Mastermath Course on Symplectic Geometry, spring semester, 2018/ 2019

Syllabus for the Mastermath Course on Symplectic Geometry, spring semester, 2018/ 2019

``MS'' refers to the book ``Introduction to Symplectic Topology'' by D. McDuff and D. A. Salamon.

``CdS'' refers to the Lectures on symplectic geometry A. by Cannas da Silva.

``Geiges'' refers to the book ``Introduction to contact topology'' by H. Geiges

The solutions to the assignments will be posted on the mastermath website.
day sections material homework
February 7 MS Sections 1-13 overview over lecture course, linear symplectic geometry Assignment 1
February 14 MS Section 2.1 review of bilinear maps, definition of linear symplectic structures and their subspaces, symplectic reduction, proof of the uniqueness of linear symplectic structures Assignment 2
February 21 MS Sections 2.2, 2.5 compatible triples consisting of a linear symplectic form, a linear complex structure, and an inner product, the linear symplectic group Assignment 3
February 28 MS Sections 3.1, 1.1, CdS Part VII, Section 20 definition, examples, properties, and orientation of symplectic manifolds, the cotangent bundle, Lagrangian mechanics, action and the Euler Lagrange equation Assignment 4
March 7 MS Sections 1.1, 3.1, CdS Part VII, Section 20 Hamilton's equation on the cotangent bundle, Legendre transform, symplectomorphism, symplectic and Hamiltonian vector fields Assignment 5
March 14 MS Sections 3.1, 3.2 symplectic isotopy and symplectic vector fields, Hamiltonian flow, diffeomorphism, isotopy, and group, the Poisson bracket Assignment 6
March 21 MS Section 3.3 examples of submanifolds of symplectic manifolds, homotopy, isotopy, Moser's trick Assignment 7
March 28 MS Section 3.3 coisotropics (particularly Liouville and cosymplectic hypersurfaces), Moser's trick and Darboux' theorem Assignment 8
April 4 MS Sections 3.2 and 3.3 Local models for submanifolds. In particular, the Weinstein tubular neighbourhood for Lagrangians. Some applications. Assignment 9
April 11 MS Chapter 10 = lecture notes Section 3.7 algebraic structure of the symplectomorphism group, symplectic flux Assignment 10
April 18 MS Sections 5.2, 5.3, 5.4 (co-adjoint) action, symplectic and Hamiltonian Lie group actions, exact action Hamiltonian, induced action on cotangent bundle, characterization of Hamiltonian actions, symmetries of a mechanical system, Noether's principle, symplectic quotient = Marsden-Weinstein quotient Assignment 11
April 25 MS Section 5.4 symplectic quotient = Marsden-Weinstein quotient, symplectic reduction and the cotangent bundle construction, reduced dynamics, application to the two-body problem Assignment 12
May 2 symplectic quotient of a regular presymplectic manifold, coadjoint orbit, Kostant-Kirillov-Souriau symplectic form, coadjoint action is Hamiltonian Assignment 13
May 9 Geiges Sections 2.1-2.5 contact and cosymplectic hypersurfaces in symplectic manifolds, contact structures, contactomorphisms, Reeb flows, and isotropic/Legendrian/transverse submanifolds Assignment 14
May 16 3-dimensional Contact Topology: Examples, Legendrian knots and their front and Lagrangian projections. Assignment 15
May 23 Construction of Legendrian knots, rotation number. Overview of flexibility in Contact and Symplectic. Assignment 16
exam: June 20