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Janko Latschev

Lecture course  Symplectic geometry, Wintersemester 2019/20

The lectures take place Tuesday 2-4 and Thursday 10-12 in H5.
The exercise class takes place Thursday 12-14 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 R2n
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 C1-neighborhood of identity in Symp(M, ω), examples of Lagrangian submanifolds in R2n

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

  Seitenanfang  Impressum 2019-11-14, Janko Latschev