Roman Fedorov, University of Massachusetts, Amherst
Langlands transform and Painleve equations

Let X be a smooth compact algebraic curve over complex numbers (a.k.a. a Riemann surface) and G be a reductive group (for example SL(n)). The (mostly conjectural) Langlands transform is an equivalence between some categories associated to the moduli space of principal G-bundles on X and the moduli space of G^v-bundles with connections. Here $G^v$ is the so-called Langlands dual group of G.

I shall explain in details what the above equivalence means and (maybe) discuss the relation with the classical Langlands correspondence. Then I will talk about the "Painleve-VI" case proved by D. Arinkin, and about other cases of Langlands transform proved recently by Arinkin and myself.

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