Lesson 00

How to use this course

Alex Eskin works where dynamics meets geometry and arithmetic. His results share one idea: a group acting on a space often has far fewer invariant objects than one would expect, and when those objects can be classified, hard counting problems become computations. He applied this idea first to integer points on varieties and quadratic forms, then to the large-scale geometry of groups, and finally to the moduli space of flat surfaces, a space that has no ambient Lie group at all.

Prerequisites for Part II

Lie groups and lattices: $\mathrm{SL}(n,\mathbb R)$, $\mathrm{SL}(n,\mathbb Z)$, Haar measure.
Measure-theoretic ergodic theory: invariant measures, ergodicity, mixing.
Riemann surfaces and holomorphic 1-forms, for Lessons 08–10.
Basic metric geometry (quasi-isometries, word metrics) for Lesson 07.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The Marina Ratner course on this site is the natural companion: Lessons 05 and 06 here use her theorems as a black box. Eskin is a living mathematician, so Part I and Part III keep to his career, his mathematics and his public honours.

Reference

Timeline

1965	Born 19 May in the Soviet Union. His father is the analyst Gregory Eskin.
1986	BA in mathematics, University of California, Los Angeles.
1991	With Zeév Rudnick and Peter Sarnak, a new proof of Siegel’s weight formula by counting points on hyperboloids.
1993	PhD at Princeton under Peter Sarnak, Counting Lattice Points on Homogeneous Spaces. Eskin–McMullen, Mixing, counting and equidistribution in Lie groups. Member of the Institute for Advanced Study, 1993–94.
1994–99	University of Chicago: Dickson Instructor, associate professor (1997), professor (1999).
1996	Eskin–Mozes–Shah, Unipotent flows and counting lattice points on homogeneous varieties, Annals.
1997	Eskin–Farb on quasi-flats in symmetric spaces. Packard Fellowship.
1998	Eskin–Margulis–Mozes on the quantitative Oppenheim conjecture, Annals. Quasi-isometric rigidity of non-uniform higher-rank lattices. Invited address at the ICM in Berlin.
2001	Eskin–Masur pointwise asymptotics on flat surfaces; Eskin–Okounkov on volumes of strata.
2003	Eskin–Masur–Zorich on the principal boundary and Siegel–Veech constants.
2005	Eskin–Margulis–Mozes on forms of signature (2,2); Eskin–Mozes–Oh on uniform exponential growth.
2007	Clay Research Award, for rational billiards and geometric group theory.
2010	Invited address at the ICM in Hyderabad, with David Fisher, on quasi-isometric rigidity of solvable groups.
2011	Elected to the American Academy of Arts and Sciences.
2012–13	Eskin–Fisher–Whyte, Coarse differentiation of quasi-isometries I and II, Annals. Eskin–Mirzakhani preprint on invariant measures (2013).
2014	Eskin–Kontsevich–Zorich formula for sums of Lyapunov exponents, Publ. IHÉS. Simons Investigator.
2015	Eskin–Mirzakhani–Mohammadi on orbit closures, Annals. Elected to the National Academy of Sciences.
2018	Eskin–Mirzakhani published in Publ. IHÉS.
2019	September: awarded the 2020 Breakthrough Prize in Mathematics.
2025	Brown–Eskin–Filip–Rodriguez Hertz preprint on measure rigidity via the factorization method. Conference in Cetraro, Italy, for his 60th birthday.

Part I · Formation

Lesson 01

From the Soviet Union to UCLA

Alex Eskin was born on 19 May 1965 in the Soviet Union. Sources differ on the city, giving Moscow or Kyiv. His father, Gregory Eskin, is an analyst known for work on pseudodifferential operators and inverse problems. According to his father’s published biography, the family left the Soviet Union in 1974 for Israel, where Gregory Eskin taught at the Hebrew University, and moved to the United States in 1982, when he joined the faculty of the University of California, Los Angeles.

Eskin took his bachelor’s degree in mathematics at UCLA in 1986. He then spent several years between physics and mathematics, studying at the Massachusetts Institute of Technology and at Stanford. In a 2015 interview with the Simons Foundation he described his path as a move from number theory to dynamical systems, which he saw as a natural continuation rather than a change of subject. He also said that what drew him was the beauty of the mathematics rather than its applications.

Lesson 02

Princeton and Peter Sarnak

Eskin wrote his doctorate under Peter Sarnak and received it from Princeton in 1993. His thesis, Counting Lattice Points on Homogeneous Spaces, belongs to a line of work that Sarnak and his collaborators were then opening up: counting integer solutions of polynomial equations whose solution sets carry a large symmetry group.

The first paper, with Zeév Rudnick and Sarnak (1991), counted integer points on a hyperboloid in two ways and compared the answers. One count used the spectral theory of automorphic forms, the other used the Hardy–Littlewood circle method. Their agreement gave a new proof of Siegel’s weight formula. In the same period William Duke, Rudnick and Sarnak proved the general counting theorem for affine symmetric varieties with spectral methods. Eskin and Curtis McMullen then found a proof that used only the mixing of the geodesic flow, with no spectral theory. Lesson 04 presents it. The shift from harmonic analysis to ergodic theory set the direction of Eskin’s career.

After the doctorate he spent the academic year 1993–94 as a member of the Institute for Advanced Study in Princeton. He has returned there often, including for the year 2019–20.

Lesson 03

Chicago, 1994–2000

Eskin went to the University of Chicago as a Dickson Instructor, became an associate professor in 1997 and a full professor in 1999. He has remained at Chicago ever since and now holds the title of Arthur Holly Compton Distinguished Service Professor. Chicago at the time was a centre for both ergodic theory and geometric group theory, and Eskin worked in both.

Two collaborations of these years shaped his next decade. With Shahar Mozes and Nimish Shah he combined Marina Ratner’s new classification of unipotent invariant measures with a technique called linearization, and counted integer points on varieties where mixing alone does not suffice. With Grigory Margulis and Mozes he turned Margulis’s qualitative proof of the Oppenheim conjecture into a counting theorem. In parallel, with Benson Farb, he studied the large-scale geometry of symmetric spaces. In 1998 he gave an invited address at the International Congress of Mathematicians in Berlin, Counting problems and semisimple groups. By the end of the decade he was turning to a new object, the moduli space of flat surfaces, through collaborations with Howard Masur, Anton Zorich and Andrei Okounkov.

Part II · The mathematics

Lesson 04

Counting lattice points by mixing

For the general reader. How many integer matrices of determinant 1 have entries of size at most $T$? The answer grows like a constant times $T^2$, and the constant is a volume. Eskin and McMullen showed that such counts follow from one dynamical fact: a flow on a finite-volume space that mixes, stirring every region evenly across the space.

Goals

State the Duke–Rudnick–Sarnak and Eskin–McMullen counting theorem.
Understand how mixing turns into equidistribution of translated orbits.
Compute the count for $\mathrm{SL}(2,\mathbb Z)$.

Setting

Let $G$ be a connected semisimple Lie group with finite centre and no compact factors, $\Gamma\subset G$ an irreducible lattice, and $H\subset G$ a closed subgroup. Suppose $H$ is symmetric: it is the fixed-point group of an involution of $G$. Assume $\Gamma_H=\Gamma\cap H$ is a lattice in $H$. The orbit $\Gamma\cdot x_0$ of the base point $x_0=H$ in $V=G/H$ is a discrete set, and one wants to count its points in large regions $B_T\subset V$.

Theorem (Duke–Rudnick–Sarnak 1993; Eskin–McMullen 1993)

In the setting above, let $B_T\subset G/H$ be a family of well-rounded sets with $\mathrm{vol}(B_T)\to\infty$. Normalize Haar measures compatibly on $G$, $H$ and $G/H$. Then

\[ \#\{\gamma\in\Gamma/\Gamma_H : \gamma x_0\in B_T\} \;\sim\; \frac{\mathrm{vol}(\Gamma_H\backslash H)}{\mathrm{vol}(\Gamma\backslash G)}\,\mathrm{vol}(B_T). \]

Well-rounded means that small perturbations of the boundary change the volume by a small fraction: for each $\varepsilon\gt0$ there is a neighbourhood $U$ of the identity with $\mathrm{vol}(U B_T^{\pm})\le(1+\varepsilon)\mathrm{vol}(B_T)$ for the inner and outer perturbations. Norm balls are well rounded.

Proof sketch

The key input is the Howe–Moore theorem: $G$ acts mixingly on $L^2(\Gamma\backslash G)$, so $\langle g\cdot\phi,\psi\rangle\to\int\phi\int\psi$ as $g\to\infty$. Eskin and McMullen deduce from mixing that translates of the closed orbit $\Gamma\backslash\Gamma H$ equidistribute: for $g\to\infty$ in $G/H$, the pushed-forward probability measure on $\Gamma\backslash\Gamma H g$ tends to Haar measure on $\Gamma\backslash G$. The deduction thickens the orbit by a small neighbourhood in the directions transverse to $H$ and uses mixing for the thickened set; here the symmetric structure gives the needed control through the Cartan decomposition $G=KAH$. Then define the counting function $F_T(g)=\sum_{\gamma\in\Gamma/\Gamma_H}\chi_{B_T}(g^{-1}\gamma x_0)$ on $\Gamma\backslash G$. Unfolding,

\[ \langle F_T,\psi\rangle=\int_{B_T}\Big(\int_{\Gamma\backslash\Gamma H g}\psi\Big)\,dv(gH), \]

and the inner integral tends to $\int\psi$ by equidistribution. Hence $F_T$ is weakly asymptotic to $\mathrm{vol}(B_T)\cdot\mathrm{vol}(\Gamma_H\backslash H)/\mathrm{vol}(\Gamma\backslash G)$. Well-roundedness lets one pass from weak convergence to the value $F_T(e)$, which is the count.

Exercise 4.1Take $G=\mathrm{SL}(2,\mathbb R)\times\mathrm{SL}(2,\mathbb R)$, $H$ the diagonal, $\Gamma=\mathrm{SL}(2,\mathbb Z)\times\mathrm{SL}(2,\mathbb Z)$. Show that $G/H\cong\mathrm{SL}(2,\mathbb R)$ and that the theorem counts elements of $\mathrm{SL}(2,\mathbb Z)$ in a ball. Which involution fixes $H$?

Show solution

The map $(g_1,g_2)\mapsto g_1g_2^{-1}$ identifies $G/H$ with $\mathrm{SL}(2,\mathbb R)$, and $\Gamma$ acts on it by $\gamma_1 x\gamma_2^{-1}$; the orbit of the identity is $\mathrm{SL}(2,\mathbb Z)$, and $\Gamma_H$ is the diagonal copy of $\mathrm{SL}(2,\mathbb Z)$. The involution is $(g_1,g_2)\mapsto(g_2,g_1)$. Here $\Gamma$ is reducible, but the equidistribution needed reduces to mixing of a single factor on $\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(2,\mathbb Z)$, and Eskin and McMullen treat this group case explicitly. So the method includes counting points of a lattice in a ball.

Exercise 4.2It is known that $\#\{\gamma\in\mathrm{SL}(2,\mathbb Z):\|\gamma\|\le T\}\sim 6T^2$, with $\|\cdot\|$ the Euclidean norm on $\mathbb R^4$. Check this heuristically: the set $\{\|g\|\le T\}$ in $\mathrm{SL}(2,\mathbb R)$ has Haar volume $\sim\pi^2T^2$ when $\mathrm{SL}(2,\mathbb R)$ is given the measure for which $\mathrm{vol}(\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(2,\mathbb Z))=\zeta(2)$. What does the theorem predict?

Show solution

The theorem predicts $\mathrm{vol}(B_T)/\mathrm{vol}(\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(2,\mathbb Z))\sim\pi^2T^2/(\pi^2/6)=6T^2$. The normalization here is the one induced from Lebesgue measure on $\mathbb R^4$ through the determinant map, $dx=d(\det)\,d\mu$, which gives $\mathrm{vol}=\zeta(2)$ for the quotient (Siegel). A direct computer count confirms the ratio approaches 6.

Lesson 05

Unipotent flows and counting

For the general reader. When the symmetry group of a variety is too small, mixing does not control the count. Eskin, Mozes and Shah used Marina Ratner’s theorems, which say that unipotent flows cannot produce wild limits, to show that integer points still spread out evenly. A striking example is the number of integer matrices with a prescribed characteristic polynomial.

Goals

State Ratner’s measure classification in the form used for counting.
Understand the role of linearization and non-divergence.
Apply the Eskin–Mozes–Shah theorem to matrices with given characteristic polynomial.

Ratner’s measure classification (1991)

Let $G$ be a connected Lie group, $\Gamma$ a lattice, and $U\subset G$ a connected subgroup generated by one-parameter unipotent subgroups. Every ergodic $U$-invariant probability measure on $G/\Gamma$ is algebraic: it is the invariant measure on a closed orbit $Lx$ of a closed subgroup $L\supseteq U$.

Mozes and Shah deduced (1995) that limits of such algebraic measures are again algebraic, or escape to infinity. To count, Eskin, Mozes and Shah needed the translates $\mu_{\Gamma_H\backslash H}\cdot g$ to equidistribute. Two things can go wrong. Mass can escape into the cusp; their non-divergence theorem rules this out under conditions on $H$. Mass can concentrate on a closed orbit of an intermediate subgroup; linearization, a method of Dani and Margulis, converts the event of staying near such an orbit into a statement about a polynomial map into a finite-dimensional representation, where it can be estimated.

Theorem (Eskin–Mozes–Shah, Annals 1996)

Let $G$ be a connected reductive algebraic group over $\mathbb Q$ with no non-trivial $\mathbb Q$-characters, $\rho:G\to\mathrm{GL}(W)$ a $\mathbb Q$-representation, and $v_0\in W(\mathbb Z)$ with closed orbit $V=v_0G(\mathbb R)$. Suppose the stabilizer $H$ of $v_0$ is reductive and not contained in any proper $\mathbb Q$-parabolic subgroup of $G$. Let $\Gamma$ be an arithmetic subgroup. Then for norm balls $B_T$,

\[ \#\{v\in v_0\Gamma : \|v\|\le T\}\;\sim\;\frac{\mathrm{vol}(\Gamma_H\backslash H)}{\mathrm{vol}(\Gamma\backslash G)}\;\mathrm{vol}(V\cap B_T). \]

Corollary (Eskin–Mozes–Shah)

Let $P$ be a monic irreducible polynomial in $\mathbb Z[\lambda]$ of degree $n\ge2$, and let $V_P(\mathbb Z)$ be the set of integer $n\times n$ matrices with characteristic polynomial $P$. There is $C_P\gt0$ with

\[ \#\{X\in V_P(\mathbb Z):\|X\|\le T\}\sim C_P\,T^{n(n-1)/2}. \]

Here $G=\mathrm{SL}_n$ acts by conjugation, and $H$ is the centralizer of a companion matrix, an anisotropic torus because $P$ is irreducible. The exponent comes from a volume count: $V_P$ is a fibre of the map sending a matrix to the coefficients of its characteristic polynomial, whose $k$-th coefficient is homogeneous of degree $k$. A ball of radius $T$ in $M_n(\mathbb R)$ has volume of order $T^{n^2}$, and the fibres over a box of unit size in coefficient space are rescaled by $T^{1+2+\dots+n}$, so the fibre volume grows like $T^{n^2-n(n+1)/2}=T^{n(n-1)/2}$. When $\mathbb Z[\alpha]$ is the full ring of integers of $K=\mathbb Q(\alpha)$, the constant is

\[ C_P=\frac{2^{r_1}(2\pi)^{r_2}\,hR}{w\sqrt{|D|}}\cdot\frac{\pi^{m/2}/\Gamma(1+\tfrac m2)}{\prod_{s=2}^{n}\pi^{-s/2}\Gamma(\tfrac s2)\zeta(s)},\qquad m=\tfrac{n(n-1)}2, \]

so class numbers and regulators of $K$ appear in a counting problem about matrices. This explicit form was worked out in later papers building on Eskin–Mozes–Shah.

Exercise 5.1Evaluate $C_P$ for $P=\lambda^2+1$ and check it by hand: such matrices are $\begin{pmatrix}a&b\\c&-a\end{pmatrix}$ with $a^2+bc=-1$.

Show solution

$K=\mathbb Q(i)$, $r_1=0$, $r_2=1$, $h=R=1$, $w=4$, $|D|=4$, so the arithmetic factor is $2\pi/8=\pi/4$. With $n=2$, $m=1$: $\pi^{1/2}/\Gamma(3/2)=2$ and the denominator is $\pi^{-1}\zeta(2)=\pi/6$. So $C_P=\frac\pi4\cdot2\cdot\frac6\pi=3$, and the count grows like $3T$. A computer count of solutions with $2a^2+b^2+c^2\le T^2$ gives ratios 2.90, 2.95, 2.96 for $T=100,400,1600$.

Exercise 5.2Let $P$ be monic irreducible of degree $n$ with root $\alpha$, and $C$ its companion matrix. Show that the centralizer of $C$ in $M_n(\mathbb Q)$ is the field $\mathbb Q[C]\cong\mathbb Q(\alpha)$, and deduce that $V_P(\mathbb C)$ has dimension $n^2-n$.

Show solution

$\mathbb Q^n$ is a cyclic $\mathbb Q[C]$-module, isomorphic to $\mathbb Q[\lambda]/(P)=\mathbb Q(\alpha)$. A matrix commuting with $C$ is a $\mathbb Q(\alpha)$-linear endomorphism of a one-dimensional $\mathbb Q(\alpha)$-vector space, hence multiplication by an element of $\mathbb Q(\alpha)$. Over $\mathbb C$, $P$ has distinct roots, so $V_P$ is the conjugacy class of a regular semisimple matrix, the orbit $\mathrm{GL}_n/T$ with $T$ a maximal torus of dimension $n$. Its dimension is $n^2-n$. The stabilizer in $\mathrm{SL}_n$ is the norm-one torus of $\mathbb Q(\alpha)$, which is $\mathbb Q$-anisotropic; this is why the hypothesis of the theorem holds.

Lesson 06

The quantitative Oppenheim conjecture

For the general reader. A quadratic form such as $x^2+y^2-\sqrt2\,z^2$ takes values arbitrarily close to any real number at integer points. Margulis proved this in 1986. Eskin, Margulis and Mozes asked how often: how many integer points of size at most $T$ give a value in a fixed interval. The answer is the obvious volume, except in two small signatures where hidden arithmetic produces an extra logarithm.

Goals

State the Oppenheim conjecture and its quantitative form.
Understand why signatures (2,1) and (2,2) are exceptional.
See the role of $\mathrm{SO}(Q)$-orbits on the space of lattices.

Theorem (Margulis, 1986)

Let $Q$ be a real non-degenerate indefinite quadratic form in $n\ge3$ variables that is not proportional to a form with rational coefficients. Then $Q(\mathbb Z^n)$ is dense in $\mathbb R$.

Margulis’s proof translates the problem into dynamics: the orthogonal group $H=\mathrm{SO}(Q)$ acts on the space $\mathrm{SL}(n,\mathbb R)/\mathrm{SL}(n,\mathbb Z)$ of unimodular lattices, and density of values follows from the fact that a bounded $H$-orbit of the standard lattice must be closed, which forces $Q$ to be rational. For a quantitative result, define for an interval $(a,b)$

\[ N_{Q}(a,b;T)=\#\{v\in\mathbb Z^n:\ \|v\|\le T,\ a\lt Q(v)\lt b\}. \]

The volume of the corresponding region is $\sim\lambda_Q(b-a)T^{n-2}$ for an explicit $\lambda_Q\gt0$. Dani and Margulis proved the lower bound $\liminf N_Q/(\lambda_Q(b-a)T^{n-2})\ge1$ for irrational $Q$.

Theorem (Eskin–Margulis–Mozes, Annals 1998)

Let $Q$ be an irrational indefinite form of signature $(p,q)$, $n=p+q$.

If $p\ge3$ and $q\ge1$, then $N_Q(a,b;T)\sim\lambda_Q(b-a)T^{n-2}$.
If the signature is $(2,1)$ or $(2,2)$, then $N_Q(a,b;T)\le C\,T^{n-2}\log T$, and there are irrational forms for which the count exceeds $T^{n-2}(\log T)^{1-\varepsilon}$ along a sequence $T_j\to\infty$. The asymptotic of (1) fails for these forms.

Theorem (Eskin–Margulis–Mozes, Annals 2005)

For signature $(2,2)$, the asymptotic $N_Q(a,b;T)\sim\lambda_Q(b-a)T^2$ holds for every interval with $0\notin[a,b]$, unless $Q$ is extremely well approximable by split rational forms (EWAS). A variant applied to forms $(x_1^2+\beta x_2^2)-(x_3^2+\beta x_4^2)$ gives the pair correlation of Laplace eigenvalues on rectangular flat tori with Diophantine aspect ratio $\beta$, as predicted by the Berry–Tabor conjecture.

Proof idea

Count with a smooth weight and rewrite the count as an integral over the $K$-orbit of a translated lattice $a_t g_Q\mathbb Z^n$, where $a_t$ is a diagonal flow in $\mathrm{SO}(Q)$. One needs equidistribution of these expanding $K$-orbits against an unbounded test function, the Siegel transform $\hat f(\Lambda)=\sum_{v\in\Lambda}f(v)$, which is large when $\Lambda$ contains a short vector. Ratner’s theorem plus linearization give equidistribution against bounded functions. The new ingredient is a family of functions $\alpha_i(\Lambda)$, built from the covolumes of sublattices of rank $i$, which satisfy a contraction inequality under averaging over $K$. This proves that $\int_K\hat f(a_tk\Lambda)\,dk$ stays bounded in $t$ when $p\ge3$ and $q\ge1$. In signatures $(2,1)$ and $(2,2)$ the average grows like $t$, which is $\log T$. The same Margulis-function technique has since become a standard tool.

Exercise 6.1Let $Q=x_1^2+x_2^2-x_3^2-x_4^2$, a rational form of signature $(2,2)$. Using $\sum_{m\le X}r_2(m)^2\sim4X\log X$, where $r_2(m)$ counts representations as a sum of two squares, show that the number of $v\in\mathbb Z^4$ with $Q(v)=0$ and $\|v\|\le T$ is $\sim4T^2\log T$. What does this say about signature $(2,2)$?

Show solution

If $Q(v)=0$ then $x_1^2+x_2^2=x_3^2+x_4^2=m$ and $\|v\|^2=2m$. The count is $\sum_{m\le T^2/2}r_2(m)^2\sim4\cdot\frac{T^2}{2}\log\frac{T^2}{2}\sim4T^2\log T$. The volume term is of order $T^{n-2}=T^2$, so the count exceeds it by a logarithm. The extra points lie on rational isotropic planes, where $Q$ vanishes identically. Irrational forms that are very close to such rational forms inherit the effect along a sequence of scales, which is why (2,2) is exceptional.

Exercise 6.2Show that Margulis’s theorem fails for $n=2$. Let $Q(x,y)=(x-\sqrt2\,y)(x+\sqrt3\,y)$. Prove that $|Q(v)|$ is bounded below by a positive constant on $\mathbb Z^2\setminus\{0\}$.

Show solution

Quadratic irrationals are badly approximable: there is $c\gt0$ with $|x-\sqrt2\,y|\ge c/|y|$ and $|x+\sqrt3\,y|\ge c/|y|$ for all integers $x$ and $y\ne0$. Put $\delta=\sqrt2+\sqrt3$. Since $(x+\sqrt3\,y)-(x-\sqrt2\,y)=\delta y$, one of the two factors has absolute value at least $\delta|y|/2$. The other is at least $c/|y|$, so $|Q|\ge c\delta/2$. For $y=0$, $Q=x^2\ge1$. The form is irrational, since its coefficients $1,\ \sqrt3-\sqrt2,\ -\sqrt6$ are not proportional to rationals, yet its values avoid a neighbourhood of 0. With two variables the group $\mathrm{SO}(Q)$ is a split torus, and the unipotent dynamics behind Margulis’s proof is absent.

Lesson 07

Quasi-isometric rigidity

For the general reader. Seen from very far away, a group drawn as a network of points and edges looks like a geometric space. Mikhail Gromov asked how much of the group this blurred picture remembers. Eskin proved that for many groups it remembers almost everything: anything that looks the same from far away is essentially the same group.

Goals

Define quasi-isometries and quasi-isometric rigidity.
State the quasi-flats theorem and rigidity for non-uniform higher-rank lattices.
Understand coarse differentiation and its use for Sol and lamplighter groups.

A map $f:X\to Y$ of metric spaces is a $(K,C)$-quasi-isometry if $\tfrac1Kd(x,x')-C\le d(f(x),f(x'))\le Kd(x,x')+C$ and every point of $Y$ lies within $C$ of the image. A finitely generated group with a word metric is well defined up to quasi-isometry, and by the Švarc–Milnor lemma it is quasi-isometric to any proper geodesic space on which it acts properly and cocompactly.

Theorem (Eskin–Farb 1997; Kleiner–Leeb independently)

Let $X$ be a symmetric space of non-compact type of rank $r$. Every $(K,C)$-quasi-isometric embedding $\mathbb R^r\to X$ has image within bounded distance of a finite union of maximal flats, the number and distance depending only on $K$, $C$ and $X$.

Theorem (Eskin, J. AMS 1998)

Let $\Gamma$ be an irreducible non-uniform lattice in a semisimple Lie group of real rank at least 2 with finite centre. Any finitely generated group quasi-isometric to $\Gamma$ is, up to finite groups, commensurable with $\Gamma$. Every quasi-isometry of $\Gamma$ lies within bounded distance of commensuration.

Richard Schwartz had proved the rank-one case. Eskin’s proof uses the quasi-flats theorem to show that a quasi-isometry of the lattice extends to a quasi-isometry of the symmetric space that preserves the pattern of horoballs removed around the cusps, and then upgrades it to an isometry.

Coarse differentiation

The group Sol is $\mathbb R^2\rtimes\mathbb R$, with $t$ acting by $\mathrm{diag}(e^t,e^{-t})$. Its lattices are the fundamental groups of torus bundles over the circle with Anosov monodromy. Benson Farb and Lee Mosher asked whether every group quasi-isometric to Sol is virtually a lattice in Sol. Eskin, David Fisher and Kevin Whyte answered yes. A quasi-isometry has no derivative, but on most long geodesic segments at a suitable scale it is close to a geodesic of the target. This coarse differentiation lets them prove that quasi-isometries of Sol coarsely preserve the foliation by hyperbolic planes.

Theorem (Eskin–Fisher–Whyte, Annals 2012, 2013)

Every quasi-isometry of Sol is within bounded distance of a map $(x,y,t)\mapsto(f(x),g(y),t)$ with $f,g$ bi-Lipschitz, or of such a map composed with the flip $(x,y,t)\mapsto(y,x,-t)$. Any finitely generated group quasi-isometric to Sol is virtually a lattice in Sol.
The Diestel–Leader graph $\mathrm{DL}(m,n)$ with $m\ne n$ is a vertex-transitive graph not quasi-isometric to the Cayley graph of any finitely generated group, answering a question of Wolfgang Woess.
The lamplighter groups $F\wr\mathbb Z$ and $F'\wr\mathbb Z$ are quasi-isometric exactly when $|F|$ and $|F'|$ are powers of a common integer.

Exercise 7.1Show that the left-invariant metric $ds^2=e^{2t}dx^2+e^{-2t}dy^2+dt^2$ on Sol restricts to the hyperbolic metric on each plane $\{y=\text{const}\}$.

Show solution

On $y=y_0$ the metric is $e^{2t}dx^2+dt^2$. Put $u=e^{-t}$: then $dt=-du/u$ and $e^{2t}dx^2=dx^2/u^2$, so the metric is $(dx^2+du^2)/u^2$, the upper half-plane. The planes $\{x=\text{const}\}$ are hyperbolic planes too, with the opposite orientation of $t$. Sol is foliated by hyperbolic planes in two transverse ways.

Exercise 7.2Show that $\mathbb Z^m$ and $\mathbb Z^n$ are quasi-isometric only if $m=n$.

Show solution

Balls of radius $R$ in $\mathbb Z^m$ contain $\asymp R^m$ points. A $(K,C)$-quasi-isometry maps distinct points that are at least $2KC$ apart to distinct points, and maps a ball of radius $R$ into a ball of radius $KR+C$. Taking a $2KC$-separated subset of the ball gives $R^m\lesssim(KR+C)^n$, so $m\le n$. By symmetry with a quasi-inverse, $m=n$. Growth is a quasi-isometry invariant.

Lesson 08

Flat surfaces, counting and volumes

For the general reader. A billiard ball on a table whose angles are rational fractions of $\pi$ can be studied by unfolding the table into a flat surface with a few cone points. Counting periodic billiard paths becomes counting straight closed lines on that surface. Eskin, with Howard Masur, Anton Zorich and Andrei Okounkov, showed how to compute these counts from the geometry of the space of all such surfaces.

Goals

Define strata of translation surfaces and the Masur–Veech measure.
State the Siegel–Veech formula and the Eskin–Masur theorem.
Understand why volumes of strata are rational multiples of powers of $\pi$.

A translation surface is a pair $(X,\omega)$ of a compact Riemann surface of genus $g$ and a non-zero holomorphic 1-form. If $\omega$ has zeros of orders $m_1,\dots,m_n$ with $\sum m_i=2g-2$, the surface belongs to the stratum $\mathcal H(m_1,\dots,m_n)$. Local period coordinates in $H^1(X,Z(\omega);\mathbb C)$ give the stratum a complex affine structure and a Lebesgue measure, and $\mathrm{SL}(2,\mathbb R)$ acts on it. Masur and William Veech proved independently (1982) that the induced measure on the unit-area hypersurface $\mathcal H_1$ is finite and ergodic for the Teichmüller flow.

Siegel–Veech formula (Veech, 1998)

Let $\mu$ be an ergodic $\mathrm{SL}(2,\mathbb R)$-invariant probability measure on $\mathcal H_1$. For a surface $\omega$, let $V(\omega)\subset\mathbb R^2$ be the set of holonomy vectors of maximal cylinders (or saddle connections), counted with multiplicity, and set $\hat f(\omega)=\sum_{v\in V(\omega)}f(v)$ for $f\in C_c(\mathbb R^2)$. Then there is a constant $c(\mu)$, the Siegel–Veech constant, with

\[ \int_{\mathcal H_1}\hat f\,d\mu=c(\mu)\int_{\mathbb R^2}f\,dx\,dy. \]

Taking $f$ the indicator of a disc of radius $L$ shows that the average number of cylinders of circumference at most $L$ is $c\,\pi L^2$. Eskin and Masur proved that this holds for almost every individual surface.

Theorem (Eskin–Masur, Ergodic Th. Dynam. Sys. 2001)

For $\mu$-almost every $\omega\in\mathcal H_1$, the number $N(\omega,L)$ of maximal cylinders of circumference at most $L$ satisfies $N(\omega,L)\sim c(\mu)\,\pi L^2$ as $L\to\infty$. The same holds for saddle connections, with their own constant.

The proof combines the Siegel–Veech formula with an ergodic theorem for circle averages $\int_0^{2\pi}\hat f(g_tr_\theta\omega)\,d\theta$, and an estimate that controls surfaces with many short saddle connections. Eskin, Masur and Zorich (Publ. IHÉS 2003) then computed the constants: a configuration of parallel saddle connections, pinched to length zero, degenerates the surface to the principal boundary, a product of simpler strata, and each Siegel–Veech constant becomes a ratio of volumes of strata.

Theorem (Eskin–Okounkov, Invent. Math. 2001)

The Masur–Veech volume of every stratum $\mathcal H_1(m_1,\dots,m_n)$ of Abelian differentials in genus $g$ is a rational multiple of $\pi^{2g}$. The generating functions counting connected torus covers with prescribed ramification are quasimodular forms.

The idea, proposed by Kontsevich and Zorich, is that square-tiled surfaces, those built from unit squares, are the integer points of the stratum in period coordinates. Counting square-tiled surfaces with at most $N$ squares therefore approximates the volume. A square-tiled surface is a branched cover of the torus, so the count becomes a sum of characters of the symmetric group. Eskin and Okounkov evaluated it using the Bloch–Okounkov theory of shifted symmetric functions.

Exercise 8.1Unramified connected covers of degree $N$ of the square torus correspond to subgroups of index $N$ in $\mathbb Z^2$. Show that there are $\sigma(N)=\sum_{d\mid N}d$ of them, and deduce $\sum_{N\le D}\sigma(N)\sim\frac{\pi^2}{12}D^2$.

Show solution

Each index-$N$ sublattice has a unique Hermite normal form basis $\{(a,0),(b,d)\}$ with $ad=N$, $0\le b\lt a$. For each divisor $a$ of $N$ there are $a$ choices of $b$, so the count is $\sigma(N)$. Then $\sum_{N\le D}\sigma(N)=\sum_{d\le D}\sum_{k\le D/d}k\approx\sum_d\frac{D^2}{2d^2}=\frac{\zeta(2)}2D^2=\frac{\pi^2}{12}D^2$. The appearance of $\zeta(2)$ is the genus-one case of Eskin–Okounkov: the volume of $\mathcal H(0)$ is a rational multiple of $\pi^2$.

Exercise 8.2On the square torus $\mathbb C/\mathbb Z[i]$ with one marked point, saddle connections correspond to primitive vectors of $\mathbb Z^2$. Compute their asymptotic count and the Siegel–Veech constant in the normalization $N\sim c\,\pi L^2$.

Show solution

The number of primitive $(p,q)$ with $p^2+q^2\le L^2$ is $\sim\frac{1}{\zeta(2)}\pi L^2=\frac6{\pi^2}\pi L^2$. So $c=6/\pi^2$. For the torus every surface in the stratum is a unimodular lattice, and this computation is the Siegel mean value theorem for primitive vectors.

Lesson 09

Lyapunov exponents of the Hodge bundle

For the general reader. Follow a straight line on a flat surface for a long time and record how often it crosses each loop. The counts grow at rates measured by numbers called Lyapunov exponents, which also govern how fast certain flows mix. Eskin, Maxim Kontsevich and Zorich found a formula for their sum that involves only the cone angles and a counting constant from Lesson 08.

Goals

Define the Kontsevich–Zorich cocycle and its Lyapunov spectrum.
State the Eskin–Kontsevich–Zorich formula.
Compute $\lambda_2$ for the two strata of genus 2.

Over a stratum, the cohomology groups $H^1(X;\mathbb R)$ form a flat vector bundle, the real Hodge bundle. The Teichmüller flow $g_t$ lifts to it by parallel transport with respect to the Gauss–Manin connection; this is the Kontsevich–Zorich cocycle. By Oseledets’ theorem, for an ergodic invariant measure it has Lyapunov exponents

\[ 1=\lambda_1\ge\lambda_2\ge\dots\ge\lambda_g\ge-\lambda_g\ge\dots\ge-\lambda_1=-1. \]

Zorich discovered numerically that these exponents govern the deviation of ergodic averages for interval exchanges and flat surfaces, and Kontsevich conjectured a formula for their sum. Giovanni Forni proved $\lambda_g\gt0$ for Masur–Veech measures, and Artur Avila and Marcelo Viana proved the spectrum is simple.

Theorem (Eskin–Kontsevich–Zorich, Publ. IHÉS 2014)

Let $\mathcal M_1$ be a closed connected $\mathrm{SL}(2,\mathbb R)$-invariant suborbifold of $\mathcal H_1(m_1,\dots,m_n)$ carrying an affine invariant measure. Then

\[ \lambda_1+\dots+\lambda_g=\frac1{12}\sum_{i=1}^{n}\frac{m_i(m_i+2)}{m_i+1}+\frac{\pi^2}3\,c_{\mathrm{area}}(\mathcal M_1), \]

where $c_{\mathrm{area}}$ is the Siegel–Veech constant that counts cylinders weighted by area.

The proof originally assumed a regularity condition on the measure; Avila, Carlos Matheus and Jean-Christophe Yoccoz showed that every invariant measure is regular. The argument runs through analytic Riemann–Roch. Kontsevich and Forni showed that the sum of exponents equals the average curvature of the Hodge bundle, an integral of $\Delta\log\det$ of the Hodge norm. EKZ compute this integral using the determinant of the flat Laplacian, whose behaviour near the boundary of the stratum, where cylinders become thin, produces the Siegel–Veech term.

Exercise 9.1Eskin, Masur and Zorich computed $c_{\mathrm{area}}(\mathcal H(2))=\frac{10}{3\pi^2}$ and $c_{\mathrm{area}}(\mathcal H(1,1))=\frac{15}{4\pi^2}$. Find $\lambda_2$ for each stratum.

Show solution

For $\mathcal H(2)$: the first term is $\frac1{12}\cdot\frac{2\cdot4}{3}=\frac29$ and the second $\frac{\pi^2}3\cdot\frac{10}{3\pi^2}=\frac{10}9$, so $1+\lambda_2=\frac43$ and $\lambda_2=\frac13$. For $\mathcal H(1,1)$: $\frac1{12}\cdot2\cdot\frac32=\frac14$ and $\frac{\pi^2}3\cdot\frac{15}{4\pi^2}=\frac54$, so $1+\lambda_2=\frac32$ and $\lambda_2=\frac12$. Both values had been observed numerically by Zorich and proved earlier by Bainbridge.

Exercise 9.2Apply the formula to the stratum $\mathcal H(0)$ of flat tori, deduce $c_{\mathrm{area}}(\mathcal H(0))$, and check the answer directly.

Show solution

Here $g=1$, so the left side is $\lambda_1=1$, and the marked point has order $m=0$, contributing 0. Hence $c_{\mathrm{area}}=3/\pi^2$. Directly: on a unit-area torus each primitive vector $\pm v$ of the lattice gives one cylinder, of area 1 and circumference $|v|$. Unordered pairs $\pm v$ of primitive vectors with $|v|\le L$ number $\sim\frac12\cdot\frac6{\pi^2}\pi L^2=\frac3{\pi^2}\pi L^2$, on average over lattices by Siegel’s formula. The two computations agree.

Lesson 10

The magic wand and beyond

For the general reader. On homogeneous spaces, Marina Ratner’s theorems say that unipotent orbits are tame. The moduli space of flat surfaces is not homogeneous, and for years it was unclear whether any similar rigidity held there. Eskin and Maryam Mirzakhani proved that it does. The result turns questions about billiards in any rational polygon into questions about a countable list of algebraic objects, which is why Anton Zorich called it a magic wand.

Goals

State the Eskin–Mirzakhani and Eskin–Mirzakhani–Mohammadi theorems.
Understand the strategy: exponential drift without a Lie group.
See the later extension to random dynamics on manifolds.

Theorem (Eskin–Mirzakhani, Publ. IHÉS 2018; preprint 2013)

Let $P\subset\mathrm{SL}(2,\mathbb R)$ be the subgroup of upper triangular matrices. Every ergodic $P$-invariant probability measure on a stratum $\mathcal H_1(\kappa)$ is $\mathrm{SL}(2,\mathbb R)$-invariant and affine: it is supported on an immersed submanifold that in period coordinates is cut out by real linear equations, and in those coordinates it is a multiple of Lebesgue measure.

Theorem (Eskin–Mirzakhani–Mohammadi, Annals 2015)

For every $x\in\mathcal H_1(\kappa)$, the closure of $\mathrm{SL}(2,\mathbb R)x$ equals the closure of $Px$ and is an affine invariant submanifold. There are only countably many affine invariant submanifolds, and the averages $\frac1T\int_0^T\frac1{2\pi}\int_0^{2\pi}\delta_{g_tr_\theta x}\,d\theta\,dt$ converge to the affine measure on the orbit closure.

Strategy

The proof follows the outline of Yves Benoist and Jean-François Quint’s work on stationary measures for random walks on homogeneous spaces. Take two nearby generic points $x$ and $x'$ on the same unstable leaf. Apply a random element of the walk and flow for a long time: the displacement between the images grows, and its direction is controlled by Lyapunov exponents of the Kontsevich–Zorich cocycle. In a Lie group the displacement would be an element of the group, and one would obtain extra invariance. In a stratum there is no group, so Eskin and Mirzakhani built the replacement by hand: a careful choice of stopping times (the “exponential drift”), Margulis-type functions that prevent escape to infinity, and a detailed Oseledets theory for the Kontsevich–Zorich cocycle. The published proof runs over two hundred pages.

Simion Filip proved (2016) that every affine invariant submanifold is a quasi-projective algebraic variety, defined over a number field. The classification of these varieties is now a research programme, with major contributions by Eskin’s students Alex Wright and Paul Apisa.

Beyond homogeneous spaces

Eskin and Elon Lindenstrauss gave a new treatment of the Benoist–Quint classification of stationary measures on $G/\Gamma$. In a 2025 preprint, Aaron Brown, Eskin, Filip and Federico Rodriguez Hertz developed a factorization method that proves measure rigidity for stationary measures of random walks by diffeomorphisms of general manifolds, under a quantitative non-integrability condition. Among its applications is the extension of Eskin–Mirzakhani to products of strata.

Exercise 10.1The stratum $\mathcal H_1(0)$ is $\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(2,\mathbb Z)$. What are its affine invariant submanifolds, and what does Ratner’s theorem say about orbit closures of the horocycle subgroup $U=\{\begin{pmatrix}1&s\\0&1\end{pmatrix}\}$?

Show solution

The only $\mathrm{SL}(2,\mathbb R)$-invariant closed subset is the whole space, since the action is transitive. For $U$, Dani’s theorem (a case of Ratner) says each orbit is either periodic, a closed horocycle that corresponds to a lattice with a horizontal vector, or dense. The magic wand is the analogue for $\mathrm{SL}(2,\mathbb R)$ and $P$ in every stratum; for $U$ alone, orbit closures in higher strata are not yet classified.

Exercise 10.2A Veech surface is one whose stabilizer $\mathrm{SL}(X,\omega)$ in $\mathrm{SL}(2,\mathbb R)$ is a lattice. Show that its $\mathrm{SL}(2,\mathbb R)$-orbit is closed, and identify it as an affine invariant submanifold. What is its complex dimension?

Show solution

The orbit map induces a proper injective immersion $\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(X,\omega)\to\mathcal H_1$ (Smillie’s theorem gives the converse: closed orbits come only from lattice stabilizers). In period coordinates the orbit of $\omega$ lies in the real span of $\mathrm{Re}\,\omega$ and $\mathrm{Im}\,\omega$, so the $\mathrm{GL}^+(2,\mathbb R)$-orbit is cut out by linear equations and has complex dimension 2; the unit-area part is a Teichmüller curve. The regular octagon of the opening figure is a Veech surface (Veech, 1989).

Part III · Life and legacy

Lesson 11

Collaborations and a school

Almost all of Eskin’s major papers are joint. His collaborators span several fields: Curtis McMullen, Shahar Mozes, Nimish Shah and Grigory Margulis in homogeneous dynamics; Benson Farb, David Fisher and Kevin Whyte in geometric group theory; Howard Masur, Anton Zorich, Maxim Kontsevich and Andrei Okounkov on flat surfaces; Maryam Mirzakhani and Amir Mohammadi on rigidity in moduli space; Elon Lindenstrauss, Aaron Brown, Simion Filip and Federico Rodriguez Hertz on random dynamics. With Hee Oh he proved equidistribution of Hecke points by ergodic methods, and with Mozes and Oh he proved that finitely generated linear groups in characteristic zero that are not virtually solvable have uniform exponential growth (Invent. Math. 2005).

The collaboration with Mirzakhani began while she was at Princeton and lasted until her death in 2017. Their first joint results counted closed geodesics in moduli space, equivalently pseudo-Anosov mapping classes, and the large-scale lattice-point problem in Teichmüller space (with Jayadev Athreya and Alexander Bufetov). The magic wand work followed. With Mohammadi they later proved an effective version of Mirzakhani’s count of simple closed geodesics. When the Breakthrough Prize was announced in 2019, Eskin said he planned to give part of the prize money to an International Mathematical Union fellowship fund for graduate students from developing countries.

At Chicago Eskin has supervised fourteen doctoral students. They include Jayadev Athreya (2006), Moon Duchin (2005), Kariane Calta (2004), Irine Peng (2008), Alex Wright (2014), Ilya Gekhtman (2014), Simion Filip (2016), Paul Apisa (2018) and Ian Frankel (2018). Several now lead the classification of orbit closures that his theorem made possible; Duchin became known for applying metric geometry to the analysis of electoral districting.

DiscussionEskin’s career moved from number theory to group theory to Teichmüller dynamics, each time carrying one method across. What made the method portable, and where did it have to be rebuilt from scratch?

Lesson 12

Honours and influence

Eskin received a Packard Fellowship in 1997 and the Clay Research Award in 2007, the latter for rational billiards and geometric group theory, with special mention of the quasi-isometric rigidity of Sol. He gave invited addresses at the International Congresses of Mathematicians in Berlin (1998) and Hyderabad (2010, with David Fisher). He was elected to the American Academy of Arts and Sciences in 2011 and to the National Academy of Sciences in 2015, and is a Fellow of the American Mathematical Society. He was named a Simons Investigator in 2014.

In September 2019 he was awarded the 2020 Breakthrough Prize in Mathematics for revolutionary discoveries in the dynamics and geometry of moduli spaces of Abelian differentials, including the proof of the magic wand theorem with Mirzakhani. In 2025 his students and collaborators organized a conference in his honour, Dynamics and Rigidity, in Cetraro, Italy, for his sixtieth birthday.

Influence

Homogeneous dynamics and number theory: the mixing method for counting, linearization and Margulis functions are standard tools, used for Hecke points, Diophantine approximation and effective equidistribution.
Geometric group theory: coarse differentiation solved the Farb–Mosher problems for Sol and lamplighters and is now used for other solvable groups.
Teichmüller dynamics: the Siegel–Veech constants, volumes of strata, the EKZ formula and the magic wand form the core of the subject, and orbit closures of the $\mathrm{SL}(2,\mathbb R)$ action are now studied as algebraic varieties.
Smooth dynamics: the factorization method extends measure rigidity from Lie groups to diffeomorphisms of manifolds.

Final projectChoose a rational polygon, such as the L-shaped table or the $(\pi/5,\pi/5,3\pi/5)$ triangle. Unfold it to a translation surface, identify its stratum and its orbit closure, and explain which theorem of Part II gives the growth rate of its periodic billiard trajectories.

Reference

Connections

People linked to Alex Eskin across this collection. A name without a link has no page here yet. See Alex Eskin in the network of all 337 lives →

Teachers and advisors

Peter Sarnak 1993

Students

Alex Wright 2014
Ian Frankel 2018
Ilya Gekhtman 2014
Irine Peng 2008
Jayadev Athreya 2006
Kariane Calta 2004
Moon Duchin 2005
Paul Apisa 2018
Simion Filip 2016

Influenced by

Marina Ratner

Colleagues and collaborators

Curtis McMullen 1993
Grigory Margulis 1998-2005
Howard Masur
Maryam Mirzakhani 2008-2017
Maxim Kontsevich
Ronald Fisher
Aaron Brown
Amir Mohammadi
Andrei Okounkov
Anton Zorich
Benson Farb
Elon Lindenstrauss
Federico Rodriguez Hertz
Hee Oh
Kevin Whyte
Nimish Shah
Shahar Mozes
Zeév Rudnick

Family

Gregory Eskin

Also mentioned in this life

Fields

Reference

Sources and further reading

Biography

Alex Eskin, Wikipedia; German Wikipedia for career dates.
University of Chicago News, UChicago mathematician, physicists win $3 million ‘Oscars of science’, September 2019.
Simons Foundation, Awardee spotlight: Alex Eskin, 2015.
Clay Mathematics Institute, Alex Eskin, Clay Research Award 2007.
Mathematics Genealogy Project, Alex Eskin.
Clay Mathematics Institute, Dynamics and Rigidity: a celebration of the mathematics of Alex Eskin, Cetraro 2025.
A. Eskin, Abstracts of papers, University of Chicago.

Mathematics

A. Eskin, C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993).
A. Eskin, S. Mozes, N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. 143 (1996).
A. Eskin, G. Margulis, S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. 147 (1998).
A. Eskin, G. Margulis, S. Mozes, Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori, Ann. of Math. 161 (2005).
A. Eskin, B. Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997).
A. Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces, J. Amer. Math. Soc. 11 (1998).
A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries I and II, Ann. of Math. 176 (2012) and 177 (2013).
A. Eskin, H. Masur, Pointwise asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001).
A. Eskin, A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001).
A. Eskin, H. Masur, A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants, Publ. Math. IHÉS 97 (2003).
A. Eskin, M. Kontsevich, A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. IHÉS 120 (2014).
A. Eskin, M. Mirzakhani, Invariant and stationary measures for the SL(2,ℝ) action on moduli space, Publ. Math. IHÉS 127 (2018).
A. Eskin, M. Mirzakhani, A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2,ℝ) action on moduli space, Ann. of Math. 182 (2015).
A. Zorich, The magic wand theorem of A. Eskin and M. Mirzakhani, arXiv:1502.05654.
A. Brown, A. Eskin, S. Filip, F. Rodriguez Hertz, Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method, arXiv:2502.14042 (2025).
S. Filip, Measure rigidity beyond homogeneous dynamics, arXiv:2512.13865.
A. Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, arXiv:1504.08290.