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### 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 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 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 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 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

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  Impressum 2020-01-31, Janko Latschev