Spring 2010, MWF 1-2, Science Center 411
Lecturer: Andrew Cotton-Clay
Office: Science Center 527
Office Hours: W 2-4, tentatively
E-mail: acotton a t math d o t harvard d o t edu
Books:
[MS1] Introduction to Symplectic Topology, by Dusa McDuff and Dietmar
Salamon
Errata
[G1] An Introduction to Contact Topology, by Hansjörg Geiges
Errata
[MS2] J-holomorphic Curves and Symplectic Topology, by Dusa McDuff and
Dietmar Salamon
Errata
Online alternatives are, respectively, [C2], [E1, Eetc, G2], and [MS3].
Online:
[C1]
Symplectic Geometry by Ana Cannas da Silva (survey article)
[C2]
Lectures on Symplectic Geometry by Ana Cannas da Silva (book)
[E1] Introductory
Lectures on Contact Geometry by John Etnyre
[G2] Contact
Geometry by Hansjörg Geiges (survey article)
[H1] Lecture notes on Morse
homology by Michael Hutchings
[MS3] J-holomorphic
Curves and Quantum Cohomology by Dusa McDuff and Dietmar Salamon (earlier
version of MS2)
[S1] Lectures on Floer
homology by Dietmar Salamon
[Eetc] Other
expository notes by John Etnyre
Articles:
[Ch1] Yuri Chekanov, Differential algebras of Legendrian links, arxiv:math/9709233.
[Gr] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones
Mathematicae, 82 (1995), 307-347. MR0809718
[H2] Michael Hutchings, Reidemeister torsion in generalized Morse theory, Forum
Mathematicum 14 (2002), 209-244. link
[RS] J.W. Robbin and D.A. Salamon, The spectral flow and the Maslov index,
Bulletin of the LMS 27 (1995), 1-33. link
[Se1] Paul Seidel, A long exact sequence for symplectic Floer cohomology, Topology
42 (2003), 1003-1063, MR1978046, math.SG/0105186
[Se2] Paul Seidel, Symplectic automorphisms of T^*S^2, math/9803084v1
More to come as we go along.
Jan 25 Physics
motivation; definitions; cotangent bundles; Hamiltonian and symplectic
vector fields; Arnold
conjecture; (nonsqueezing theorem Jan 27). See [MS1, 1.1-2, 2.1,
3.1] or [C2, 1-2, 18].
Jan 27 Symplectic linear
algebra; complex structures; symplectic vector bundles; (Moser's trick Jan
29). See [MS1, 2.1-2, 2.5-6, (3.2)] or [G1, 1.3, (2.2), 2.4] or [C2, 1,
(6-7), 12-13].
Jan 29 Moser's trick;
Darboux's theorem; neighborhood theorems. See [MS1, 3.2-3] or [G1, 2.2,
2.5, Appendices A and B] or [C2, 6-9].
Feb 1 Fubini-Study metric on
CP^n; non-Kähler symplectic manifolds; symplectic fibrations. See
[MS1, 4.3, 6.1] or [C2, HW12 (no material on fibrations)].
Feb 3 Symplectomorphism
groups; flux. See [MS1, 10.1-2].
Feb 5 Symplectic connections,
locally Hamiltonian fibrations, Lefschetz fibrations, Dehn twists. See
[MS1, 6.3-4] and [Se1, Section 1].
Feb 8 More on Dehn twists, blow ups,
Lefschetz pencils. See [MS1, 6.3, 7.1] and [Se1, Section 1].
Feb 10 Finished up Feb 8 notes,
introduction to contact structures, examples, hypersurfaces of contact type. See [G1,
1.1, 1.4] or [E1, Section 2].
Feb 12 Examples, Weinstein conjecture, Neighborhood theorems. See [G1, 2].
Feb 17 Legendrian and transverse knots; front and lagrangian projections; Thurston-Bennequin
number and rotation number. See [G1, 3].
Feb 19 Combinatorial legendrian contact homology; augmentations and linearized homology;
examples. See [Ch1].
Feb 22 Morse homology; compactification and ad-hoc gluing; d-squared is zero. See [H1, 2].
Feb 24 Bifurcations and invariance in Morse
theory and combinatorial legendrian contact homology. See [H2] and [Ch1].
Feb 26 Fredholm operators; implicit function
theorem; Sard-Smale theorem. See [MS2, Appendix A] or [H1, 5].
Mar 1 Genericity/transversality for Morse
functions and Morse-Smale pairs; statement of spectral flow. See [H1, 5] and [RS, 2].
Mar 3 Spectral flow; continuation maps; chain homotopies. See [RS, 2] and [H1, 4]. (See Mar 1
notes for the first and Feb 26 for the latter two.)
Mar 5 Introduction to holomorphic curves; quick review of index/transversality/compactness
for Morse flow-lines; linearization of d-bar; statement of index for compact J-holomorphic
curves with lagrangian boundary. See [MS2, 2.2, C.1, C.3].
Mar 8 Sketch of index (see Mar 5); Aronsajn's theorem and unique continuation; finiteness of
critical points. See [MS2, C.4, 2.3-4].
Mar 10 (see Mar 8) Somewhere injective curves; transversality; energy. See [MS2, 2.5,
3.1-2, 4.1].
Mar 12 Isoperimetric inequality,
monotonicity lemma, removable singularities, example of bubbling. See [MS2, 4].
Mar 31 Bubbling, compactness,
nonsqueezing. See [MS2, 4, 9.3, 9.5].
Apr 2 (see Mar 31 notes) Nonsqueezing details, symplectomorphisms of S^2 x S^2.
See [MS2, 9.3, 9.5].
Apr 5 Symplectomorphisms of T^*S^2;
Floer theory intro. See [Se2] and [S1].