Mathematics of Conformal Field Theory

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String theory is our best hope for a `theory-of-everything' in physics. How accurately string theory actually describes the universe is still completely uncertain, but its impact on math is clear, and probably unparalleled in history.**My interest is in studying the mathematics inspired by string theory. Instead of string theory itself, I work with a more or less equivalent theory called conformal field theory, because it's much more accessible mathematically. More precisely, there are two main approaches to conformal field theory, but superficially they look very different. Some things are much easier to understand in one approach, other things are much easier in the other. I have become an expert in both. What would be extremely valuable would be a mathematically precise dictionary between them, allowing us to carefully pass constructions, insights, theorems from one to the other. The physics says this should be possible, but this is very hard to see mathematically. Much of my work surrounds understanding, testing, strengthening that dictionary.**Canadians have already featured prominently in aspects of this. For example, John McKay discovered a bridge (called Moonshine) between two seemingly unrelated areas: certain algebraic symmetries, and `modular functions' (i.e. functions that live on surfaces). Our best understanding of Moonshine interprets it using conformal field theory. Robert Moody co-discovered what are now called Kac-Moody algebras; perhaps there most important realisation is as symmetries of certain very special string theories. I am very interested in both these aspects.