office: 528 Machray Hall
phone: (204) 474-6926
Machray Hall, Dept. of Mathematics,
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.
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.