Janko Latschev
Lecture course Symplectic geometry, Wintersemester 2019/20
The lectures take place Tuesday 24 and Thursday 1012 in H5.
The exercise class takes place Thursday 1214 in room Geom 430.
There will be oral exams at the end of the course. Active participation in the exercise classes is a prerequisite for admission to the exam.
The aim of this course is to give an introduction to modern symplectic geometry. We will start from its origins in Hamiltonian dynamics, cover linear symplectic geometry as well as basic properties of symplectic and contact manifolds (and their relation), and move on to discuss some of the typical questions and the methods which can be used to address them.
The exercise sheets will be posted here:
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Other material will also appear here as needed.
Log of lecture content
15.10. 
brief introduction; linear symplectic geometry: basic definitions: symplectic form, orthogonal complements, types of subspaces, existence of symplectic basis (adapted to a subspace), consequences, relation of standard symplectic structure to standard euclidean structure and standard complex structure on R^{2n} 
18.10. 
linear symplectic group, properties of symplectic matrices, relation of Sp(2n,R) to O(2n) and GL(n,C), fundamental group of Sp(2n,R), axiomatic characterization of the Maslov index for loops of symplectic matrices (statement) 
22.10. 
proof of existence and uniqueness of Maslov index for loops of symplectic matrices, Lagrangian Grassmannian is U(n)/O(n), fundamental group, axiomatic characterization of the Maslov index for loops of Lagrangian subspaces

24.10. 
example and remarks on Maslov index, tamed and compatible complex structures: contractible set of choices, hermitian metrics and relation to symplectic forms, affine nonsqueezing theorem

29.10. 
basic properties of symplectic manifolds, examples: surfaces, cotangent bundles, complex projective space, symplectomorphisms, Hamiltonian vector fields and flows

05.11. 
examples of Hamiltonian flows, Poincaré recurrence for symplectomorphisms, symplectic vector fields, symplectic isotopies, Hamiltonian diffeomorphisms

07.11. 
long aside on Legendre transform, Moser's argument: application ot volume forms and to symplectic forms on closed manifolds

12.11. 
Darboux' theorem, types of submanifolds with examples, tubular neighborhood theorem of differential topology, symplectic vector bundles

14.11. 
neighborhood theorem for Lagrangian submanifolds, remark on generalizations to other types of submanifolds, description of C^{1}neighborhood of identity in Symp(M, ω), examples of Lagrangian submanifolds in R^{2n}

The following books and lecture notes are useful study material for various parts of the course.
For background on manifolds, flows, Lie derivative, etc.:
F. Warner  Foundations of differentiable manifolds and Lie groups  Springer Verlag 
M. Spivak  A comprehensive introduction to differential geometry, vol. 1  Publish or Perish 
I. Madsen, J.Tornehave  From calculus to cohomology  Cambridge University Press 
For background on differential topology (tubular neighborhood theorem, intersection theory, etc.):
J. Milnor  Topology from the differentiable viewpoint  The University of Virginia Press 
V. Guillemin, A. Pollack  Differential topology  Prentice Hall 
M. Hirsch  Differential topology  Springer Verlag 
For general topics in symplectic geometry:
D. McDuff, D. Salamon  Introduction to symplectic topology  Oxford University Press 
A. Canas da Silva  Lectures on Symplectic Geometry  Springer Lecture Notes in Mathematics 1764 
H. Hofer, E. Zehnder  Symplectic Invariants and Hamiltonian dynamics  Birkhäuser 
L. Polterovich  The Geometry of the Group of Symplectic Diffeomorphisms  Birkhäuser 
For some relations to physics:
V.I. Arnold  Mathematical methods of classical mechanics  Springer Verlag 
V. Guillemin, S. Sternberg  Symplectic techniques in physics  Cambridge University Press 
