*office*: 528 Machray Hall

*phone*: (204) 474-6926

*mailing address*:

Machray Hall, Dept. of Mathematics,

e-mail:
eric.schippers@umanitoba.ca

__Geometric Function Theory__

Geometric function theory is a branch of complex analysis concerning
classes of
holomorphic functions of one variable. I am particularly
interested in the
class of
one-to-one holomorphic functions on the unit disc (univalent
functions).
This is an important class of functions, firstly because of the Riemann
mapping
theorem, and secondly because it provides a model of the universal
Teichmueller
space. An outstanding problem is to characterize the
class of
univalent functions in terms of their power series. One approach
is to
this problem is to solve extremal problems over the class of
functions.
This approach leads naturally to many geometric ideas, involving
for
example extremal conformal metrics, semigroups of transformations, the
Dirichlet principle, and variational methods. My work in function
theory
has focussed directly or indirectly on conformal invariants and their
role in
extremal problems.

__Teichmueller Theory__

A Teichmueller space is a covering of the Riemann moduli space of a
surface M. The Riemann moduli space of M is the set of
Riemann surfaces which are topologically equivalent to M, where one
does not distinguish between Riemann surfaces which are
biholomorphically equivalent. The invention of Teichmueller
space made possible a comprehensive theory of moduli spaces and their
complex structures. It is an important part of many fields,
including hyperbolic geometry, complex dynamics, algebraic geometry and
conformal field theory. My interest is particularly in
quasiconformal Teichmueller theory.

__Conformal Field Theory__

I have also been working in two-dimensional conformal field theory,
which arises both in condensed matter physics and string theory.
A mathematical definition of 2D conformal field theory was sketched by
Segal and Kontsevich. The attempt to understand the
"details" of this model has spawned an incredible range of
interesting mathematics.

My work in two-dimensional conformal field theory is based on the
discovery with David Radnell that a moduli space of Riemann surfaces
with boundary parametrizations, arising in conformal field theory, is
(up to a discrete group action) the quasiconformal Teichmuller space of
a bordered surface. This discovery appeared during work on the
program of Yi-Zhi Huang to construct conformal field theory from vertex
operator algebras. Applying this insight, we have made progress
on several analytic problems in the rigorous construction of conformal
field theory. Conversely, we are able to apply algebraic and
geometric ideas of conformal field theory to derive unexpected results
in Teichmuller theory.