Symplectic Geometry ( Spring 2005)
Instructor: Chris Woodward, ctw@math, Hill 336
Course Number: 16:640:547 Topology of Manifolds
Time and Place: Tentatively MTh3 11:30-12:50
This will be a course on symplectic geometry. Symplectic geometry is
a kind of skew-symmetric sister to Riemannian geometry, in which one
studies manifolds equipped with a closed non-degenerate two-form.
Originally a framework for Hamiltonian mechanics, symplectic geometry
now places a role in most areas of modern mathematics. Many of the
recent activity has centered around Floer theory for symplectic
manifolds, an infinite dimensional version of Morse theory, developed
to prove Arnold's conjectural lower bound on the number of periodic
orbits of a time-dependent Hamiltonian dynamical system. Even more
recently, Fukaya's categorical framework for Floer theory has been
conjectured by Konstevich to play a role in mirror symmetry of quantum
field theories.
The first part of the course is planned as an introduction to the
basics, the second will be a survey of various topics, including
Fukaya-Floer theory.
Topics:
I Introduction to symplectic geometry: Darboux's theorem, Poisson
brackets, Hamiltonian flows, and examples in classical mechanics.
Symmetries of symplectic manifolds: moment maps, symplectic reduction.
II Topics in Floer-Fukaya theory: J-holomorphic curves, Floer theory
of Lagrangian submanifolds, Fukaya category for monotone symplectic
manifolds.
The texts for the course are for Part I: Notes on
Symplectic Geometry, by Eckhard Meinrenken, ; for Part II: Lectures
on Floer theory, by Dietmar Salamon ; for Part III: Homological mirror
symmetry for the quartic surface , by Paul Seidel.
The reserve list at the Math Research Library
is: Foundations of Mechanics / R. Abraham, J. Marsden. J-holomorphic
curves and symplectic topology / Dusa McDuff, Dietmar Salamon.
Methods of homological algebra / Sergei I. Gelfand, Yuri I. Manin.
Prerequisites: Differential forms
on manifolds.
More detailed outline (tentative):
Symplectic linear algebra
Review of manifolds
Symplectic manifolds
Hamiltonian vector fields and Poisson brackets
Darboux and other local form theorems
Hamiltonian group actions and geometric invariant theory
Morse-Smale-Witten theory
Arnold conjecture and pseudo-holomorphic curves
Floer homology
Gromov compactness and stable maps
Quantum cohomology
Riemann-Roch for surfaces with boundary
Floer homology for Lagrangian intersections
Products in Lagrangian Floer homology
Fukaya category
Derived Fukaya category
Seidel's exact triangle
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