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Explore an alternative approach to the classical Borel-Écalle resummation method for factorially divergent series in this lecture by Maxim Kontsevich from IHES. Delve into the concept of analytic wall-crossing structures, introduced by Kontsevich and Yan Soibelman, which defines a holomorphic bundle over a small disc in the original coordinate. Learn how this method differs from working in the Borel plane by using gauge transformations with convergent series in exponentially small terms to glue trivialized bundles on overlapping sectors. Discover the global geometric interpretation as a bundle over a neighborhood of a wheel of 1-dimensional torus orbits in a higher-dimensional toric variety. Examine several illustrative examples, including exponential integrals, a generalization to closed 1-forms (encompassing the Stirling formula), and the quantum dilogarithm, to gain a deeper understanding of this innovative approach to resurgence theory.
