Sommaire:

After a brief introduction to the theory of Painlevé equations, I will present in more detail the classification of algebraic solutions to Painlevé VI. The approach will be based on the study of finite orbits of a braid group action on the monodromy of the associated linear system. The resulting list is a nonlinear analog of the Schwarz's table, describing 15 cases when hypergeometric functions can be expressed algebraically.

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