Alex Eskin

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

• Invariant and stationary measures for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> action on Moduli space

  Publications mathématiques de l IHÉS · 2018-04-17 · 98 citations

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  We prove some ergodic-theoretic rigidity properties of the action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>SL</mml:mi> <mml:m…

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• Right-angled billiards and volumes of moduli spaces of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$

  French digital mathematics library (Numdam) · 2016-01-01 · 54 citations

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  We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on CP 1 and counting functions of the number of (bands of) simple closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago.Applying ergodic techniques to the Teichmüller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories…

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• Symplectic and isometric SL(2,#x211D;)-invariant subbundles of the Hodge bundle

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2015-04-14 · 32 citations

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  Abstract Suppose N is an affine <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>ℝ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{SL}(2,{\mathbb{R}})} -invariant submanifold of the moduli space of pairs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>ω</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> (M,\omega) , where M…

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• Billiards, quadrilaterals and moduli spaces

  Journal of the American Mathematical Society · 2020 · 30 citations

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• Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space

  Annals of Mathematics · 2015-07-29 · 28 citations

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  We prove results about orbit closures and equidistribution for the SL(2, R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed SL(2, R) invariant manifolds developed in this paper.

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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

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)

Similar researchers at University of Chicago

• Jiajie Chenh-98
• Vladimir Drinfeldh-39
• Victor Ginzburgh-38