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RESEARCH PRODUCT

Integrable systems and moduli spaces of curves

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This document has the purpose of presenting in an organic way my research on integrable systems originating from the geometry of moduli spaces of curves, with applications to Gromov-Witten theory and mirror symmetry. The text contains a short introduction to the main ideas and prerequisites of the subject from geometry and mathematical physics, followed by a synthetic review of some of my papers (listed below) starting from my PhD thesis (October 2008), and with some open questions and future developements. My results include: • the triple mirror symmetry among P 1-orbifolds with positive Euler characteristic , the Landau-Ginzburg model with superpotential −xyz + x p + y q + z r with 1 p + 1 q + 1 r > 1 and the orbit spaces of extended ane Weyl groups of type ADE, • the mirror symmetry between local footballs (local toric P 1-orbifolds) and certain double Hurwitz spaces together with the identication of the corresponding integrable hierarchy as a rational reduction of the 2DToda hierarchy (with A. Brini, G. Carlet and S. Romano). • a series of papers on various aspects of the double ramication hierarchy (after A. Buryak), forming a large program investigating integrable systems arising from cohomological eld theories and the geometry of the double ram-ication cycle, their quantization, their relation with the Dubrovin-Zhang hierarchy, the generalizations of Witten's conjecture and relations in the co-homology of the moduli space of stable curves (with A. Buryak, B. Dubrovin and J. Guéré).