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Qualifying Exam Syllabus



Committee: Michael Hutchings, Rob Kirby, Alan Weinstein, Robert Littlejohn (physics)


Exam to take place on Wednesday, September 10, 2003, 1-4 p.m. in Evans 959
\begin{itemize}

\item{Major topic: Geometry and Topology of Manifolds}

-fundamental group and covering spaces, higher homotopy groups, long exact sequence of a fiber bundle, Hurewicz and 
Whitehead theorems

-singular homology, long exact sequence of a pair.  Brouwer and Lefschetz fixed point theorems.

-de Rham cohomology, Mayer-Vietoris

-vector fields, flows, Morse homology

-Poincare duality, Lefschetz duality

-vector bundles, principal bundles, Stiefel-Whitney classes, Euler class, Chern classes, connections, curvature of a connection

\item{Major topic: Symplectic Geometry}

-symplectic linear algebra

-symplectic structures.  examples: cotangent bundles, complex projective spaces

-Lagrangian submanifolds

-Moser trick, Darboux theorem

-the flux homomorphism, symplectic versus Hamiltonian isotopies

-almost complex structures, the space of compatible almost complex structures

-pseudoholomorphic curves, statement of Gromov compactness theorem.  Applications: recognition of $\bf{R}^4$, Gromov's nonsqueezing theorem, the topology of the symplectomorphism group of $S^2 \times S^2$

-Floer homology

\item{Algebraic Geometry}

-complex varieties and manifolds.  

-Dolbeault cohomology.

-sheaves,  Cech and sheaf cohomology,  the exponential sequence and the classification of line bundles,  De Rham cohomology $\cong$ singular cohomology

-subvarieties of a complex manifold carry a fundamental homology class

-Kahler varieties
%%--Wirtinger's theorem, Hodge decomposition, Lefschetz decomposition

-divisors, line bundles, Chern classes of line bundles

-tangent spaces, degree, blow-ups

\end{itemize}
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