Conformal field theory and the nature of symmetry

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String theory is our best hope for a `theory of everything' in physics. How relevant string theory will be to the physics of tomorrow is still uncertain, but its impact on mathematics is clear, and probably unparalleled in history. I study 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. It is an especially friendly example of a quantum field theory, the main language of modern physics. According to Fields' medalist Witten, this is the century when mathematics finally comes to terms with quantum field theory. It is expected that this will be at least as significant for mathematics, as was the coming to terms with calculus in the 18th and 19th centuries. I am exploring the extent to which some of our classical notions, like symmetry, are being disrupted as we understand conformal field theory better. I want to know how significant is this disruption. Though we are being confronted by these new ideas and examples, classical notions still permeate today's theory. Is this a hint that the classical notions are truly foundational and will continue to be relevant, or will we eventually see this as a reactionary bias, an embarrassing prejudice? I will map out some of the range of possibilities allowed in conformal field theory, and judge for myself the role the classical notions appear to deserve. What I am finding is that the coming changes will be profound. Canadian mathematicians have already featured prominently in aspects of this evolving theory. For example, John McKay discovered a bridge (called Moonshine) between two seemingly unrelated areas: certain classical symmetries, and `modular functions' (i.e. functions which live on surfaces). Our best understanding of Moonshine interpolates them using conformal field theory. Robert Moody co-discovered what are now called Kac-Moody algebras; perhaps their most important realization is as symmetries of certain very special (and very classical) conformal field theories. I am very interested in both of these aspects. The thought that I am continuing their legacy inspires me.