Resurgent functions: examples and perspectives

Maxim Kontsevich, Institut des Hautes Études Scientifiques

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Theory of resurgent functions was developed by Jean Écalle starting from early 80-ies. Very often in physics and mathematics we have an asymptotic expansion in small parameter which is divergent because the coefficients have factorial growth an  n! exp(O(n)). The property of resurgence means that the Borel transform ∑n anζn/n! admits an endless analytic continuation, which allows to reconstruct the exact value of the original divergent series.

I will give some examples of resurgent series (classical special functions, Écalle-Voronin theory, heat kernel, WKB approximation), and propose some new viewpoints (Hodge structures of infinite rank, analytic wall-crossing structures).