About

Alex Eskin is a professor at the Department of Mathematics at the University of Chicago. His research focuses on dynamical systems, ergodic theory, and their applications to geometry and number theory. Eskin has contributed significantly to the understanding of measure rigidity, Lyapunov exponents, and the dynamics of flat surfaces, moduli spaces, and Teichmüller theory. His work includes studies on invariant measures, orbit closures, and counting problems related to moduli spaces of Abelian differentials and quadratic differentials. Throughout his career, Eskin has collaborated on numerous influential papers and has been involved in advancing the theory of unipotent flows, homogeneous spaces, and the spectral properties of Laplacians on modular surfaces. His research has led to important developments in the understanding of the structure and dynamics of moduli spaces, as well as applications to counting closed geodesics and the geometry of hyperbolic surfaces. Eskin's contributions are recognized as foundational in modern mathematical research at the intersection of dynamics, geometry, and arithmetic.

Research topics

• Mathematics
• Mathematical analysis
• Pure mathematics
• Physics
• Statistics
• Geometry

Selected publications

• Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

  ArXiv.org · 2025-02-19

  preprintOpen access

  We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.

  PublisherOA PDFDOI

• The work of G. A. Margulis

  ˜The œAbel prize · 2024-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Continuity of the Lyapunov exponents of random matrix products

  arXiv (Cornell University) · 2023 · 6 citations

  • Mathematics
  • Pure mathematics
  • Mathematical analysis

  We prove that the Lyapunov exponents of random products in a (real or complex) matrix group depends continuously on the matrix coefficients and probability weights. More generally, the Lyapunov exponents of the random product defined by any compactly supported probability distribution on $GL(d)$ vary continuously with the distribution, in a natural topology corresponding to weak$^*$-closeness of the distributions and Hausdorff-closeness of their supports.

  PublisherOA PDFDOI

• Geometric properties of partially hyperbolic measures and applications to measure rigidity

  arXiv (Cornell University) · 2023-02-25 · 2 citations

  preprintOpen access1st authorCorresponding

  We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of $uu$-states and the existence of physical measures.

  PublisherOA PDFDOI

• Effective counting of simple closed geodesics on hyperbolic surfaces

  Journal of the European Mathematical Society · 2021-10-19 · 11 citations

  preprintOpen access1st authorCorresponding

  We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most L on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

  PublisherOA PDFDOI

• Self-joinings for 3-IETs

  Journal of the European Mathematical Society · 2021-05-03 · 2 citations

  preprintOpen accessSenior author

  We show that typical interval exchange transformations on three intervals are not 2-simple answering a question of Veech. Moreover, the set of self-joinings of almost every 3-IET is a Poulsen simplex.

  PublisherOA PDFDOI

• Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

  Astérisque · 2020-01-01 · 5 citations

  articleOpen access

  We consider the action of $SL(2,\\mathbb{R})$ on a vector bundle $\\mathbf{H}$\npreserving an ergodic probability measure $\\nu$ on the base $X$. Under an\nirreducibility assumption on this action, we prove that if $\\hat\\nu$ is any\nlift of $\\nu$ to a probability measure on the projectivized bunde\n$\\mathbb{P}(\\mathbf{H})$ that is invariant under the upper triangular subgroup,\nthen $\\hat \\nu$ is supported in the projectivization $\\mathbb{P}(\\mathbf{E}_1)$\nof the top Lyapunov subspace of the positive diagonal semigroup. We derive two\napplications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle\ndepend continuously on affine measures, answering a question in [MMY]. Second,\nif $\\mathbb{P}(\\mathbf{V})$ is an irreducible, flat projective bundle over a\ncompact hyperbolic surface $\\Sigma$, with hyperbolic foliation $\\mathcal{F}$\ntangent to the flat connection, then the foliated horocycle flow on\n$T^1\\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the\nfoliated geodesic flow is simple. This generalizes results in [BG] to arbitrary\ndimension.\n

  PublisherOA PDFDOI

• Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

  Astérisque · 2020-01-01 · 3 citations

  article

  We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $\nu$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat\nu$ is any lift of $\nu$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat \nu$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $\Sigma$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.

  PublisherDOI

• Billiards, quadrilaterals and moduli spaces

  Journal of the American Mathematical Society · 2020 · 30 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Mathematical analysis
  PublisherDOI

• Semisimplicity of the Lyapunov spectrum for irreducible cocycles

  Israel Journal of Mathematics · 2019-01-17 · 3 citations

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

Recent grants

• Averaging Methods in Coarse Geometry

  NSF · $218k · 2006–2010

• The SL(2,R) action on moduli space

  NSF · $314k · 2012–2016

• Measure rigidity in Teichmuller space and beyond

  NSF · $375k · 2015–2020

• Measure Rigidity and Smooth Dynamics

  NSF · $270k · 2018–2022

• Coarse Differentiation and Teichmuller Dynamics

  NSF · $382k · 2009–2013

Frequent coauthors

• Anton Zorich

  Institut de Mathématiques de Jussieu-Paris Rive Gauche

  49 shared

• Maryam Mirzakhani

  Mazandaran University of Medical Sciences

  23 shared

• Jon Chaika

  12 shared

• Amir Mohammadi

  12 shared

• Jayadev S. Athreya

  University of Washington

  11 shared

• Howard Masur

  University of Chicago

  10 shared

• Hee Oh

  Yale University

  9 shared

• Shahar Mozes

  8 shared

Education

• B.S.

  UCLA

  1986

• Ph.D.

  Princeton

  1993

Awards & honors

• Breakthrough Prize in Mathematics (2019)