250131 VO Symplectic Geometry (2023W)

This is a masters course in symplectic geometry. It will be taught by lectures (blackboard, no printed notes). It will assume some knowledge of algebraic topology and differential geometry (at the level of an advanced undergraduate course); a little knowledge of algebraic geometry would be useful but is not required.

The course will start with some fundamental notions in symplectic topology: symplectic linear algebra; Hamilton’s equations, cotangent bundles; Lagrangian submanifolds; symplectic submanifolds; Moser’s trick, Darboux and Weinstein neighbourhood theorems; almost complex structures and compatible triples; basic examples and properties of Kaehler manifolds.

Time allowing, and depending on the background and intellectual interests of the audience, the lectures will cover some of the following topics:
-- Surgery constructions: blow ups, symplectic fibre sums.
-- Lefschetz pencils, Gompf's theorem on fundamental groups of symplectic 4-manifolds
-- Weinstein manifolds, Lefschetz fibrations
-- introduction to symplectomorphism groups
-- introduction to Floer theory
-- introduction to homological mirror symmetry
-- further possible topics as determined in consultation with the audience