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The work of Maryam Mirzakhani

18 August, 2014

Abstract

Maryam Mirzakhani has been awarded the Fields Medal for her out- standing work on the dynamics and geometry of Riemann surfaces and their moduli spaces.

1 Introduction

Mirzakhani has established a suite of powerful new results on orbit closures and invariant measures for dynamical systems on moduli spaces. She has also given a new proof of Witten's conjecture, which emerges naturally from a counting problem for simple closed geodesics on Riemann surfaces. This note gives a brief discussion of her main results and their ramications, including the striking parallels between homogeneous spaces and moduli spaces that they suggest.

2 The setting

We begin with a resume of background material, to set the stage. LetMgdenote the moduli space of curves of genusg2. This space is both a complex variety, with dim

CMg= 3g3, and a symplectic orbifold.

Its points are in bijection with the isomorphism classes of compact Riemann surfacesXof genusg. The dimension ofMgwas known already to Riemann. Rigorous con- structions of moduli space were given in the 1960s, by Ahlfors and Bers in the setting of complex analysis and by Mumford in the setting of alge- braic geometry. Today the theory of moduli spaces is a meeting ground for mathematical disciplines ranging from arithmetic geometry to string theory. 1 The symplectic form!onMgarises from the hyperbolic metric onX. As shown by Wolpert, in the length{twist coordinates coming from a pair of pants decomposition ofX, one can write !=3g3X 1d` i^di: The complex structure onMgarises from the natural isomorphism T

XMg=Q(X) =fholomorphic formsq=q(z)dz2onXg

between the cotangent space toMgatXand the space of holomorphic quadratic dierentials onX. TheTeichmuller metriconMgalso emerges from its complex structure: on the one hand, it is dual to theL1norm kqk=Z X jq(z)jjdzj2= area(X;jqj) on T XMg; on the other hand, it agrees with the intrinsic Kobayashi metric onMg(Royden). Moduli space can be presented as the quotientMg=Tg=Modgof Te- ichmuller space | its universal cover, a contractible bounded domain in C

3g3| by the action of the mapping{class group of a surface.

One of the challenges of working with moduli space is that it istotally inhomogeneous: for example, the symmetry group ofTg(as a complex man- ifold) is simply the discrete group Mod g(forg >2). One of Mirzakhani's remarkable contributions is to show that, nevertheless, dynamics on moduli space displays many of the same rigidity properties as dynamics on homo- geneous spaces (seex4).

3 From simple geodesics to Witten's conjecture

We begin with Mirzakhani's work on simple geodesics. In the 1940s, Del- sarte, Huber and Selberg established theprime number theoremfor hyper- bolic surfaces, which states that the number of (oriented, primitive) closed geodesics onX2 Mgwith lengthLsatises (X;L)eLL (The usual prime number theorem says that the number of prime integers with 01(E1)d1c1(En)dn to a power series solution to the KdV hierarchy (an innite system of dier- ential equations satisfying the Virasoro relations). HereM g;nis the Deligne{ Mumford compactication of the moduli space of Riemann surfacesXwith marked points (p1;:::;pn), andc1(Ei) denotes the rst Chern class of the line bundleEi!M g;nwith bers TpiX. Mirzakhani's investigation of(X;L) also leads to formulas for the fre- quencies of dierent topological types of simple closed curves onX; for example, a random simple curve on a surface of genus 2 has probability

1=7 of cuttingXinto two pieces of genus 1. These frequencies are always

rational numbers, and they depend only ong, notX. At the core of these results is Mirzakhani's novel, recursive calculation of the volume of the moduli space of Riemann surfaces of genusgwithn geodesic boundary components with lengths (L1;:::;Ln). This volume is dened by P g;n(L1;:::;Ln) =Z M g;n(L1;:::;Ln)!3g3+n; for example, one can show thatP1;1(L1) = (1=24)(L21+ 42). In general, P g;nis a polynomial whose coecients (which lie inQ()) can be related to frequencies and characteristic classes, yielding the results discussed above. Previously only the values ofPg;n(0;:::;0) were known. The proofs depend on intricate formulas for dissections of surfaces along hyperbolic geodesics; see [Mir3], [Mir1] and [Mir2]. Mirzakhani has also studied the behavior of M gasg! 1; see [Mir4],[Mir6]. 3

4 Complex geodesics in moduli space

We now turn to Mirzakhani's work on moduli spaces and dynamics. Her con- tributions to this area include a prime number theorem for closed geodesics inMg, counting results for orbits of ModgonTg, and the classication of Mod g{invariant measures on the space of measured laminationsMLg. But perhaps her most striking work | which we will present here | is a version of Ratner's theorem for moduli spaces. Complex geodesics.It has been known for some time that the Teichmuller geodesic ow is ergodic (Masur, Veech), and hence almost every geodesic Mgis dense. It is dicult, however, to describe the behavior ofevery single geodesic ; already on a hyperbolic surface, the closure of a geodesic can be a fractal cobweb, and matters only get worse in moduli space. Teichmuller showed that moduli space is also abundantly populated by complexgeodesics, these being holomorphic, isometric immersions

F:H! Mg:

In fact there is a complex geodesic through everyX2 Mgin every possible direction. In principle, the closure of a complex geodesic might exhibit the same type of pathology as a real geodesic. But in fact, the opposite is true. In a major breakthrough, Mirzakhani and her coworkers have shown: The closure of any complex geodesic is an algebraic subvariety

V=F(H) Mg.

This long sought{after rigidity theorem was known previously only forg= 2, with some restrictions onF[Mc]. (In the case of genus two,Vcan be an isometrically immersed curve, a Hilbert modular surface, or the whole space M 2.) Dynamics over moduli space.The proof of this rigidity theorem involves the natural action of SL

2(R) on the sphere bundle

Q

1Mg! Mg;

consisting of pairs (X;q) withq2Q(X) andkqk= 1. To describe this action, consider a Riemann surfaceX=P=presented as the quotient of a polygonPCunder isometric edge identications between pairs of parallel sides. Such identications preserve the quadratic dierentialdz2jP, so a polygonal model forXactually determines a pair 4 (X;q)2QMgwithkqk= area(P). Conversely, every nonzero quadratic dierential (X;q)2QMgcan be presented in this form.

Since SL

2(R) acts linearly onR2=C, givenA2SL2(R) we can form a

new polygonA(P)C, and use the corresponding edge identications to dene

A(X;q) = (XA;qA) = (A(P);dz2)=:

Note that [XA] = [X] ifA2SO2(R). Thus the mapA7!XAdescends to give a map

F:H=SL2(R)=SO2(R)! Mg;

which is the complex geodesicgeneratedby (X;q). The proof thatF(H) Mgis an algebraic variety involves the following three theorems, each of which a substantial work in its own right.

1. Measure classication (Eskin and Mirzakhani).Every ergodic,SL2(R){

invariant probability measure onQ1Mgcomes from Euclidean measure on a special complex{analytic subvarietyAQMg(The varietyAis linear in period coordinates). This is the deepest step in the proof; it uses a wide variety of tech- niques, including conditional measures and a random walk argument inspired by the work of Benoist and Quint [BQ].

2. Topological classication (Eskin, Mirzakhani and Mohammadi).The

closure of anySL2(R)orbit inQ1Mgis given byA\Q1Mgfor some special analytic subvarietyA.

3. Algebraic structure (Filip).Any special analytic subvarietyAis in

fact an algebraic subvariety ofQMg.Thus its projection toMg,

V=F(H), is an algebraic subvariety as well.

See [EM], [EMM] and [Fil] for these developments.

Ramications: Beyond homogeneous spaces.This collection of re- sults reveals that the theory of dynamics on homogeneous spaces, developed by Margulis, Ratner and others, has adenite resonancein the highly inho- mogeneous, but equally important, world of moduli spaces. The setting for homogeneous dynamics is the theory of Lie groups. Given a lattice in a Lie groupG, and a Lie subgroupHofG, one can consider the action H;G= 5 by left multiplication, just as in the setting of moduli spaces we have con- sidered the action SL

2(R);Q1Tg=Modg:

One of the most powerful results in homogeneous dynamics isRatner's the- orem. It implies that ifHis generated by unipotent elements, then every orbit closureHxG= is aspecial submanifold| in fact, it has the formHx=JxG= for some Lie subgroupJwithHJG. A similar statement holds for invariant measures. Since SL

2(R) is generated by unipotent elements

(matrices such as (

1t0 1) and its transpose), one might hope for a version of

Ratner's theorem to hold in moduli spaces. This is what Mirzakhani's work conrms. Hodge theory versus geometry.For another perspective, recall that M gembeds into the moduli space of Abelian varietiesAg=Hg=Sp2g(Z), a locally symmetric space amenable to the methods of homogeneous dynamics. But the complex geodesics inMgbecome inhomogeneous when mapped into A g, so they cannot be analyzed by these methods. Mirzakhani's work shows that one can work eectively and directly withMgrather than withAg, by geometric analysis on Riemann surfaces themselves. Ramications: Billiards.The SL2(R) action onQ1Mgis also connected with the theory ofbilliards in polygons| an elementary branch of dynamics in which dicult problems abound. LetTCbe a connected polygon with angles inQ. The behavior of billiard paths inTis closely related to the behavior of the complex geodesic generated by a quadratic dierential (X;q) obtained by `unfolding' the table T. Indeed, the rst examples of complex geodesics such thatV=F(H) M gis an algebraic curve | i.e. the image of the complex geodesic is as small as possible | were constructed by Veech in his analysis of billiards in regular polygons. In this case the stabilizer of the corresponding quadratic dierential is a lattice SL(X;q)SL2(R), which serves as therenormaliza- tion groupfor the original billiard ow. The work of Mirzakhani has bearing on several open conjectures in the eld of billiard dynamics. For example, it provides progress on the open problem of showing that, for any tableT, there is an algebraic numberCT such that the numberN(T;L) of types ofprimitive, periodicbilliard paths 6 inTof lengthLsatises

N(T;L)CTL2area(T)

Eskin and Mirzakhani have shown that an asymptotic equation of this form holds after averaging overL, and thatCTcan assume only countably many values.

5 Dynamics of earthquakes

We conclude by discussing Mirzakhani's work on the earthquake ow, and a measurable bridge between the symplectic and holomorphic aspects ofMg. A classical construction of Fenchel and Nielsen associates to a simple closed geodesic

X2 Mgandt2Ra new Riemann surface

X t= twt (X)2 Mg; obtained by cuttingXopen along , twisting by lengthtto the right, and then regluing. The resultingtwist pathinMgis periodic; if has lengthL, thenXt+L=Xt. On the other hand, one can also twist alonglimitsof weighted simple geodesics, calledmeasured laminations. As shown by Thurston, the space of measured laminations forms a PL manifoldMLg=R6g6with a natural volume form, and the limiting twists, calledearthquakes, are dened for all time. Earthquakes are a natural feature of the symplectic geometry of moduli space. While they can be dened geometrically by fracturing and reglu- ingXalong the (possibly fractal) support of2 MLg, they also arise more conventionally as theHamilton owsassociated to the functionsY7! length(;Y).

The earthquake

ow lives on the bundleL1Mgof unit length laminations overMg. Mirzakhani has shown that, with respect to the natural measure onL1Mg:

Thurston's earthquake

ow is ergodic. Prior to this result, the dynamics of earthquakes seemed completely opaque. Not a single example of a dense earthquake path inMgwas known; we can now assert that almost every earthquake path is dense and uniformly distributed. 7 Bridging the symplectic/holomorphic divide.The proof of ergodicity of the earthquake ow uses a remarkable bridge between the symplectic and holomorphic sides of moduli space.

In more detail, recall that thehorocycle

owonQ1Mgis dened by the action of the 1-parameter groupN=f(1t0 1) :t2Rg SL2(R). Drawing on ideas from Thurston's work on stretch maps, Mirzakhani shows there is a measure{preserving map:L1Mg!Q1Mgwhich transports the earth- quake ow to the horocycle ow. In other words, we have a commutative diagram of the form earthquake owL1Mg +3Q 1Mg horocycle ow M gMg:

But the horocycle

ow onQ1Mgis well{known to be ergodic (this is a formal corollary of ergodicity of the geodesic ow [Zim, Thm. 2.4.2]), so the same is true for the earthquake ow [Mir5]. (It is an open problem to establish Ratner{type rigidity for these ows.) Summary.Mirzakhani's research has integrated, with great originality, a broad range of mathematical disciplines | including algebraic and sym- plectic geometry, low{dimensional topology, and random processes. Her breakthroughs have transformed our perspective on moduli spaces, and led the way to mathematical frontiers where striking developments are still un- folding.

Curtis T. McMullen, Cambridge, 2014.

References

[BQ] Y. Benoist and J.-F. Quint. Mesures stationnaires et fermes invari- ants des espaces homogenes.Annals of Math.174(2011), 1111{1162. [EM] A. Eskin and M. Mirzakhani. Invariant and stationary measures for the SL

2(R) action on moduli space.Preprint, 2/2014.

[EMM] A. Eskin, M. Mirzakhani, and A. Mohammadi. Isolation, equidis- tribution, and orbit closures for the SL

2(R) action on moduli space.

To appear, Annals of Mathematics.

8 [Fil] Simion Filip. Splitting mixed Hodge structures over ane invariant manifolds.Preprint, 11/2013. [Mc] C. McMullen. Dynamics of SL

2(R) over moduli space in genus two.

Annals of Math.165(2007), 397{456.

[Mir1] M. Mirzakhani. Simple geodesics and Weil{Petersson volumes of moduli spaces of bordered Riemann surfaces.Inv. math.167(2007),

179{222.

[Mir2] M. Mirzakhani. Weil{Petersson volumes and intersection theory on the moduli spaces of curves.J. Amer. Math. Soc.20(2007), 1{23. [Mir3] M. Mirzakhani. Growth of the number of simple closed geodesics on hyperbolic surfaces.Annals of Math.168(2008), 97{125. [Mir4] M. Mirzakhani. Growth of Weil{Petersson volumes and random hyperbolic surfaces of large genus.J. Dierential Geom.94(2013),

267{300.

[Mir5] M. Mirzakhani. Ergodic theory of the earthquake ow.Int. Math.

Res. Not.(2008), Art. ID rnm116, 39 pp.

[Mir6] M. Mirzakhani and P. Zograf. Towards large genus asymtotics of intersection numbers on moduli spaces of curves.Preprint, 12/2011. [Zim] R. Zimmer.Ergodic Theory and Semisimple Groups. Birkhauser, 1984.
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