A new approach to conformal invariants in complex function theory

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This proposal involves three fields of mathematics: complex analysis, Teichmuller theory, and conformal field theory.  Here is an explanation of these fields, together with an explanation of how my research fits in the larger context.  This is aimed at a layperson.***In brief, this proposal contains a new approach to conformal invariance (explained below), which unifies ideas in the three fields. ******Complex analysis is the study of calculus in the setting of complex numbers.  Like all branches of mathematics, it has many unexpected consequences for many fields.  Complex analysis is particularly ubiquitous: it is an indispensable tool in engineering, physics, astronomy, and geology among other fields, for example because of its use in the theory of Fourier series and approximations.  Research in pure mathematics in general can and usually does proceed without concern for applications of this kind; however, the research spins off unexpected applications every few decades.  Much of complex analysis research concerns the study of families of complex analytic mappings.  Teichmuller theory is the study families of Riemann surfaces, which are two-dimensional surfaces along with a definition of angle.  Riemann surfaces are central to the geometric understanding of complex analysis.  ***My own research is on conformal invariants. The idea of an invariant is a powerful tool in mathematics (and physics); it is a quantity which is unchanged under some transformation, and is usually associated with geometric information. Conformal invariants are quantities invariant under complex analytic transformations.  This proposal regards an approach to conformal invariance which unifies ideas in complex analysis, Teichmuller theory, and conformal field theory.  ******In physics, conformal field theory is the study of quantum and mechanical systems which possess conformal invariance.  The mathematical study of conformal field theory began in the late 80s and early 90s, and continues to generate exciting new ideas in pure mathematics, especially in algebra and geometry.  I study certain families of Riemann surfaces which arise in conformal field theory.  David Radnell and I proved that these families equivalent to known families in a branch of complex analysis and geometry called Teichmuller theory.  This allowed us to solve some outstanding analytic issues in the mathematical formulation of conformal field theory, and also led to many new results in geometric function theory and Teichmuller theory.  This proposal investigates further connections in light of the new approach to conformal invariance.  ******The work in this proposal will advance both complex analysis and the mathematical formulation of conformal field theory.  Furthermore it creates new unexpected connections between these fields.  **