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