invertible sheaves.  
Computation of the divisor groups of P^n.
Ample and very ample invertible sheaves.