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 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 STiNE-mail 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 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
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| 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
|
| 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, Borman-Eliashberg-Murphy theorem, open books, 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
|
| 03.12. |
almost complex manifolds vs. complex manifolds: Nijenhuis tensor and Newlander-Nirenberg theorem, Kähler structures on manifolds: equivalent characterizations, examples
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| 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, C0 rigidity of symplectomorphisms, nonsqueezing theorem (statement), Hofer's metric on Ham(M,ω), Arnold's conjecture on fixpoints of Hamiltonian diffeomorphisms
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| 17.12. |
J-holomorphic 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 J-holomorphic map, real and complex linear Cauchy-Riemann 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 real-linear Cauchy-Riemann operators: Calderon-Zygmund inequality (statement, consequences and idea of proof), existence of local solutions to Du=0 for real linear Cauchy-Riemann operators
|
| 09.01. |
proof of local existence theorem, Carleman similarity principle and consequences, moduli spaces of j-holomorphic 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
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| 21.01. |
compactness of moduli spaces of J-holomorphic 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 J-holomorphic 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 J-holomorphic curves
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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 | 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 |
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