Alex Eskin has been awarded the 2020 Breakthrough Prize in Mathematics. The short citation reads: *For revolutionary discoveries in the dynamics and geometry of moduli spaces of Abelian differentials, including the proof of the “magic wand theorem” with Maryam Mirzakhani. *The full citation highlights, in particular, their paper “Invariant and stationary measures for the ${\rm SL}(2,\Bbb R)$ action on moduli space”, *Publ. Math. Inst. Hautes Études Sci.* 127 (2018), 95–324. The review of it on MathSciNet is copied below. Congratulations!

You may also check out the video of his talk at IAS on the work, available on YouTube.

**MR3814652**

Eskin, Alex(1-CHI); Mirzakhani, Maryam(1-STF)

Invariant and stationary measures for the ${\rm SL}(2,\Bbb R)$ action on moduli space. (English summary)

*Publ. Math. Inst. Hautes Études Sci.* 127 (2018), 95–324.

37D40 (22E50 37C85)

This monumental work has a deceptively simple objective. There is a natural action of ${\rm{SL}}_2(\Bbb{R})$ on the space ${\rm{GL}}_2(\Bbb{R})/{\rm{SL}}_2(\Bbb{Z})$; its ergodic and dynamical properties are well understood, and there is an extensive arsenal of tools from entropy theory, conditional measure techniques, measure rigidity, and Ratner theory available to study it. Here this action is thought of as the natural action of ${\rm{SL}}_2(\Bbb{R})$ on the space of flat tori, and this action is generalized to an action of ${\rm{SL}}_2(\Bbb{R})$ on the space ${{\mathcal H}}(\alpha)$ of translation surfaces, parameterized by a partition $\alpha=(\alpha_1,\dots,\alpha_n)$ of $2g-2$ for a fixed genus $g\geqslant1$. The main emphasis is on finding analogous rigidity and stationarity results in this setting, subsuming and generalizing much earlier work. While some of the results are inspired by the Ratner theory of unipotent flows on homogeneous spaces, much is different in this setting. In particular, the dynamical properties of the unipotent (upper triangular) flow are not understood well enough to be used, so the fundamental ‘polynomial divergence’ technique from unipotent flows on homogeneous spaces is not available. Instead, and in a setting where there is little control over the Lyapunov spectrum of the geodesic (diagonal) flow, new ideas are brought in to allow the ‘exponential drift’ technique of Y. Benoist and J.-F. Quint [Ann. of Math. (2) 174 (2011), no. 2, 1111–1162; MR2831114] to be used. This enormously understates the complexity of the work, which in fact makes use of many of the most significant results in the ergodic and rigidity theory of homogeneous dynamics. The authors have gone to great lengths to explain the overall view of the proofs, and take pains to explain where and why the main technical problems arise.

Reviewed by Thomas Ward