Annual Report 2005 - Fields Institute - University of Toronto

Hitchin, Savilian Professor of Geometry at Oxford University.
His research interests are remarkably broad, and
have led him to highly original discoveries which have
had significant impact on both mathematics and theoretical
physics. His membership on a frightening number of
editorial and planning boards also indicates his deep commitment
to fostering the development of mathematics.
Hitchin is well-known for his deep contributions to the
theory of harmonic spinors, hyperkähler geometry, gauge
theory of instantons and monopoles, minimal surface
theory, geometrical solutions of Painlevé equations, Einstein
metrics, moduli spaces of Higgs bundles, and special
Lagrangian geometry in mirror symmetry, among others.
He is also a gerbe hunter par excellence. Any one of these
topics would have been eminently relevant to the thematic
program on the Geometry of String Theory being held at
the Fields Institute. However, Hitchin laid these topics aside
in favour of introducing a new field he has recently developed:
that of generalized geometric structures.
In his first lecture, entitled Open Orbits and Geometrical
Structures, Hitchin described a completely different method
for studying special geometrical structures (such as Kähler,
Calabi-Yau, or G 2 geometry). Traditionally they are viewed
from the point of view of Riemannian holonomy groups or
G-structures on the tangent bundle. Instead, he explained
that these geometries can be viewed as the critical points
of a diffeomorphism-invariant functional defined on open
orbits of differential forms, or more generally as structures
on the sum of the tangent and cotangent bundle. By viewing
geometry in this way, Hitchin showed how one is led to
define new (generalized) geometrical structures.
In his second lecture, entitled Instantons and Bihermitian
Metrics, Hitchin presented in detail some examples of
generalized geometries: generalized complex and Kähler
geometry. He regaled the audience with a flurry of integrations
by parts to prove the remarkable result that the
moduli space of instantons on a bihermitian four-manifold
carries a natural generalized Kähler structure.
For the final Coxeter lecture, Hitchin took the show on the
road – to Waterloo, to be precise. The impressive new site
of Perimeter Institute for Theoretical Physics was the ideal
setting for his lecture on Geometry with B-fields, in which
he detailed the correspondence between natural objects
and operations in generalized geometry with certain geometrical
phenomena in string theory. Following his lecture
there was much excitement and discussion, among both
mathematicians and physicists.
T h e m a t i c P r o g r a m s
Robbert Dijkgraaf
January 17, 18 and 20, 2005
Robbert Dijkgraaf (Amsterdam)
The Mathematics of String Theory
Topological String Theory I and II
The winter 2005 Coxeter Lectures were delivered by Robbert
Dijkgraaf on January 17 and 20 at Fields Institute, and
on January 18 at Perimeter Institute.
Dijkgraaf was awarded a Ph.D. cum laude under the supervision
of the Nobel laureate Gerard ‘t Hooft in 1989. He has
been at the forefront in research in string theory, and his
research greatly contributed to the remarkable interplay
experienced by theoretical physics and mathematics in the
last 15 years. His ideas have proved fundamental in a vast
array of research directions including (topological) string
theory, quantum gravity, Gromov–Witten theory, mirror
symmetry and matrix models, to name just a few. A true
Renaissance man, Dijkgraaf also studied painting at the
Gerrit Rietveld Academie. He is an accomplished artist as
well as a columnist for the newspaper NRC Handelsblad
and the magazine Folia. In 2003, he was awarded the Spinoza
Prize, the highest scientific award in the Netherlands.
The first lecture, entitled The Mathematics of String Theory,
was a delightfully informal presentation of the main
research directions that have brought together (reluctantly,
one might say, at least in the beginning) physicists and
mathematicians. Punctuated by historical quotes and pictures,
as well as his own vivid drawings, Dijkgraaf took the
audience on a tour that started with classical mechanics,
touched upon the fundamental ideas in quantum mechanics
and quantum field theory, and moved to string theory
and quantum gravity. Paraphrasing the famous quote of
Wigner, Dijkgraaf talked about the “Unreasonable Effec-
Fields Institute 2005 ANNUAL REPORT 32