Math Math 268: Reading and Homework Assignments

Day 01: 08 January

Lagrangians on manifolds, the Euler-Lagrange equations, canonical coordinates on tangent bundles, the energy and canonical 1-form associated to a Lagrangian, symmetries
  1. Reading: Lecture 4, Introduction to Lie groups and symplectic geometry.
  2. Exercises: You should work on Exercises 1–5 at the end of this lecture. Note: Ilarion has pointed out that there is an error in Exercise 1. Where it claims that the Lagrangians L1 and L2 must differ by an exact 1-form df, one can actually only prove that the difference is a closed 1-form.

Day 02: 10 January

Non-degenerate Lagrangians, the Euler-Lagrange vector field on the tangent bundle, Noether's Theorem.
  1. Reading: Lecture 4, Introduction to Lie groups and symplectic geometry.
  2. Exercises: You should work on Exercises 6–12 at the end of this lecture.

Day 03: 13 January

No class: (Instructor in DC for meeting of VEF board)

Day 04: 15 January

The cotangent bundle, canonical coordinates, the canonical 1-form, the Legendre transformation, Hamiltonian form and Hamiltonians, Liouville metrics.
  1. Reading: Lecture 4, Introduction to Lie groups and symplectic geometry.
  2. Exercises: You should work on Exercises 6–12 at the end of this lecture.
  3. Further reading: Liouville metrics. An example of the use of the Hamiltonian point of view to integrate the equations for geodesic flow of certain Riemannian metrics.

Day 05: 17 January

Symplectic linear algebra, symplectic vector spaces, the symplectic group, normal forms, rank of a 2-form, subspaces of symplectic vector spaces, classification.
  1. Reading: Lecture 5, Introduction to Lie groups and symplectic geometry.
  2. Exercises: You should work on Exercises 1–7 at the end of this Lecture.

Day 06: 20 January

No class (Martin Luther King, Jr. holiday observed)

Day 07: 22 January

Symplectic manifolds and mappings. Examples: cotangent bundles, surfaces, products, symplectic submanifolds, coadjoint orbits.
  1. Reading: Lecture 5, Introduction to Lie groups and symplectic geometry. (up to p. 101, Darboux' Theorem)
  2. Exercises: You should work on Exercises 8–11 at the end of this Lecture.

Day 08: 24 January

Darboux' Theorem (classical proof), symplectic and Hamiltonian vector fields, the group of symplectomorphisms, its transitivity on sets of points, the Poisson bracket
  1. Reading: Lecture 5, Introduction to Lie groups and symplectic geometry. (finish the Lecture)
  2. Exercises: You should work on Exercises 12–14 at the end of this Lecture.

Day 09: 27 January

Obstructions to symplectic structure existence: cohomology ring and characteristic classes. Almost symplectic structures and almost complex structures. Darboux-Moser-Weinstein theorem and the local moduli of symplectic structures.
  1. Reading: Lecture 6, Introduction to Lie groups and symplectic geometry. (up to "Submanifolds of symplectic manifolds")
  2. Exercises: You should work on Exercises 1–6 at the end of this Lecture.

Day 10: 29 January

Weinstein's Theorem, symplectic submanifolds, Lagrangian submanifolds.
  1. Reading: Lecture 6, Introduction to Lie groups and symplectic geometry. (to the end)
  2. Exercises: You should work on Exercises 7–12 at the end of this Lecture.

Day 11: 31 January

Group actions, Hamiltonian and Poisson actions, Lie algebra cohomology, the momentum mapping
  1. Reading: Lecture 7, Introduction to Lie groups and symplectic geometry. (up to Reduction on p. 134)
  2. Exercises: You should work on Exercises 1–8 at the end of this Lecture.

Day 12: 03 February

The momentum mapping of a linear symplectic action, remarks on group actions, clean values of smooth mappings, the Marsden-Weinstein reduction theorem.
  1. Reading: Lecture 7, Introduction to Lie groups and symplectic geometry. (to the end of the Lecture)
  2. Exercises: You should work on Exercises 9–13 at the end of this Lecture.

Day 13: 05 February

No class! (Instructor in DC for meeting of VEF board)
Also: No office hours today!

Day 14: 07 February

Statement of the convexity theorem of Atiyah-Guillemin-Sternberg. Compatible almost complex structures and metrics. Almost Kähler, almost Hermitian, and Kähler structures. Examples of manifolds admitting symplectic structures but no complex structure, complex structure but no symplectic structure, complex structure or symplectic structure but no Kähler structure
  1. Reading: Lecture 8, Introduction to Lie groups and symplectic geometry. (to Kähler reduction)
  2. Exercises: You should work on Exercises 1–2 at the end of this Lecture.

Day 15: 10 February

Kähler reduction at 0. The shifting trick. Kähler reduction at clean momentum values for which the coadjoint orbit supports a Kähler structure.
  1. Reading: Lecture 8, Introduction to Lie groups and symplectic geometry. (the section on Kähler reduction)
  2. Exercises: You should work on Exercises 3–4 at the end of this Lecture.

Day 16: 12 February

Examples and non-examples of Kähler structures on coadjoint orbits. The coadjoint orbits of SL(2,R). The coadjoint orbits of the unitary group U(n). Existence of a Kähler structure on any coadjoint orbit of any compact semi-simple Lie group.
  1. Reading: Lecture 8, Introduction to Lie groups and symplectic geometry. (the section on Kähler reduction)
  2. Exercises: Calculate the Kähler structures on the (co)adjoint orbits of SO(n).

Day 17: 14 February

Toric Varieties I, examples, orbifold examples, compact examples, Kahler holonomy, quaternion linear algebra, quaternion Hermitian structures, Sp(n), the invariant 3-forms, hyperKähler structures on manifolds
  1. Reading: Lecture 8, Introduction to Lie groups and symplectic geometry. (the section on hyperKähler reduction)
  2. Exercises: Work on Exercises 3–4 at the end of this Lecture.

Day 18: 17 February

No class (Classes cancelled by provost due to inclement weather.)

Day 19: 19 February

HyperKähler manifolds, integrability, hyperKähler reduction, examples.
  1. Reading: Lecture 8, Introduction to Lie groups and symplectic geometry. (the section on hyperKähler reduction)
  2. Exercises: Work on the remaining exercises at the end of this lecture. (Please note that there is a new version on the web of Lectures 7 and 8.

Day 20: 21 February

Review of Morse theory, the Hessian at a critical point of a smooth function on M, nondegenerate critical points, the Morse Lemma, Morse-Bott functions and the Morse-Bott Lemma. Construction of a vector field F associated to a Morse-Bott function f on a compact manifold whose flow lines have unique alpha and omega limit points and along which the function is strictly decreasing.
  1. Reading: pp. 180-191 of McDuff and Salamon (the section on Morse-Bott functions, particularly).
  2. Exercises: Write out an explicit proof of the Morse-Bott Lemma along the lines of the proof of the Morse Lemma.

Day 21: 24 February

No class (Instructor at MIT for colloquium talk)

Day 22: 26 February

Connectedness of level sets of a Morse-Bott function. Existence of G-invariant compatible metrics when a compact group G acts on a symplectic manifold M. The symplectic nature of the fixed point set.

Day 23: 28 February

No class (Instructor at BIRS for MSRI Board of Trustees Meeting)

Day 24: 03 March

The equivariant Darboux-Weinstein Theorem. Hamiltonian functions of Hamiltonian compact group actions are Morse-Bott. A normal form for the momentum mapping near a component of the fixed locus of a Hamiltonian compact torus action.

Notes: Convexity notes. This is a beta release. Please inform me of corrections or any confusing passages.

Day 25: 05 March

Stabilizer subgroups of Hamiltonian torus actions, the convexity theorem of Atiyah and Guillemin-Sternberg (at last)!

Day 26: 07 March

Toric manifolds, restrictions on the momentum polytope, integrality conditions, location of singular values, existence and topology of toric manifolds. The push-forward of the Liouville measure under the momentum mapping.

Day 27: 17 March

Day 28: 19 March

Day 29: 21 March

Day 30: 24 March

Day 31: 26 March

Day 32: 28 March

Day 33: 31 March

Day 34: 02 April

Day 35: 04 April

Day 36: 07 April

Day 37: 09 April

Day 38: 11 April

Day 39: 14 April

Day 40: 16 April

Day 41: 18 April

Note: This is the official end of graduate classes, but...

Day 42: 21 April

Day 43: 23 April