Math 242: Symplectic geometry

Logistics:
Lectures: TuTh 2:00-3:30pm, Room 71 Evans Hall
Course Control Number: 55248
Office: 1033 Evans Hall
Office Hours: TuTh at tea
Prerequisites: Math 214 or equivalent.
GSI: Olga Radko
Text: Ana Canas da Silva's notes (see below)
Additional Reading:"Supersymmetry and equivariant de Rham theory", by Guillemin and Sternberg
Grading: Homework and a final
Homework: weekly

The focus:

I will emphasize symplectic geometry from the Kahler geometry (rather than Poisson) viewpoint, and most of my examples will be from the algebraic-geometry world. While symplectic techniques are proving useful in 4-manifold theory, I will instead be thinking of them as nice places to see Lie groups act (no prior knowledge of Lie groups will be assumed, however).

Definitely we will see Atiyah/Guillemin-Sternberg's convexity theorem, and Delzant's reconstruction theorem, relating symplectic geometry to toric geometry (and hence to combinatorics). With luck, we'll see how moment maps fit into "perfect Morse theory" and equivariant topology/cohomology.


Errata to the final (one serious!)

In question 3, there is repeated confusion between p (in the manifold) and Phi(p) (in the moment polytope). From this you can figure out which one is supposed to be which.

Question 3 is supposed to be about connected manifolds. Correction: Phi(p) is not just on the boundary, but actually a vertex. For part b you can assume q a regular value if you want.

Important correction: In question 4, parts b and c, you must assume that the action of S^1 is free on the fiber over 0, not just locally free, or else the problem is false! (Example: let S^1 act on C^x with weight n, instead of weight 1. Then the DH measure will be 1/n times Lebesgue measure.)

The text is Ana Canas da Silva's "Lectures on Symplectic Geometry" and is available from Copy Central Southside, at 2560 Bancroft Way, (510) 848-9600. The price is about $16.

Homeworks will in general be due on Thursdays.

Olga's email address is radko@math.berkeley.edu.

Homework 1 is due on Thursday Sep 13; sorry I forgot to define "Witt's Theorem" in class. It is this:

Let A be a vector space with symplectic form w.
Let B and C two subspaces (not necessarily symplectic!).
Let f:B->C be a linear isomorphism of B with C such that w(b1,b2) = w(f(b1),f(b2)) for any b1,b2 in B.
Show that there exists a symplectic isomorphism g: A->A that equals f when restricted to B.

Homework 2:

  • p48 #1,2,3
  • A volumetric form of Moser: show that if v_t is a family of volume forms on a compact manifold M, all with the same total volume, then there exists a smooth family of autodiffeomorphisms phi_t such that phi_t^*(v_t) = v_0.
  • Let P be a polytope such that every corner is an orthant, so the corresponding toric manifold M_P is smooth. Define a "porthant", for portion-of-orthant, as a contractible open set of an orthant containing the corner.
  • Find a way to write P as a disjoint union of porthants of various dimensions. (For example, a triangle is a point = a 0-d porthant, union a segment minus a point = a 1-d porthant, union the triangle minus a side = a 2-d porthant.)
  • Show that the preimage of a porthant, in M_P, is an even-dimensional cell and compute the Betti numbers of M_P.
  • Let T be a torus, and H a finite subgroup. Then the Lie algebra of T and of T/H, which we will call T', can be identified. Consider the kernel of exp: t -> T, a lattice in the Lie algebra t, and let L be the dual lattice in t^* (those vectors with integer pairing against the kernel).
  • What is the relation between L and L', the corresponding lattice for T'?
  • Given a polytope P in t^*, we can construct toric spaces by quotienting PxT or PxT' by the equivalence relation defined in class. What is the relation between these two spaces?
  • Let P be the quadrant in R^2, below y=x and above y=-x, with x nonnegative. Let T be the 2-torus such that L is the integer lattice; with respect to this L, P is not an orthant. Show that M_P is C^2 modulo the action of Z_2 acting by +/-1. Hint: in the foregoing problems, this Z_2 appears as H.
  • If P is a rational cone in R^n centered at the origin, and generated by n linearly independent integer vectors {v_i}, show that M_P is a quotient of C^n by a finite group, and is smooth iff the {v_i} generate the integer lattice. ("If" is pretty easy; for "only if", think about pi_1 of M_P minus the origin.)
  • What are the dimensions of all the coadjoint orbits of U(n)?

    • Here's the sequence of topics I consider de rigeur, so they'll come first, to be sure they get done. (Chapter references are to Ana's book.)

  • Linear symplectic geometry (ch. 1)
  • classification of symplectic vector spaces
  • the volume form
  • Hermitian structures
  • a description of the linear symplectic group
  • Cotangent bundles (ch. 2)
  • Some Hamiltonian classical mechanics (ch. 18)
  • Some examples (a first pass):
  • coadjoint orbits (exercises to ch. 22)
  • toric manifolds (ch. 28-29)
  • Darboux's theorem (ch. 6-8, but I'll give an abbreviated treatment)
  • Compatible almost complex structures (ch. 12)
  • Group actions and moment maps
  • Existence and uniqueness (ch. 26)
  • Symplectic reduction
  • The Marsden-Weinstein theorem (ch. 23-24)
  • Polygon spaces
  • Symplectic cuts
  • Toric manifolds redux (ch. 28-29)
  • Local forms and Atiyah/Guillemin-Sternberg convexity (ch. 27)
    • We're here as of Oct 9 Maybe a gauge theory example (ch. 25).
  • The Duistermaat-Heckman theorem (ch. 30)
  • Complex geometry
  • Kahler manifolds and Dolbeault theory (ch. 14-17)
  • Degree vs. symplectic volume for projective varieties
  • Coadjoint orbits redux (for compact groups only)
  • Classification
  • Complex structures on coadjoint orbits
  • Topology and Schubert calculus
  • The Borel-Weil-Bott-Kostant theorem

    • The four things I would like to focus on, if there is extra time, are

  • Kahler "geometric quantization", relating group actions on symplectic manifolds to group actions on Hilbert spaces, enriching both subjects. This is NOT to be confused with "deformation quantization", which deforms the symplectic manifold to a noncommutative space (and is the study of Math 277 this term).
  • Lagrangian vs. Hamiltonian mechanics, and path integrals. This would get very touchy-feely, as path integrals are not on a secure mathematical footing, but it would be nice to see why physicists like and use them so much anyway.
  • Equivariant cohomology as the right way to state Duistermaat-Heckman and many other theorems. Also, why do people want to compute the cohomology of symplectic reductions, and what to do with equivariant cohomology? The most ambitious dream would be to define "equivariant formality" and prove that Hamiltonian actions are equivariantly formal.
  • Lagrangian fibrations. There is much interest in this now particularly due to the Strominger-Yau-Zaslow conjecture in mirror symmetry, and of course it relates nicely to integrable systems.

    Another topic that would be nice to present in detail is the Gel'fand-Cetlin completely integrable system on coadjoint orbits of U(n), and the application to polygon spaces.