Symplectic Geometry

Meetings

MW, 1:00-2:20pm, in Javits 101, simultaneously broadcasted over Zoom

Recordings of lectures are posted below under password protection: e-mail the instructor for access

Instructor

Kevin Sackel, kevin.sackel(at)stonybrook.edu

Office Hours: M 5-7 PM; also by appointment!

Sources

There will be no unique textbook or reference from which the material will be drawn. Here is a list of some texts I expect to follow at some point in the term:

• Mathematical Methods of Classical Mechanics - V.I. Arnold

• Lectures on Symplectic Geometry - A.C. da Silva

• Introduction to Symplectic Topology - D. McDuff and D. Salamon

• J-holomorphic Curves and Symplectic Topology - D. McDuff and D. Salamon

• An Introduction to Contact Topology - H. Geiges

• Introduction to the h-principle - Y. Eliashberg and N. Mishachev

Syllabus

Syllabus

Description from the Bulletin: Hamilton’s equations and their physical origin, symplectic manifolds and various submanifolds, Moser arguments including Darboux theorem and Moser neighborhood theorems, contact manifolds, contact hypersurfaces, symplectizations, Legendrian front diagrams, topological Legendrian knot invariants, almost complex structures compatible with symplectic form, Hamiltonian group actions and symplectic reduction, symplectic toric manifolds, h principle with emphasis on holonomic approximation theorem along with applications to symplectic and contact geometry, Gromov non squeezing theorem and a summary of pseudoholomorphic curve theory.

Homework

Homework (82 Problems, FINAL UPDATE (November 28))

Schedule / Lecture Notes / Lecture Slides

VIDEOS OF LECTURES AVAILABLE BY REQUEST

LECTURE NOTES (last updated November 10)

Lecture 1 (M, Aug 24): Newtonian Mechanics, Lagrangian Mechanics, the Legendre Transform

Lecture 2 (W, Aug 26): The Riemannian setting for physics, Hamiltonian mechanics, the emergence of symplectic geometry (video)

Lecture 3 (M, Aug 31): Symplectic linear algebra (video1, video2; slides)

Lecture 4 (W, Sept 2): Relations to complex linear algebra, U(n) is homotopy equivalent to Sp(2n) (video; slides)

Lecture XXX (M, Sept 7): NO CLASS (Labor Day)

Lecture 5 (W, Sept 9): Linear complex structures, compatibility and tameness, compatible triples (video; slides)

Lecture 6 (M, Sept 14): Lagrangian Grassmannian, Maslov class, symplectic vector bundles, G-structures (video; slides, additional_slides)

Lecture 7 (W, Sept 16): Classifying spaces, obstruction theory, existence of almost symplectic structure (video, slides)

Lecture 8 (M, Sept 21): Symplectic form in H^2; Examples of symplectic manifolds; CP^n (video, slides)

Lecture 9 (W, Sept 23): More examples of symplectic manifolds; Hamiltonian diffeomorphisms (video, slides)

Lecture 10 (M, Sept 28): Symplectomorphisms vs Hamiltonian diffeomorphisms; flux; the Arnold Conjecture (video, slides)

Lecture 11 (W, Sept 30): Submanifolds; conormal bundles; exact Lagrangians (video, slides)

Lecture 12 (M, Oct 5): Contact manifolds, coorientation, isotropic and Legendrian submanifolds, hypersurfaces of contact type (video, slides)

Lecture 13 (W, Oct 7): Strong and exact fillings, symplectization, Reeb field, Weinstein conjecture (video, slides)

Lecture 14 (M, Oct 12): Contactomorphisms, contact vector fields, contact Hamiltonians, contactization (video, slides)

Lecture 15 (W, Oct 14): Lagrangian and front projections, Legendrian knots, classical invariants (video, slides)

Lecture 16 (M, Oct 19): Bennequin inequality, overtwisted R^3, Moser's trick, symplectic Moser theorem (video, slides)

Lecture 17 (W, Oct 21): Gray stability, Darboux theorem, contact Darboux theorem (video, slides)

Lecture 18 (M, Oct 26): Tubular neighborhoods in symplectic (and contact) geometry (video, slides)

Lecture 19 (W, Oct 28): Isotropic, Lagrangian, and Legendrian neighborhood theorems (video, slides)

Lecture 20 (M, Nov 2): Coisotropic submanifolds, characteristic foliation, coisotropic reduction, symplectic Lie group actions (video, slides)

Lecture 21 (W, Nov 4): Lie theory recollections, Hamiltonian G-spaces, moment maps (video, slides)

Lecture 22 (M, Nov 9): Marsden-Weinstein reduction, Noether's principle, Hamiltonian torus actions, symplectic toric manifolds, Arnold-Liouville theorem (special case) (video, slides)

Lecture 23 (W, Nov 11): Atiyah/Guillemin-Sternberg Theorem (statement only), Delzant polytopes (statement only), Riemann surfaces, Dirichlet energy (video, slides)

Lecture 24 (M, Nov 16): Energy identity, pseudo-holomorphic curves, topological nature of energy, del-bar operator (video, slides)

Lecture 25 (W, Nov 18): Properties of pseudo-holomorphic curves, simple curves, regular almost complex structures, moduli spaces of pseudo-holomorphic curves (video, slides)

Lecture XXX (M, Nov 23): NO CLASS (Thanksgiving week)

Lecture XXX (W, Nov 25): NO CLASS (Thanksgiving week)

Lecture 26 (M, Nov 30):

Lecture 27 (W, Dec 2):

Lecture 28 (M, Dec 7):