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 exam opportunities on February 17/18 and on March 30/31. Active participation in the exercise classes is a prerequisite for admission to the exam. Please consult your STiNEmail for details about your personal exam time.
Here is a summary of the main exam topics, together with some general remarks concerning 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
Sheet 5
Sheet 6
Sheet 7
Sheet 8
Sheet 9
Sheet 10
Sheet 11
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}

19.11. 
remarks on Maslov index and symplectic energy for Lagrangian submanifolds; contact manifolds: definition and basic examples, hypersurfaces of contact type, symplectization

21.11. 
invariant description of symplectization, aside on origins of contact geometry: the method of characteristics for first order pde

26.11. 
contactomorphisms, Reeb vector field associated to a contact form, general contact vector fields and relation to contact Hamiltonians, Darboux' theorem for contact forms

28.11. 
Gray stability theorem, uniqueness of induced contact structure on hypersurfacves of contact type; existence of contact structures, BormanEliashbergMurphy theorem, open books, contact forms supported by an open book, ThurstonWikelnkempner theorem in dimension 3, existence of open books in dimension 3, existence of contact structures as a consequence

03.12. 
almost complex manifolds vs. complex manifolds: Nijenhuis tensor and NewlanderNirenberg theorem, Kähler structures on manifolds: equivalent characterizations, examples

05.12. 
analysis on complex manifolds: complexified tangent and cotangent bundles, ∂ and ∂ operators, Kähler forms are (1,1)forms with hermitian positive definite coefficient matrix, local Kähler potentials

10.12. 
examples of Kähler potentials, Stein manifolds, relation to Weinstein manifolds; the existence problem for symplectic structures

12.12. 
more motivating questions: the uniqueness problem for symplectic structures, C^{0} rigidity of symplectomorphisms, nonsqueezing theorem (statement), Hofer's metric on Ham(M,ω), Arnold's conjecture on fixpoints of Hamiltonian diffeomorphisms

17.12. 
Jholomorphic curves: definition and reformulations of the defining equation, energy of a map and energy of holomorphic curves in symplectic manifolds, preliminary remarks for the linearization of the ∂_{J} operator

19.12. 
computation of the linearization of the ∂_{J} operator at a Jholomorphic map, real and complex linear CauchyRiemann operators, relation of complex linear CR operators to holomorphic structures on the underlying complex vector bundle

07.01. 
crash course on Sobolev spaces, elliptic regularity for reallinear CauchyRiemann operators: CalderonZygmund inequality (statement, consequences and idea of proof), existence of local solutions to Du=0 for real linear CauchyRiemann operators

09.01. 
proof of local existence theorem, Carleman similarity principle and consequences, moduli spaces of jholomorphic curves, statement of Fredholm property for linearized ∂_{J} operator

14.01. 
proof of the Fredholm property for linearized ∂_{J} operator and the index formula

16.01. 
manifold structure on universal moduli space of simple curves and existence of regular almost complex structures

21.01. 
compactness of moduli spaces of Jholomorphic curves: uniform gradient bounds yield convergence of a subsequence, failure of gradient bounds leads to bubbling 
23.01. 
a simple example for bubbling, nonconstant spheres have a lower bound on their energy, rough statement of Gromov compactness for curves with a fixed underlying domain and energy bounds

28.01. 
proof of the nonsqueezing theorem using holomorphic curves: existence of Jholomorphic spheres in the fiber class through every point for every J, monotonicity lemma

30.01. 
some consequences of nonsqueezing and examples of other results from Gromov's paper introducing Jholomorphic curves

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 contact topology:
H. Geiges  An introduction to contact topology  Cambridge University Press 
For holomorphic curves in symplectic geometry:
D. McDuff, D. Salamon  Jholomorphic curves in symplectic topology  AMS Colloquium Series 
C. Wendl  Lectures on holomorphic curves 
M. Audin, J. Lafontaine (eds.)  Holomorphic curves in symplectic geometry  Birkhäuser Progress in Math. 117 
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 
