The first lecture will provide a quick overview of symplectic geometry
and its main tools: symplectic manifolds, almost-complex structures,
pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology.
The
second and third lectures will focus on symplectic Lefschetz pencils:
their
construction (following Donaldson), the corresponding monodromy
invariants,
and their applications to symplectic topology, in particular the
connection
to Gromov-Witten invariants of symplectic 4-manifolds (following Smith)
and
to Fukaya categories (following Seidel). In the last lecture, we will
offer
an alternative description of symplectic 4-manifolds by viewing them as
branched covers of the complex projective plane; the corresponding
monodromy
invariants and their potential applications will be discussed.
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