April 19, 2018

Infinite Dimensional Lie Theory and its Applications
Graduate Courses 2000-2001

Courses offered in association with the program on Infinite-Dimensional Lie Theory and Its Applications:

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 1-3 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)


Introduction to Extended Affine Lie Algebras

Held in conjunction with the program on Infinite-Dimensional Lie Theory and Its Applications

Instructor: Yun Gao, York University

Time: Tuesdays and Thursdays at 10:45 am - 12:15 pm
Start Date: September 12, 2000
Location: Room 210 at the Fields Institute

This course is an introduction to the affine Kac-Moody Lie algebras and the newly developed extended affine Lie algebras. Topics to be covered include:

  • Kac-Moody Lie algebras and loop algebras
  • Toroidal Lie algebras and universal central extensions
  • Basics on extended affine Lie algebras
  • Vertex operator representations for the extended affine Lie algebras over quantum tori.

References and suggested reading:

Frenkel & Lepowsky & Meurman (1989). Vertex Operator Algebras and the Monster. (Academic)
Kac (1990). Infinite Dimensional Lie Algebras. (Cambridge)
Moody & Pianzola (1995). Lie Algebras with Triangular Decomposition. (Wiley:New York, 1995)


Topics in Representation Theory

Held in conjunction with the program on Infinite-Dimensional Lie Theory and Its Applications

This is a series of minicourses, given by organizers and participants. It includes, in particular, the minicourse on Lie Methods in Soliton Theory.

Instructor: Y. Billig, University of New Brunswick

Time: Wednesdays at 9:30 am - 11:00 am
Start Date: September 13, 2000
Location: Room 210 at the Fields Institute

The symmetries of many important PDEs are described by the Kac-Moody algebras and groups. Applying the group action to the trivial solution, one recovers all solition solutions. Starting with a representation of an affine Kac-Moody algebra it is possible to construct an infinite hierarchy of the soliton partial differential equations. Topics to be covered include:

  • Representations of $a_\infty$ and the boson-fermion correspondence
  • Vertex operator constructions
  • Korteweg-de Vries, sine-Gordon, non-linear Schroedinger equations and the basic representation of of sl(2)

References and suggested reading:

V.G. Kac (1990). Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University press, 1990


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.