Annual Report 2005 - Fields Institute - University of Toronto

extends to a holomorphic function on the complexified
quadric, which has only one component. Physicists call this
property of the S-matrix crossing symmetry and Witten
explained how it can be used to deduce the existence of
antimatter.
In the second lecture, the audience was treated to an eminently
understandable account of gauge symmetry breaking
in the context of superconductors. Witten described the
Landau-Ginzburg model of superconductivity, where the
material is endowed with a section of the square of the
electromagnetic line bundle. Witten then explained that
although a superconducting ring maintains a current for
an extremely long time, the current does decay due to the
nucleation of flux lines, which correspond to zeros of the
section. The flux lines may attract or repel depending on
the properties of the superconductor, and in the limiting
case the equations describing the superconductor’s 4dimensional
worldvolume approach Taubes’ perturbation
of the Seiberg-Witten equations for 4-manifolds. In this
way, Witten showed that smooth invariants of 4-manifold
theory pertain to the properties of superconductors! He
then turned to the description of the weak force in directly
analogous terms, but for a U(2) connection instead of the
U(1) of electromagnetism. The breaking of U(2) to U(1)
is described by a section of the U(2) bundle, and the same
mechanism which keeps magnetic fields out of superconductors
is responsible for the fact that the W and Z particles
of the weak force are visible only at high energies and short
distances.
In the third lecture, Witten introduced the quantum Hall
effect, whereby a thin conducting surface, when exposed
to perpendicular electric and magnetic fields, generates a
measurable current whose strength is quantized. Witten
then described a topological mechanism which explains the
quantization of this current. The remarkable thing is that
the model describing the quantum Hall effect involves no
degrees of freedom associated to the surface whatsoever. In
other words it is a pure gauge theory. Witten then produced
the Lagrangian describing this model, which is a U(1)
Chern-Simons theory on the 3-volume of the conducting
surface. Generalizing this situation to an arbitrary gauge
group, Witten described other invariants of 3-manifolds
as well as the Jones polynomial invariant of knots in a 3manifold.
In the spirit of the previous lectures, we learned
that esoteric 3-manifold invariants are directly linked to
measurable physical phenomena.
L e c t u r e s a n d S p e c i a l E v e n t s
Sir David Cox
DISTINGUISHED LECTURE SERIES
IN STATISTICAL SCIENCE
Sir David Cox
Graphical Models
September 14 and 15, 2004
On September 14 and 15 Sir David Cox, Professor Emeritus
at Nuffield College and the Department of Statistics, University
of Oxford, gave the Fields Institute’s Distinguished
Lecture Series in Statistical Science for 2004. This annual
series was established in 2000, with previous lectures given
by Peter Hall, D.A.S. Fraser and Donald Dawson.
Cox has made singular contributions to statistics in almost
every area of theory and application. His focus for the
Distinguished Lecture Series was “Graphical Models”.
Graphical models provide a structure for thinking about
multivariate data, particularly but not exclusively gathered
in observational studies. This is an area of active current
interest in statistics, computer science, sociology, and
econometrics. The 1996 book Multivariate Dependencies
by Cox and Wermuth provides a useful starting point for
learning about the approach described in these lectures.
In the analysis of observational data, it is usual that information
on several variables is collected for each unit in
the study. An example was introduced in the first lecture
related to a study of diabetics, where information was collected
on a primary response variable, glucose control, a
number of intermediate variables, such as knowledge about
the illness, and some purely explanatory variables, such as
gender and duration of schooling. In general, for any pair
of measured variables, we might regard one of the pair as a
response to another, or we might think of both members of
Fields Institute 2005 ANNUAL REPORT 36