Integrable systems and moduli spaces of curves

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)