Janko Latschev
Lecture course Symplectic geometry, Winter semester 2024
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.
Here is a summary of the main exam topics, together with some general remarks concerning the exam.
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
Sheet 12
Sheet 13
Other material will also appear here as needed.
Log of lecture content
| Oct 15 |
brief introduction; linear symplectic geometry: basic definitions: symplectic form, orthogonal complements, types of subspaces, existence of symplectic basis (adapted to a subspace), first consequences, relation of standard symplectic structure to standard euclidean structure and standard complex structure on R2n, linear symplectic group
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| Oct 16 |
properties of symplectic matrices, relation of Sp(2n,R) to O(2n), GL(n,C) and U(n), tamed and compatible complex structures: basic definitions and remarks, the set of compatible complex structures is contractible, hermitian metrics and relation to symplectic forms |
| Oct 22 |
definition and basic properties of symplectic manifolds, examples: R2n, surfaces, tori, products, cotangent bundles; start of discussion of complex projective space
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| Oct 23 |
the symplectic form on complex projective space; symplectomorphisms, Hamiltonian vector fields and flows, invariance of the symplectic form, Hamilton's equations
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| Oct 29 |
example: height function on the sphere, symplectic vector fields, commutator of symplectic vector fields is Hamiltonian, definition of Poisson bracket
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| Oct 30 |
properties of the Poisson bracket, integrable systems, Arnold-Liouville theorem
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| Nov 05 |
the (planar) mathematical pendulum, relation of Lagrangian formulation of classical mechanics to Hamiltonian formulation via Legendre transform, geodesic flow as example
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| Nov 06 |
conclusion on Legendre transform of pure kinetic energy on a Riemann manifold; the spherical mathematical pendulum; definition of Hamiltonian isotopy
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| Nov 12 |
Hamiltonian diffeomorphisms; Moser's argument, application to volume forms, isotopy of cohomologous forms
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| Nov 13 |
Darboux' theorem on local standard coordinates, first consequences; special submanifolds of symplectic manifolds: first examples
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| Nov 19 |
(local) generating functions for symplectomorphisms of R2n and of cotangent bundles, example: convex billard
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| Nov 20 |
convex billard (continued); review: vector bundles, tubular neighborhood theorem for submanifolds
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| Nov 26 |
normal bundle of a Lagrangian submanifold, Weinstein neighborhood theorem for Lagrangian submanifolds
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| Nov 27 |
first consequences of the Weinstein neighborhood theorem; Lagrangian submanifolds in R2n: classification questions and examples
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| Dec 3 |
Lagrangian Grassmannian L(n) is U(n)/O(n), the Maslov index of a loop in L(n), characterizing properties, Maslov class of a Lagrangian submanifold in R2n
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| Dec 4 |
remarks on Maslov class and symplectic area class, Chekanov's classification of product tori in R2n, monotone Lagrangian submanifolds; symplectic group actions, Hamiltonian S1-actions, symplectic reduction for S1-actions
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| Dec 10 |
symplectic cuts, example of symplectic blow-up; Lie group actions
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| Dec 11 |
Hamiltonian action of a Lie group on a symplectic manifold, moment map
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| Dec 17 |
examples of Hamiltonian actions and their moment maps
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| Dec 18 |
brief discussion on coadjoint orbits; isotropic leaves in a coisotropic submanifold, quotient of a regular coisotropic submanifold is symplectic, Marsden-Weinstein reduction, example
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| Jan 07 |
toric symplectic manifolds: definition, examples and basic facts; almost complex vs. complex manifolds: Nijenhuis tensor and Newlander-Nirenberg theorem
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| Jan 08 |
Kähler manifolds: definitions and examples, analysis on complex manifolds, describing the Kähler form in local complex coordinates
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| Jan 14 |
local Kähler potentials, examples; J-holomorphic curves: definition and reformulations of the defining equation, energy of a map and energy of holomorphic curves in symplectic manifolds
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| Jan 15 |
consequence of formula for energy, spaces of J-holomorphic curves, regularity theorems, aside on first Chern class
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| Jan 21 |
sketch of proof strategy for generic regularity theorem; discussion of compactness assuming gradient bounds, illustration of breakdown of compactness for quadrics in CP2
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| Jan 22 |
general argument that failure of gradient bounds leads to sphere bubbles;
motivation and statement of the nonsqueezing theorem, start of the proof of the nonsqueezing theorem using holomorphic curves: existence statement for J-holomorphic spheres in the fiber class in VxS2 through every point for every compatible J (assuming V has no nonconstant holomorphic spheres)
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| Jan 28 |
proof of the nonsqueezing theorem, assuming the monotonicity lemma
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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, 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 |
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