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Ane Kac-Moody algebras give rise to interesting systems of dif- ferential equations, so-called Knizhnik-Zamolodchikov equations. The mon- odromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the ane Lie algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the existence of the latter on world sheets of arbitrary topology. 1. Some venerable dierential equations One of the surprises in the theory of Kac-Moody algebras is the observation that they give a new and powerful handle on problems that, a priori, do not have an algebraic flavor, including such which are not even genuinely infinite-dimensional. In this contribution we discuss one such application of structures related to ane Kac-Moody algebras: properties of solutions to dierential equations. In the middle of the 18th century Euler, Cauchy and other analysts, the succes- sors of the founding fathers like Newton, Leibniz, and Johann and Jakob Bernoulli, developped a strong interest in "special functions" and series. Arguably, the sim- plest non-trivial series one can think of is the geometric series

Lie algebras, Fuchsian differential equations and CFT correlation functions