After the works of A. Zorich and G. Forni, it is known that the Lyapunov exponents of the so-called Kontsevich-Zorich cocycle are interesting quantities related to the deviations of ergodic averages of interval exchange transformations and translation flows. In a recent breakthrough article, A. Eskin, M. Kontsevich and A. Zorich obtained a formula for the sum of non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle with respect to ergodic SL(2,R)-invariant measures satisfying a certain regularity condition. Concerning the hypothesis for the validity of Eskin-Kontsevich-Zorich, the regularity condition is always satisfied in all known examples of ergodic SL(2,R)-invariant measures. In particular, partly motivated by this scenario, A. Eskin, M. Kontsevich and A. Zorich conjectured that all ergodic SL(2,R)-invariant measures are regular. In this talk, we will discuss our joint work with A. Avila and J.-C. Yoccoz where the Eskin-Kontsevich-Zorich regularity conjecture is confirmed.
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