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# Painleve Equations, Coverings of the Sphere and Belyi Functions

IIIUL, B1-01
2012-11-09 14:30 .. 15:30
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Davide Masoero (GFMUL)

The six Painleve equations are "integrable" nonlinear ODE in the complex plane. They are integrable in the sense that solutions only develop pole-singularities.

The main interest lies in the computation and classification of poles of solutions.

In the case of Painleve first equation, poles are classified by infinitely-sheeted branched coverings of the sphere. The pole is then computed through the holomorphic uniformization of the coverings, that is a meromorphic function with a prescribed monodromy representation.

We tackle the problem of constructing these coverings as the limit of a sequence of finitely-sheeted coverings, i.e. we approximate the sought meromorphic function by rational functions.

For a special solution of Painleve first equation, these rational functions are ramified over three points: they are Belyi functions and can be defined over some number fields.

We show how they can be explicitly constructed.