University of Toronto

MAT1347HS: Hamiltonian Group Actions, winter 2010.

Lectures: Mondays 3 - 4 and Wednesdays 12 - 2,   BA 2185.
Instructor: Professor Yael Karshon
Email:   karshon@math.toronto.edu
Office hours: Wednesday after class, or by appointment
Office:   BA 6119   (416) 978-7895

Presentation summaries

Crash course on manifolds:   pdf.
Crash course on flows:   pdf.
Proper actions of Lie groups (a chapter from my book): pdf.

This course is about Hamiltonian actions of compact Lie groups on symplectic manifolds.
Applications include using polytopes and graphs to read "classical" geometric information (volumes; symplectic packings)
and "quantum" information (dimension/multiplicities of group representations).


The following lecture notes are full of typos (and probably mistakes too).
The plan is to later turn them into something more coherent so please let me know of any mistakes that you find.
Problem set 1: pdf.
  • Notes of Monday Jan.4, 2010: pdf.
    (Group actions, orbit type strata.)
  • Notes of Wednesday Jan.6, 2010: pdf.
    (Momentum maps; Xrays; infinitesimal stabilizers.)
    Problem set 2: pdf.
  • Notes of Monday Jan.11, 2010: pdf.
    (Coadjoint action. Averaging.)
  • Notes of Wednesday Jan.13, 2010, excluding the ``linear algebra": pdf.
    (Smooth local linearization; consequences.)
  • ``Linear algebra" notes for Wednesday Jan.13 and Monday Jan.18, 2010: pdf.
    (Sp-equivariant strong deformation retraction from innner products to compatible inner products.)
  • Notes of Monday Jan.18, 2010: pdf.
    (Quotient spaces; slice theorem for free group actions.)
  • Notes of Wednesday Jan.20, 2010, only the part about the slice theorem: pdf.
    Problem set 3: pdf.
  • Notes of Jan.20,25,27 - the point set topology ``lokal global prinzip": pdf.
  • Notes of Jan.27, 2010: pdf.
    (Hamiltonian torus orbits are isotropic; for a torus Hamilton's equation implies invariance;
    existence and uniqueness of momentum maps; def of the symplectic slice representation modulo linear algebra;
    some symplectic linear algebra (subspaces, subquotients).)
  • Notes of Feb.1, 2010: pdf.
    (Preparation for the local normal form for a neighbourhood of an isotropic orbit; including
    discussion of push forward (fibrewise integration) of differential forms.)
  • Notes of Feb.3, 2010: pdf.
    (The symplectic slice representation determines a neighbourhood of the orbit.
    Properties of symplectic forms: Darboux theorem; Liouville measure; cohomology.
    Hard Lefschetz property. Exact momentum maps. Cotangent bundles. The case $T^*G$.
    Basic differential forms. Duality properties of the momentum map; corollaries. Symplectic reduction.)
  • Notes of Feb.8, 2010: pdf.
    (Linear symplectic actions and their quadratic momentum maps; the abelian case; Hamiltonian $G$ models with isotropic zero section.
    End of proof of the convexity package for Hamiltonian torus actions.)
    In order to get to more interesting material, from this point onwards I'll probably give fewer details.
    (But people are welcome to ask me for details in class or in person.) Also, I might back off from posting notes.
    Problem set 4: pdf.
    Problem set 5: pdf.

    Spring/summer 2010: the course is over, but I am back to typing my lecture notes. Tara Holm and I have been playing with the idea of writing a book
    about Hamiltonian actions of compact groups on symplectic manifolds. If this ever happens, these notes might be a first step.
  • Notes of Feb.10, 2010: pdf . (Fubini Study; Delzant uniqueness for toric models; momentum level sets = orbits for proper toric manifold;
    map from quotient to image is homeo for proper toric manifold.)
    The week of Feb.15 was winter break.
  • Notes of Feb.22, 2010: pdf . (Delzant construction.)
  • Notes of Feb.24, 2010: pdf . (Delzant uniqueness.)
    I *might* post here a review of sheaves and cohomology (which I didn't do in class).
  • Notes of March 1, 2010: pdf . (Application of Delzant uniqueness to symplectic embeddings. Hamiltonian G model for G a torus. Toric case.
    Consequences: Duistermaat-Heckman; symplectic area corresponds to rational length. Complex Delzant construction. Example: Hirzebruch. The bundle O(l).
  • Supplement to the notes of March 1, 2010: pdf . (The Lerman-Tolman-Woodward approach to Delzant uniqueness.)
  • Another supplement to the notes of March 1, 2010: pdf . (Schwarz lemma in the toric case.)
  • Notes of March 3, 2010: pdf . (The bundle O(l) over CP^1; blowups and their effect on topology and combinatorics; Cutting; first Chern class.
  • Notes of March 8, 2010, minus the proof: pdf . The proof: pdf . (The effect of blowup on homology; toric surfaces are blowups of CP2.)
    Notes of March 10, 2010 pdf . (Morse theory for momentum maps. Homology of toric surfaces. A compact symplectic four manifold admits only finitely many toric actions. Coadjoint orbits.)
    Notes of March 15, 2010 pdf . Continued: pdf . (More on coadjoint orbits.)
    Notes of March 17, 2010: pdf . (Duistermaat-Heckman.)
    On the week of March 22nd I was away. (I gave extra lectures on some Mondays to compensate for this.)
    Notes of March 29, 2010: pdf. (Equivariant cohomology.)
    Notes of March 31, 2010: pdf. (Kirwan surjectivity.)

    Requirements for credit: participation, assignments, presentations and written summaries.

    Non-inclusive list of relevant books and resources: