Janko Latschev
Lecture course Symplectic geometry, Summer semester 2023
The lectures take place Tuesday 8-10 in H2 and Thursday 8-10 in H5.
The exercise class this week will take place Thursday 10-12. Starting the week of April 17, the exercise class will take place Tuesday 2-4 in room Geom 434.
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 linear symplectic geometry and follow with the discussion of basic properties of symplectic and contact manifolds (and their relation). We then move on to discuss some of the typical questions and the methods which can be used to address them, most notably holomorphic curves.
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
Other material will also appear here as needed.
Log of lecture content
| 11.04. |
brief introduction; linear symplectic geometry: basic definitions: symplectic form, orthogonal complements, types of subspaces, existence of symplectic basis (adapted to a subspace), first consequences |
| 13.04. |
relation of standard symplectic structure to standard euclidean structure and standard complex structure on R2n, linear symplectic group, properties of symplectic matrices, relation of Sp(2n,R) to O(2n), GL(n,C) and U(n), fundamental group of Sp(2n,R), axiomatic characterization of the Maslov index for loops of symplectic matrices (statement) |
| 18.04. |
proof of existence and uniqueness of Maslov index for loops of symplectic matrices, Lagrangian Grassmannian is U(n)/O(n), fundamental group of U(n)/O(n) is Z
|
| 20.04. |
axiomatic characterization of the Maslov index for loops of Lagrangian subspaces, example and remarks on Maslov index, tamed and compatible complex structures: basic definitions and remarks
|
| 25.04. |
the set of compatible complex structures is contractible, hermitian metrics and relation to symplectic forms; definition and basic properties of symplectic manifolds, examples: R2n, surfaces, products, tori, cotangent bundles
|
| 27.04. |
complex projective space as a complex manifold, construction of the Fubini-Study form, expression in homogeneous coordinates
|
| 02.05. |
symplectomorphisms, Hamiltonian vector fields and flows, examples of Hamiltonian flows, relation to calculus of variations via Legendre transform between tangent and cotangent bundles
|
| 04.05. |
commutator of symplectic vector fields is Hamiltonian, aside on Poisson bracket, symplectic and Hamiltonian isotopies, definition of Hamiltonian diffeomorphism; Poincaré recurrence
|
| 09.05. |
Moser's argument, isotopies for volume forms of the same total volume, Moser stability for symplectic forms in the same cohomology class, Darboux' theorem, transitivity of the group of Hamiltonian diffeomorphisms
|
| 11.05. |
types of submanifolds with examples, method of generating functions for constructing symplectomorphisms of (subsets of) cotangent bundles, billard as an example
|
| 23.05. |
discussion of convex billard, for more, see section 8.3 in McDuff/Salamon or the book by S. Tabachnikov; tubular neighborhood theorem of differential topology
|
| 30.05. |
neighborhood theorem for Lagrangian submanifolds, remark on generalizations to other types of submanifolds, description of C1-neighborhood of identity in Symp(M, ω), discussion of Lagrangian submanifolds in R2n
|
| 01.06. |
examples of Lagrangian embeddings in R2n, Maslov index and symplectic area for Lagrangian submanifolds; contact manifolds: definitions and first remarks
|
| 06.06. |
basic examples of contact manifolds, hypersurfaces of contact type, symplectization
|
| 08.06. |
contactomorphisms, Reeb vector field associated to a contact form, general contact vector fields and relation to contact Hamiltonians
|
| 13.06. |
Darboux' theorem for contact forms, Gray stability theorem, uniqueness of induced contact structure on hypersurfaces of contact type
|
| 15.06. |
existence of contact structures, Borman-Eliashberg-Murphy theorem, overtwisted contact structures in dimension 3, open book decompositions
|
| 20.06. |
contact forms supported by an open book, Thurston-Wikelnkempner theorem in dimension 3, existence of open books in dimension 3, existence of contact structures as a consequence; current research directions in contact topology
|
| 22.06. |
some motivating questions for research in symplectic topology
|
| 27.06. |
Kähler manifolds as symplectic manifolds with compatible complex structure: Nijenhuis tensor and Newlander-Nirenberg theorem on integrability of almost complex structures, characterization of Kähler manifolds in terms of NJ
|
| 29.06. |
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, examples
|
| 04.07. |
J-holomorphic curves: definition and reformulations of the defining equation, energy of a map and energy of holomorphic curves in symplectic manifolds
|
| 06.07. |
spaces of J-holomorphic maps, generic regularity theorem, aside on first Chern class; discussion of compactness assuming gradient bounds, illustration of breakdown of compactness for quadrics in CP2
|
| 11.07. |
general argument that failure of gradient bounds leads to sphere bubbles; start of the proof of the nonsqueezing theorem using holomorphic curves: existence of J-holomorphic spheres in the fiber class in VxS2 through every point for every compatible J (assuming V has no nonconstant holomorphic spheres)
|
| 13.07. |
completion of the proof of non-squeezing, the Gromov width as an example of a symplectic capacity, further results from Gromov's 1985 paper with a sketch of proof: non-existence of exact Lagrangian embeddings of closed manifolds into R2n, existence of exotic symplectic structures on R2n, symplectic forms standard at infinity in R4 are standard |
The following books and lecture notes are useful study material for various parts of the course.
For background on manifolds, flows, Lie derivative, differential forms, 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 background on algebraic topology (fundamental group, homology, cohomology, etc.):
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 | J-holomorphic 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 |
|
