June 25, 2018

Symplectic Topology, Geometry, and Gauge Theory Program
Graduate Courses 2000-2001

Courses offered in association with the Program on Symplectic Topology, Geometry and Gauge Theory

Infinite-Dimensional Lie Groups and Gauge Theory

Held in conjunction with the program on Infinite-Dimensional Lie Theory and Its Applications and the Program on Symplectic Topology, Geometry and Gauge Theory

Instructor: Boris Khesin, University of Toronto

Time: Wednesdays 10:30-11:30 am and Fridays 1:00-3:00 pm
Start Date: January 15, 2001
Location: Room 210 or 230 at the Fields Institute

The course is an introduction to the classical theory of gauge groups and connections on real low-dimensinonal manifolds and new techniques in theory of double loop groups and connections on K3 surfaces and Calabi-Yau manifolds. Topics to be covered include:

  • Geometry of loop groups, affine Kac-Moody groups, Virasoro groups, groups of double loops, and their orbits. Introduction to Leray residues.
  • Basics in differential geometry of vector bundles; flat connections and holomorphic bundles; Poisson structures on their moduli spaces. Hitchin systems.
  • The Chern-Simons functional on connections on real and complex three-folds. Its relation to linking number and holomorphic linking number. Polor homology of complex manifolds.

References and suggested reading:

Pressley & Segal (1986). Loop Groups. (Oxford)
Atiyah. Collected Works, 5th Volume, Gauge Theory.
Koyobashi (1987). Differential Geometry of Complex Vector Bundles, (Iwanami Shoten and Princeton University Press)


Symplectic Geometry and Hamiltonian Group Actions

Held in conjunction with the program on Infinite-Dimensional Lie Theory and Its Applications and the program on Symplectic Topology, Geometry and Gauge Theory.

Instructor: L. Jeffrey, University of Toronto

Time: Tuesday and Thursday 3:30 - 5 pm.
Start Date: week of January 15, 2001
Location: Room 210 or 230 at the Fields Institute

A symplectic manifold (a manifold equipped with a nondegenerate closed two-form) is the natural mathematical generalization of the phase space considered in classical machanics. Hamiltonian group actions are a special case of Hamiltonian flows, which are a natural generalization of Hamilton's equations. Coadjoint orbits are natural examples of symplectic manifolds equipped with Hamiltonian group actions. The course treats the following topics:

  • Moment maps; symplectic quotients
  • The symplectic structure on coadjoint orbits
  • The Atiyah-Guillemin-Sternberg convexity theorem
  • Delzant's theorem and introduction to toric geometry from the synthetic point of view
  • Geometric quantization: applications to representation theory (survey)
  • Equivariant cohomology and applications to symplectic geometry: (a) the localization theorem of Berline-Vergne, the Duistermaat-Heckman theorem, (b) Recent results on cohomology rings of symplectic quotients, obtained using localization (survey)
  • An infinite dimensional symplectic quotient: the moduli space of flat connections on a Riemann surface (following Atiyah-Bott 1982 and Goldman 1984)

References and suggested reading:

Audin (1991). The Topology of Torus Actions on Symplectic Manifolds. (Birhauser)
Berline & Getzler & Vergne(1992). Heat Kernels and Dirac Operators. (Springer-Verlag)
Guillemin & Sternberg (1984). Symplectic Techniques in Physics. (Cambridge)
Guillemin (1994). Moment Maps and Combinatorial Invariants of Hamiltonian T^n-spaces. (Birkhauser)
Guillemin & Lerman & Sternberg (1996). Symplectic Fibrations and Multiplicity Diagrams. (Cambridge)


Moduli Spaces of Flat Connections

Held in conjunction with the Program on Symplectic Topology, Geometry and Gauge Theory.

Instructor: E.Meinrenken, University of Toronto

Time: Tuesdays and Thursdays 1:30 - 3 pm
Start Date: January 16, 2001
Location: Room 210 or 230 at the Fields Institute

Moduli Spaces of flat connections on 2-manifolds are encountered in areas so diverse as Algebraic Geometry, Representation Theory, Knot Theory, Quantum Cohomology, and Conformal Field Theory. In this course we will give an overview from the perspective of Symplectic Geometry and (to a lesser extent) Mathematical Physics. Among other things, we will describe fixed point formulas for loop group actions which imply formulas for volumes, intersection pairings, and quantum dimensions of the moduli spaces.

Topics to be covered include:

  • Connections, Curvature and Holonomy
  • Gauge Theory construction of moduli spaces
  • Holonomy description of moduli spaces
  • Hamiltonian loop group actions
  • Witten's volume formulas, intersection pairings
  • Verlinde dimension formulas

References and suggested reading:

Atiyah (1990). The Geometry and Physics of knots. (Cambridge)
Thaddeus (1995). 'An introduction to the topology of the moduli space' and In: Geometry


Introduction to Symplectic Field Theory

Held in conjunction with the Program on Symplectic Topology, Geometry and Gauge Theory

February 12, 19: Preparatory lectures (by M.Abreu)
*February 26-March 2: Ya. Eliashberg, Stanford University
March 12-14: A. Givental, University of California at Berkeley - Coxeter lectures

*Schedule for February 26-March 2
Monday 1:00 - 3:00 pm
Tuesday 3:30 - 5:30 pm
Wednesday 1:00 - 3:00 pm
Thursday 3:30 - 5:00 pm
Friday 1:00 - 3:00 pm

Location: Room 230 at the Fields Institute

This course covers the following areas:

  • Basic contact and symplectic geometry: local properties, Hamiltonian dynamics, Lagrangian and Legendrian manifolds, relations between symplectic and contact manifolds. Maslov index and its properties.
  • Overview of the theory of J-holomorphic curves: Fredholm properties, Gromov compactness, gluing. Floer homology.
  • Contact homology theory and its applications.
  • Stable curves. Deligne-Mumford compactifications. Topology of the moduli spaces of Riemann surfaces.
  • Quantum cohomology and Gromov-Witten invariants.
  • Symplectic field theory.


Symplectic Geometry

Held in conjunction with the Program on Symplectic Topology, Geometry and Gauge Theory

Instructor: E.Meinrenken, University of Toronto

Time: Mondays, Wednesdays, Fridays at 11:00 am - 12:00 pm
Start Date: September 11, 2000
Location: Sidney Smith Hall, Room 2128. Offered by the Math Dept.

This course will be an introduction to basic concepts of symplectic geometry, covering the following topics:

  • Linear symplectic geometry
  • Darboux-Weinstein theorems
  • Poisson brackets, Hamiltonian systems
  • Completely inetgrable systems
  • Hamiltonian group actions, moment maps
  • Convexity theorems, Duistermaat-Heckman theory

The course will be accompanied by lecture notes. References and suggested reading:

McDuff & Salamon (1997). Introduction to Symplectic Topology. (Oxford)
Libermann & Marle (1987). Symplectic Geometry and Analytical Mechanics. (Reidel)
Guillemin & Sternberg (1990). Symplectic Techniques in Physics (2nd.Ed.). (Cambridge)


Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.


Financial Assistance

As part of the Affiliation agreement with some Canadian Universities, graduate students are eligible to apply for financial assistance to attend graduate courses. Two types of support are available:
  • Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).
  • Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to $1000 per month. Send an application letter, curriculum vitae and letter of reference from a thesis advisor to the Director, Attn.: Course Registration, The Fields Institute, 222 College Street, Toronto, Ontario, M5T 3J1.

Applications for financial support should be received by the following deadlines: April 15, 2000 for the Fall term and September 15, 2000 for the Winter term.