Annual Report 2005 - Fields Institute - University of Toronto

homotopy category of Tate or complete resolutions, to Serre
duality, the theory of algebraic cycles and to the variational
Hodge conjecture were discussed. It was a very lively
experience that offered the opportunity for a dialogue at a
sophisticated level, involving participants at all stages of a
research career.
Course on Symplectic Topology and Integrable Systems
January–April 2005
Instructor: Boris Khesin (Toronto)
This course was an introduction to the principal notions
and methods in several active research areas surrounding
symplectic topology. It was also designed to be a bridge
between the introductory course on symplectic geometry
given at the University of Toronto Mathematics Department
in the Fall semester and the thematic program on
String Theory. As a result it attracted graduate students and
postdocs from both sides, as well as students from institutions
outside of Toronto, such as McMaster University and
the University of Saskatchewan.
The course consisted of three major parts. The first one
was devoted to the main tools and ramifications of symplectic
topology and Morse theory. The topics developed
ranged from symplectic properties of billiards to Floer
homology and recent developments in Arnold’s conjectures.
These conjectures, which to a large extent shaped this field
of mathematics, predicted that the number of fixed points
of Hamiltonian diffeomorphisms for symplectic manifolds
is bounded below by the minimal number of critical
points of a smooth function on this manifold. The latter is
governed by Morse inequalities and the Lusternik–Schnirelmann
category. Consequently, a large part of the course
dealt with Morse theory and its various applications
and ramifications, which are of independent interest:
applications to geodesics, the Morse–Novikov theory for
T h e m a t i c P r o g r a m s
Boris Khesin Marco Gualtieri
multivalued functionals, the Morse–Witten complex, etc.
The techniques discussed in the course also allowed one
to describe the geometry and geodesics on symplectomorphism
groups, to outline various symplectic non-squeezing
results, the notion of symplectic capacities, Floer homology,
the Hofer metric, etc.
Another closely related large topic was contact geometry
and topology. The course described several key results on
Legendrian knots and their invariants, as well as giving the
definition of Chekanov–Eliashberg contact homology of
Legendrian knots.
The last part of the course was devoted to the main tools in
integrable systems. The key constructions, such as the Lax
form, compatible brackets, and the shift argument method
were illustrated by numerous examples. The principal
one, the Korteweg-de Vries equation, was also used as the
launching pad to describe infinite-dimensional methods in
the theory, including the formalism of pseudo-differential
operators and the Adler–Gelfand–Dickey brackets.
Mini-course on Generalized Geometries in String Theory
February 15–17, 2005
March 1–3, 2005
Organizer: Robert Myers (Perimeter)
Instructors: Marco Gualtieri (Fields), Yi Li (Caltech) and
Mariana Grana (Ecole Polytechnique & Ecole Normale
Supérieure)
Held at Perimeter Institute
On more than one occasion during the theme year on The
Geometry of String Theory, jointly hosted by the Fields
and Perimeter Institutes, visiting lecturers mentioned with
excitement the work being done by Nigel Hitchin at Oxford
on generalized Calabi-Yau manifolds, and how surprisingly
well it worked as a natural language for the mathematics of
Fields Institute 2005 ANNUAL REPORT 18