Fall 2009 Course Announcement: MATH 8230

Symplectic geometry (i.e. the geometry of a manifold equipped with a closed, nondegenerate 2-form) has its origins as the appropriate geometric setting for classical mechanics, but in the last 25 years it has expanded its reach significantly, with important relationships to subjects like three- and four-dimensional topology and complex algebraic geometry, in addition to some remarkable internal developments. This course will give a general introduction to the subject, mostly following Chapters 1-7 of McDuff and Salamon's text "Introduction to Symplectic Topology," and will then sample some more advanced topics, the precise choice of which will depend on the interests of the students.

Introductory topics:

  • Motivation and examples from physics and differential topology
  • Symplectic linear algebra and symplectic vector bundles
  • Basics about symplectic manifolds and symplectic diffeomorphisms (Darboux's theorem, Hamiltonian vector fields, Lagrangian and other special submanifolds)
  • Group actions, moment maps, and toric manifolds from the point of view of symplectic geometry
  • Topological constructions of symplectic manifolds (symplectic structures on fiber bundles, blow-ups, Gompf's symplectic sum)
  • Basics about almost complex structures and pseudoholomorphic curves

    • Possible more advanced topics (depending on who signs up):

  • Gromov-Witten invariants, with applications to four-manifolds and/or to enumerative algebraic geometry
  • Hamiltonian dynamics (e.g., Poincare's last geometric theorem, the Arnold Conjecture, Hofer's geometry on the Hamiltonian diffeomorphism group)
  • The construction of Heegaard Floer homology, with applications to 3-manifolds and/or knots.

    • Prerequisites: Familiarity with the basics of smooth manifolds—4220/6220 would be more than sufficient. You should know what a closed 2-form on a smooth manifold is, and should know the differential-form version of Stokes' theorem.
    Please feel free to contact me at [my surname]@math.uga.edu with any questions.

    Michael Usher's home page.