# A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

## Abstract

We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of nonexperts.

## 1. Introduction

This survey aims to be a tour through Maryam Mirzakhani’s remarkable work on Riemann surfaces, dynamics, and geometry. The star characters, all across mathematics and physics as well as in this survey, are the moduli spaces of Riemann surfaces.

Sections 2 through 10 all relate to Mirzakhani’s study of the size of these moduli spaces, as measured by the Weil–Petersson symplectic form (see the left side of Figure 1.1). Goldman has shown that many related moduli spaces also have a Weil–Petersson symplectic form, so this can be viewed as part of a broader story Reference Gol84. Even more important than the broader story, Mirzakhani’s study unlocks applications to the topology of random surfaces of large genus, and even geodesics on individual hyperbolic surfaces. ,

Sections 11 to 19 reflect the philosophy that despite being a totally inhomogeneous object, enjoys many of the dynamical properties of nicer spaces and even some of the dynamical miracles characteristic of homogeneous spaces (see the right side of Figure ,1.1). The dynamics of group actions in turn clarify the geometry of and produce otherwise unattainable counting results.

Our goal is not to provide a comprehensive reference, but rather to highlight some of the most beautiful and easily understood ideas from the roughly 20 papers that constitute Mirzakhani’s work in this area. Very roughly speaking, we devote comparable time to each paper or closely related group of papers. This means in particular that we cannot proportionately discuss the longest paper Reference EM18, but on this topic the reader may see the surveys Reference Zor15Reference Wri16Reference Qui16. We include some open problems, and hope that we have succeeded in conveying the thriving legacy of Mirzakhani’s research.

We invite the reader to discover for themselves Mirzakhani’s five papers on combinatorics Reference MM95Reference Mir96Reference Mir98Reference MV15Reference MV17, where the author is not qualified to guide the tour.

We also omit comprehensive citations to work preceding Mirzakhani, suggesting instead that the reader may get off the tour bus at any time to find more details and context in the references and reboard later or to revisit the tour locations at a later date. Where possible, we give references to expository sources, which will be more useful to the learner than the originals. The reader who consults the references will be rewarded with views of the vast tapestry of important and beautiful work that Mirzakhani builds upon, something we can only offer tiny glimpses of here.

We hope that a second year graduate student who has previously encountered the definitions of hyperbolic space, Riemann surface, line bundle, symplectic manifold, etc., will be able to read and appreciate the survey, choosing not to be distracted by the occasional remark aimed at the experts.

Other surveys on Mirzakhani’s work include Reference Wol10Reference Wol13Reference Do13Reference McM14Reference Zor14Reference Zor15Reference Hua16Reference Wri16Reference Qui16Reference Mar17Reference Wri18. See also the issue of the *Notices of the AMS* that was devoted to Mirzakhani Reference Not18.

## 2. Preliminaries on Teichmüller theory

We begin with the beautiful and basic results that underlie most of Mirzakhani’s work.

### 2.1. Hyperbolic geometry and complex analysis

All surfaces are assumed to be orientable and connected. Any simply connected surface with a complete Riemannian metric of constant curvature is isometric to the upper half-plane

endowed with the hyperbolic metric

Perhaps the most important miracle of low-dimensional geometry is that the group of orientation-preserving isometries of hyperbolic space is equal to the group of biholomorphisms of (Both are equal to the group . of Möbius transformations that stabilize the upper half-plane.)

Every oriented complete hyperbolic surface has universal cover and the deck group acts on , via orientation-preserving isometries. Since these isometries are also biholomorphisms, this endows with the structure of a Riemann surface, namely an atlas of charts to whose transition functions are biholomorphisms.

Conversely, every Riemann surface that is not simply connected, not and not a torus has universal cover , and the deck group acts on , via biholomorphisms. Since these biholomorphisms are also isometries, this endows with a complete hyperbolic metric.

### 2.2. Cusps, geodesics, and collars

Suppose is a complete hyperbolic surface. Each subset of isometric to

is called a cusp. Each cusp has infinite diameter and finite volume. Distinct cusps are disjoint, and if has finite area, then the complement of the cusps is compact.

Each cusp is biholomorphic to a punctured disc via the exponential map. If has finite area, then is biholomorphic to a compact Riemann surface minus a finite set of punctures, and the punctures are in bijection with the cusps.

Any closed curve on not homotopic to a point or a loop around a cusp is isotopic to a unique closed geodesic. Unless otherwise stated, all closed curves we consider will be of this type. A closed geodesic is called simple if it does not intersect itself.

Gauss–Bonnet gives that any closed hyperbolic surface of genus has area This is the first indication that a hyperbolic surface cannot be “small”. Moreover, the Collar Lemma gives that any closed geodesic of length less than a universal constant is simple, and every short simple closed geodesic must be surrounded by a large embedded annulus known as its collar. As the length of the simple closed geodesic goes to zero, the size of its collar goes to infinity. See Figure .2.1.

### 2.3. Building a surface out of pants

A significant amount of this survey will concern hyperbolic surfaces with boundary. We will always assume that any surface with boundary that we consider can be isometrically embedded in a complete surface so that the boundary consists of a finite union of closed geodesics.

A hyperbolic sphere with three boundary components is known as a pair of pants, or simply as a pants. A fundamental fact gives that, for any three numbers there is a unique pants with these three boundary lengths. Each , may also be allowed to be zero, in which case a pants, now somewhat degenerate, has a cusp instead of a boundary component.

One of the simplest ways to build a closed hyperbolic surface is by gluing together pants. For example, given two pants with the same boundary lengths, we may glue together the corresponding boundaries to obtain a closed genus surface, as in Figure 2.2. In fact, the corresponding boundaries can be glued using different isometries from the circle to the circle, giving infinitely many genus hyperbolic surfaces. More complicated surfaces can be obtained by gluing together more pants.

### 2.4. Teichmüller space and moduli space

We define moduli space formally as the set of equivalence classes of oriented genus hyperbolic surfaces with cusps labeled by where two surfaces are considered equivalent if they are isometric via an orientation-preserving isometry that respects the labels of the cusps. Equivalently, , can be defined as the set of equivalence classes of genus Riemann surfaces with punctures labeled by where two surfaces are considered equivalent if they are biholomorphic via a biholomorphism that respects the labels of the punctures. ,

We will follow the almost universal abuse of referring to a point in as a hyperbolic or Riemann surface, leaving out the notational bookkeeping of the equivalence class.

Teichmüller space is defined to be the set

of points in which as indicated we think of as hyperbolic or Riemann surfaces, equipped with a homotopy class , of orientation-preserving homeomorphisms from a fixed oriented topological surface of genus with punctures. The homotopy class is called a marking, and one says that parametrizes marked hyperbolic or Riemann surfaces.

Let denote the group of orientation-preserving homeomorphisms of that do not permute the punctures. This group acts on by precomposition with the marking. The subgroup of homeomorphisms isotopic to the identity acts trivially, so the quotient

acts on This countable group is called the mapping class group, and .

Given we can similarly define , to be the Teichmüller space of oriented genus hyperbolic surfaces with boundary components of length and we define , to be the corresponding moduli space. Here is replaced with a genus surface with boundary circles, and we define to be the orientation-preserving homeomorphisms that do not permute the boundary components.

It is sometimes convenient to allow in the definitions above, in which case the corresponding boundary is replaced by a cusp. For example, using this convention .

### 2.5. Classification of simple closed curves

Let and be two different simple closed curves on that are nonseparating, in that cutting either curve does not disconnect the surface. In this case the result of cutting either or is homeomorphic to a genus surface with two boundary curves, and hence they are homeomorphic to each other. This homeomorphism can be modified to give rise to a homeomorphism of that takes to In particular, we conclude that there is some . such that where , denotes the homotopy class of .

Next suppose that is a separating simple closed curve. In this case, has two components, one of which is a surface of genus with one boundary component, and the other of which is a surface of genus with one boundary component. If is another separating curve, then there is some such that if and only if the set arising from is the same as for .

In summary, there is a single mapping class group orbit of nonseparating simple closed curves on and there are , mapping class group orbits of separating simple closed curves.

### 2.6. The twist flow

Let be a simple closed curve on that is not a loop around a cusp. We now introduce the twist flow on Teichmüller space as follows. It may be conceptually helpful to start by assuming is small and positive.

For each point we can consider the geodesic representative of , Cut this geodesic to obtain a surface with two geodesic boundary components of equal length. Both of these components inherit an orientation from the surface (see Figure .2.3), and we will call the positive direction “left”.

Reglue the two components by the original identification, composed with a rotation by so that if two points , on the two boundary components were originally identified, now is identified with the point to the left of and vice versa. See Figure ,2.4.

If we use the notation this regluing defines , The marking . is more subtle, and we will omit its definition. Here it will suffice to accept that, despite the fact that the twist path is injective, so , if and only if .

### 2.7. Fenchel–Nielsen coordinates

Fix a pants decomposition of This is a collection of disjoint simple closed curves, such that cutting these curves gives a collection of topological pants. It turns out any such collection has . curves, and we denote these curves If . some of the pants will be degenerate, in that they will have a puncture instead of boundary circle. See Figure ,2.5.

Given a marked hyperbolic surface we can consider the curves , on Let . denote the length of the geodesic homotopic to For short, we write . to denote .

Each hyperbolic surface in can be obtained by gluing together the pants with the correct boundary lengths in the correct combinatorial pattern, but additional parameters are required in the construction to keep track of how the boundary curves are glued together.

#### Theorem 2.1 (Fenchel and Nielsen).

There are functions

such that the map defined by

is a homeomorphism, and so that for each and all ,

and all the other coordinates of and are the same.

The twist parameters and the length parameters are called Fenchel–Nielsen coordinates for Teichmüller space.

One can show that the mapping class group acts properly discontinuously on In particular, the stabilizer of each point is finite. The quotient . is thus an orbifold, which is similar to a manifold except that some points have neighborhoods homeomorphic to a neighborhood of the origin in quotiented by a finite group action.

Fenchel–Nielsen coordinates work similarly for Note there are no twist or length parameters for the geodesic boundary curves, since they have fixed lengths . and are not glued to anything.

### 2.8. The Weil–Petersson symplectic structure

Fix a choice of Fenchel– Nielsen coordinates, and define

to be the standard symplectic form in these coordinates on .

Wolpert proved that this symplectic form is invariant under the action of the mapping class group. Hence it descends to a symplectic form on or .Footnote^{1} We define on moduli space so that its pullback to Teichmüller space is the standard symplectic form defined above. In other words, it is the standard symplectic form in local Fenchel–Nielsen coordinates on moduli space. This is sometimes called the “topologist’s definition”, and it ignores that may be considered as a stack, which is the algebro-geometric version of an orbifold. To reconcile with the algebro-geometric perspective without using stacks, one could also define the Weil–Petersson volume form on as the local pushforward of the Weil–Petersson volume form on Teichmüller space. The definitions give volume forms that are equal except for and where the reconciled volume form is half of the topologist’s. Every surface in those two moduli spaces has an involution symmetry. , Independently of this issue, it is also common to include a separate factor of in the definition of for all and Reference Wol07, Section 5.^{✖}

Wolpert also showed that in the case of this symplectic form is twice the one arising from the Weil–Petersson Kähler structure on , This result is sometimes called Wolpert’s Magic Formula, since .

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the definition of the Weil–Petersson Kähler structure, although very natural, gives no hint of a relationship to Fenchel–Nielsen coordinates, and

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it is surprising that the two-form obtained from a pants decomposition, does not depend on which pants decomposition is used. ,

The associated Weil–Petersson volume form, which is the standard volume form in local Fenchel–Nielsen coordinates, is the most natural known notion of volume on moduli space. The Weil–Peterson volume of each moduli space is finite.

### 2.9. References

More details can be found in the books Reference FM12, Chapters 10, 12 and, for the Weil–Petersson symplectic structure, Reference Wol10, Chapter 3.

## 3. The volume of

Mirzakhani discovered an elegant new computation of the volume of We reproduce this computation, which is highlighted in the introduction to .Reference Mir07b, since it was perhaps the first seed for her thesis. The starting point is the remarkable identity

of McShane Reference McS98, which gives that a certain sum involving the lengths of all simple closed geodesics on is independent of In Section .5 we will explain where this identity comes from.

Let denote the infinite cover of parametrizing pairs where , and is a simple closed geodesic on Mirzakhani’s computation begins .

This *unfolding* is justified because the fibers of the map are precisely the set of simple closed geodesics, which is the set being summed over in McShane’s identity.

Given a point we may cut , along to get a hyperbolic sphere with two boundary components of length and one cusp. It is helpful to view this as a degenerate pants (see Figure ,3.1), where one of the boundary curves has been replaced with a cusp.

For any there is a unique such pants, so the point , is uniquely determined by and a twist parameter which controls how the two length boundary curves of the degenerate pants are glued together to give .

Using this parametrization

and Wolpert’s formula Mirzakhani concludes ,

## 4. Integrating geometric functions over moduli space

In this section we give a key result from Reference Mir07b that gives a procedure for integrating certain functions over moduli space, generalizing the unfolding step in the previous section.

### 4.1. A special case

Let be a simple nonseparating closed curve on a surface of genus For a continuous function . we define a function , by

Here the sum is over the mapping class group orbit of the homotopy class of Soon we will generalize this notation, but for the moment the subscript may seem strange: since there is only one mapping class group orbit of nonseparating simple closed curves, for the moment . does not depend on .

Recall that is the moduli space of genus hyperbolic surfaces with two labeled boundary geodesics of length We will give an outline of Mirzakhani’s proof that .

Define to be the set of pairs where , and is a geodesic with The fibers of the map .

correspond exactly to the set that is summed over in the definition of and in fact ,

Cutting along almost determines a point of except that the two boundary geodesics are not labeled. However, since there are two choices of labeling, we can say that there is a two-to-one map ,

where the map glues together the two boundary components of with a twist determined by Wolpert’s Magic Formula determines the pullback of the Weil–Petersson measure, and we get .

The case of is special, because every has an involution exchanging the two boundary components. Because of this involution, one cannot distinguish between the two choices of labeling the two boundary components, and the map that was two-to-one is now one-to-one. Thus, the same formula holds in genus with the factor of removed.

### 4.2. The general case

A simple multi-curve, often just multi-curve for short, is a finite sum of disjoint simple closed curves with positive real weights, none of whose components are loops around a cusp. If is a multi-curve, its length is defined by

We define for multi-curves in the same way as above and note that in fact only depends on the mapping class group orbit of .

Suppose that cutting the geodesic representative of decomposes into connected components and that ,

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has genus ,

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has boundary components, and

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the lengths of the boundary components of are given by .

If we set then all the entries of each , are from (if they correspond to the original boundary of or ) (if they correspond to the new boundary created by cutting ).

#### Theorem 4.1 (Mirzakhani’s Integration Formula).

For any multi-curve ,

where is an explicit constant.Footnote^{2} Slightly different values of the constant have been recorded in different places in the literature. We believe the correct constant is where is the number of such that bounds a torus with no other boundary components and not containing any other component of , is the stabilizer of the *weighted* multi-curve , is the subgroup of the mapping class group that fixes and its orientation, and is the kernel of the action of the mapping class group on Teichmüller space. (Note that is trivial except in the case when is or in which case it has size , and is central.) Given two subgroups we write , for the subgroup they generate. The number arises as the degree of a measurable map from to the space of pairs where , and is a multi-geodesic with This natural map factors through the space . of (ordered) tuples where , and the , are disjoint oriented geodesics with The degree of . is and the degree of is the remaining index factor.^{✖}

## 5. Generalizing McShane’s identity

The starting point for Mirzakhani’s volume computations is the following result proven in Reference Mir07b. It relies on two explicit functions whose exact definitions are omitted here.

### Theorem 5.1.

For any hyperbolic surface with geodesic boundary circles of lengths ,

where the first sum is over all pairs of closed geodesics bounding a pants with and the second sum is over all simple closed geodesics , bounding a pants with and (see Figure 5.1

By studying the asymptotics of this formula when some it is possible to derive a related formula in the case when the boundary , is replaced with a cusp. In the case when all are replaced with cusps, Mirzakhani recovers identities due to McShane Reference McS98, including the identity given in Section 3. Thus, Mirzakhani refers to Theorem 5.1 as the generalized McShane identities.

### Idea of the proof of Theorem 5.1.

Let be the set of points on from which the unique geodesic ray beginning at and perpendicular to the boundary continues forever without intersecting itself or hitting the boundary. By a result of Birman and Series, has measure 0, reflecting the fact that most geodesic rays intersect themselves Reference BS85.

It is easy to see that is open and, hence, is a countable union of disjoint intervals .

Mirzakhani shows that the geodesics and both spiral toward either a simple closed curve or a boundary component other than as in Figure ,5.2. There is a unique pants with geodesic boundary containing and .

Each pants is associated with one or more intervals and the sum of the lengths of these intervals characterizes the functions , and Having computed these functions, the identity is equivalent to . .

■### 5.1. References

See Reference BT16 for a survey of related identities that have been proven since Mirzakhani’s work. Of special note is that there is a related identity for closed surfaces Reference LT14.

## 6. Computation of volumes using McShane identities

We now outline how Mirzakhani used her integration formula and the generalized McShane identities to recursively compute the Weil–Petersson volumes of Reference Mir07b. Except in the case of , and these volumes were unknown before Mirzakhani’s work. ,

As in Section 3, we begin by integrating the generalized McShane identity to obtain

In fact, and are nicer functions than and so Mirzakhani considers the , derivative of this identity.

Let us consider just the sum

over all simple closed geodesics which bound a pants with and The set of such . is one mapping class group orbit, so we may apply Mirzakhani’s Integration Formula to get

Note that the surfaces in are smaller than those in in that they have one less pants in a pants decomposition.

The sum over pairs is similar, but more complicated because the set of multi-curves that arise consists of a finite but possibly large number of mapping class group orbits.

This produces an expression for as a finite sum of integrals involving volumes of smaller moduli spaces. Mirzakhani was able to compute these integrals, allowing her to compute recursively. These computations imply in particular that is a polynomial in the whose coefficients are positive rational multiples of powers of which we will reprove from a different point of view in the next section. ,

### 6.1. References

Mirzakhani’s recursions are concisely presented in terms of the coefficients of the polynomials in Reference Mir13, Section 3.1 and Reference MZ15, Section 2.Footnote^{3} Due to a different convention, in these papers the volumes of and are half what our conventions give. In the second line of Reference Mir13, Section 3.1, there is a typo that should be corrected as In .Reference MZ15, Equation 2.13, there is a typo that should be corrected as .^{✖} Using these recursions to compute the volume polynomials is rather slow, because of the combinatorial explosion in high genus of the number of different moduli spaces that arise from recursively cutting along geodesics. Zograf has given a faster algorithm Reference Zog08.

Mirzakhani’s results do not directly allow for the computation of However these volumes were previously known via intersection theory. They can also be recovered via the remarkable formula .

proven in Reference DN09.

It would be interesting to recompute using Mirzakhani’s strategy and the identity for closed surfaces in Reference BT16.

Mirzakhani’s recursions fit into the framework of *topological recursions* Reference Eyn14.

## 7. Computation of volumes using symplectic reduction

We now give Mirzakhani’s second point of view on Weil–Petersson volumes, from Reference Mir07c.

### 7.1. A larger moduli space

Consider the moduli space of genus Riemann surfaces with geodesic boundary circles with a marked point on each boundary circle. This moduli space has dimension greater than that of because the length of each boundary circle can vary, and the marked point on each boundary circle can vary. ,

admits a version of local Fenchel–Nielsen coordinates, where in addition to the usual Fenchel–Nielsen coordinates there is a length parameter for each boundary circle and a parameter that keeps track of the position the each marked point on each boundary circle. The parameters keeping track of the marked points are thought of as twist parameters. The space also has a Weil–Peterson form which is still described by Wolpert’s Magic Formula, meaning that it is the standard symplectic form in any system of local Fenchel–Nielsen coordinates. ,

Consider now the function defined by

The reason for this definition will become clear in the next subsection.

Let and consider the , action that moves the marked points along the boundary circles. Each level set is invariant under the action, and the quotient is the space with fixed boundary lengths and no marked points on the boundary. That is,

### 7.2. Symplectic reduction

We now review a version of the Duistermaat–Heckman Theorem in symplectic geometry, as it applies to .

The symplectic form on provides a nondegenerate bilinear form on each tangent space to This gives an identification between the tangent space and its dual, and hence between vector fields and one-forms. .

Any function on determines a one-form and hence also a vector field defined via this duality. This duality is recorded symbolically as

The flow in the vector field is called the Hamiltonian flow of and , is called the Hamiltonian function.

To begin, take Let . denote the length of the unique boundary circle, and let denote the twist coordinate giving the position of the marked point. The action on discussed above is simply given by where , and hence is generated by the vector field , Wolpert’s Magic Formula gives .

If then , so the , action is Hamiltonian with Hamiltonian function .

Now take Then the . coordinate th action on which moves the position of the marked point on the , boundary circle, is Hamiltonian with Hamiltonian function th Using a natural way to combine different Hamiltonian functions into a single function called the moment map, one says that the . action on is Hamiltonian with moment map given above.

For every the space , is the manifold parametrizing surfaces in with a marked point on each boundary circle, and as we have discussed

The fact that this quotient is a symplectic manifold is an instance of a general phenomenon called symplectic reduction.

Return to the case Then . is a principal circle bundle over (A principal . bundle is a bundle with an action of that is simply transitive on fibers.) Let us call this circle bundle .

Generalizing this to we see that , is a product of circle bundles Here . can be defined as the spaces of surfaces in together with just a single marked point on the boundary circle (and no marked points on any of the other boundary circles). th

#### Theorem 7.1 (Duistermaat–Heckman Theorem).

For any fixed and for small enough, there exists a diffeomorphism

such that

where is the first Chern class of the circle bundle over Here, on the left-hand side . refers to the Weil–Petersson form on and on the right-hand side it refers to the Weil–Petersson form on , .

The reader unfamiliar with Chern classes may in fact take this theorem to be the definition for this survey; we will not use any other properties of Chern classes.

Part of Theorem 7.1 is powered by a relative of the Darboux Theorem. The Darboux Theorem states that a neighborhood of any point in a symplectic manifold is symplectomorphic to the simplest thing you could guess it to be, namely a neighbourhood in a vector space with the standard symplectic form.

Here, a neighborhood of is topologically and one can create a guess for what the symplectic form , might look like on using , on and curvature forms for the circle bundles. The Equivariant Coisotropic Reduction Theorem, which is the relative of the Darboux Theorem we referred to, says that this guess is in fact symplectomorphic to a neighborhood of in .

### 7.3. Computations of volumes

From Theorem 7.1, is a polynomial in a small neighborhood of any and hence is globally a polynomial. By considering , we get ,

Note that Theorem 7.1 only directly gives this for close to but since the volume is a polynomial, it must in fact be true for all , Note also that . denotes the Weil–Petersson symplectic form on in the first integral and on in the second integral.

Taking a limit as Mirzakhani obtains ,

Here can be defined as the circle of points on a horocycle of size 1 about the cusp. th

One can first interpret this integral in terms of the differential forms representing produced by the proof of Theorem 7.1. These differential forms, and the circle bundles extend continuously to a natural compactification , constructed by Deligne and Mumford, and so one can and typically does replace with as the space to be integrated over. This allows a more topological interpretation of the integral as the pairing of a class in with the fundamental class of .

In summary, we have the following.

#### Theorem 7.2.

The volume of is a polynomial

whose coefficients are rational multiples of integrals of powers of the Chern classes and the Weil–Petersson symplectic form. Here , and , .

We will discuss a number of interesting results and open problems about these polynomials in Section 10.

### 7.4. References

For more on the material relating to symplectic reduction, see, for example, Reference CdS01, Chapters 22, 23, 30.2.

The work of Mirzakhani suggests some similarities between moduli spaces of Riemann surfaces and spaces of representations of surface groups into compact Lie groups modulo conjugacy. These spaces of representations are also known as character varieties or moduli spaces of stable bundles. Mirzakhani points out the connection between her techniques and those used previously by Witten and others in this context Reference Wit92. See the citations in Mirzakhani’s papers and the survey Reference Jef05 for more details.

## 8. Witten’s conjecture

We now describe Mirzakhani’s proof of Witten’s conjecture Reference Mir07c. This brings us to the algebro-geometric perspective on the coefficients from Theorem 7.2.

### 8.1. Intersection theory

Let and denote the extensions of and from to .

The class is typically denoted and it is much studied. One often defines , as the first Chern class of a line bundle called the relative cotangent bundle at the marked point. th

Wolpert showed that the cohomology class is equal to where , is the much-studied first kappa class Reference Wol83. As a result, all of the coefficients from Theorem 7.2 are in .

The compactification is an algebraic variety and a smooth orbifold, and the classes and can be thought of as dual to (equivalence classes of) divisors, which are linear combinations of subvarieties of complex codimension 1. The intersection of two such classes, if transverse, has complex codimension 2, and similarly the intersection of them, if transverse, is a finite collection of points. Integrals of a product of of the and classes, which up to factors are exactly the coefficients count the number of points of intersection. Thus, they are called intersection numbers. See the book ,Reference LZ04, Chapter 4.6 for some example computations using this point of view.

It is hard for the uninitiated to fathom how much useful information such intersection numbers can contain, so we pause to give just a few points of motivation.

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They are central to the study of the geometry of .

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By Theorem 7.1 they determine Weil–Petersson volumes. Later we will see that these volumes can be used to understand the geometry of Weil–Petersson random surfaces.

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They appear in theoretical physics Reference Wit91.

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They determine counts of combinatorial objects called ribbon graphs Reference Kon92.

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They determine Hurwitz numbers, which count certain branched coverings of the sphere or, equivalently, factorizations of permutations into transpositions Reference ELSV01.

### 8.2. A generating function for intersection numbers

Make the notational convention

Unless this is defined to be zero. Note that ,“ should be considered as a single mathematical symbol, and the order of the ” doesn’t matter. ’s

Define the generating function for top intersection products in genus by

where the sum is over all nonnegative sequences such that One can then form the generating function .

which arises as a partition function in two-dimensional quantum gravity. Note that is a generating function in infinitely many variables: keeps track of the genus, and keeps track of the number of powers of psi classes. th

Witten’s conjecture is equivalent to the fact that is annihilated by a sequence of differential operators

satisfying the Virasoro relations

To give an idea of the complexity of these operators, we record the formula for :

The equations encode recursions among the intersection numbers, which appear as the constant terms in Mirzakhani’s volume polynomials. These recursions allow for the computation of all intersection numbers of psi classes. Mirzakhani showed that these recursions follow from her recursive formulas for the volume polynomials, thus giving a new proof of Witten’s conjecture.

### 8.3. A brief history

Witten’s conjecture was published in 1991, motivated by physical intuition that two different models for two-dimensional quantum gravity should be equivalent Reference Wit91. Kontsevich published a proof in 1992, using a combinatorial model for arising from Strebel differentials, ribbon graphs, and random matrices Reference Kon92. This work was central in his 1998 Fields Medal citation.

It was not until 2007 that Mirzakhani’s proof was published, and around the same time other proofs appeared. Later, Do related Mirzakhani’s and Kontsevich’s proofs, recovering Kontsevich’s formula for the number of ribbon graphs by considering asymptotics of the Weil–Petersson volume polynomials, using that a rescaled Riemann surface with very large geodesic boundary looks like a graph Reference Do10.

## 9. Counting simple closed geodesics

Let be a complete hyperbolic surface, and let be the number of primitive closed geodesics of length at most on “Primitive” means simply that the geodesic does not traverse the same path multiple times. The famous Prime Number Theorem for Geodesics gives the asymptotic .

as (The factor of . disappears if one counts primitive oriented geodesics, since there are two orientations on each closed geodesic.) Amazingly, this does not depend on which surface we choose or even the genus of .

In Reference Mir08b, Mirzakhani proved that the number of closed geodesics of length at most on that do not intersect themselves is asymptotic to a constant depending on times That this asymptotic is polynomial rather than exponential reflects the extreme unlikeliness that a random closed geodesic is simple, in the same spirit as the result of Birman and Series mentioned in Section .5.

In fact Mirzakhani proved a more general result. For any rational multi-curve she considered ,

In other words, counts the number of closed multi-geodesics on of length less than that are “of the same topological type” as .

The set of simple closed curves forms finitely many mapping class group orbits. So by summing finitely many of these functions one gets the corresponding count for all simple closed curves. ,

### Theorem 9.1.

For any rational multi-curve ,

where , is a proper, continuous function with a simple geometric definition, and .

Eskin, Mirzakhani, and Mohammadi have recently given a new proof of Theorem 9.1 that gives an error term, which we will discuss in Section 18, and Erlandsson and Souto have also given a new proof in Reference ES19. Here we outline the original proof, after first commenting on one application.

### 9.1. Relative frequencies

Consider, for example, the case of closed genus 2 surfaces, just to be concrete. The set of simple closed curves consists of two mapping class groups orbits: the orbit of a nonseparating curve and the orbit of a curve that separates the surface into two genus subsurfaces.

The fact that the limit in Theorem 9.1 is the product of a function of and a function of has the following consequence: A very long simple closed curve on chosen at random among all such curves, has probability about ,

of being separating. Remarkably, this probability is computable and does not depend on !

Even more remarkably, recent discoveries prove that the same probabilities appear in discrete problems about surfaces assembled out of finitely many unit squares Reference DGZZReference AH19.

### 9.2. The space of measured foliations

The space of rational multi-curves admits a natural completion called the space of measured foliations. Later we will delve into measured foliations, but here we only need a few properties of this space.

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is homeomorphic to

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does not carry a natural linear structure. The most superficial indication of this is that any closed curve gives a point of but there is no ,“ in ” because multi-curves are defined to have positive coefficients. There is however a natural action of , on which on multi-curves simply multiplies the coefficients by , .

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has a natural piecewise linear integral structure, that is, an atlas of charts to whose transition functions are piecewise in .

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Define the integral points of as the set of points mapping to under the charts. Define the rational points similarly. Then integral (resp., rational) points of parametrize homotopy classes of integral (resp., rational) multi-curves on the surface.

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Any defines a continuous length function

whose restriction to multi-curves gives the hyperbolic length of the geodesic representative of the multi-curve on In particular, . for all .

### 9.3. Warmup

How many points of have length at most It is equivalent to ask about the number of points of ? contained in the unit ball.

Recall that the Lebesgue measure can be defined as the limit as of

where denotes the point mass at Hence the number of points of . contained in the unit ball is asymptotic to times the Lebesgue measure of the unit ball.

### 9.4. The Thurston measure

Let us start with the easy question of asymptotics for the number

of all integral multi-curves of length at most Using . we observe that

It is now useful to define the “unit ball”

and the measures

With these definitions,

As in our warm up, the measures converge to a natural Lebesgue class measure on This measure . is called the Thurston measure and is Lebesgue measure in the charts mentioned above. If we define we get the asymptotic ,

### 9.5. The proof

Mirzakhani’s approach to Theorem 9.1 similarly defines measures

As above, to prove Theorem 9.1, it suffices to show the convergence of measures

Using the Banach–Alaoglu Theorem, it is not hard to show that there are subsequences such that converges to some measure which might a priori depend on which subsequence we pick. To prove Theorem ,9.1 it suffices to show that, no matter which such subsequence we use, we have

By definition, since the mapping class group orbit of , is a subset of Since . converges to the Thurston measure, we get that .

Since is mapping class group invariant, the same is true for A result of Masur in ergodic theory, which we will discuss in Section .13, gives that any mapping class group invariant measure on that is absolutely continuous to must be a multiple of Reference Mas85.

So for some At this point in the argument, as far as we know, . could depend on the subsequence of .

Unraveling the definitions, we have that

for any Writing .

we recognize the type of function that Mirzakhani’s Integration Formula applies to. By integrating the left-hand side of Equation 9.5.1 over moduli space, Mirzakhani is able to prove that as desired. On the one hand, the integral of is On the other hand, the limit . of the integral of is easily expressed in terms of the leading order term in one of Mirzakhani’s volume polynomials.

### 9.6. Open problems

We will return to counting later, but for now we mention the following.

#### Problem 9.2.

Prove an analogue of Theorem 9.1 for nonorientable hyperbolic surfaces.

An example is known already with asymptotics with nonintegral Reference Mag17. See Reference Gen17 for a more precise conjecture, as well as a number of related open problems and an analogy between moduli spaces of nonoriented hyperbolic surfaces and infinite volume geometrically finite hyperbolic manifolds.

## 10. Random surfaces of large genus

Given a random graph with many vertices, what is the chance that it contains a short loop? Is a random graph easy to cut in two? What properties can be expected of the graph Laplacian? -regular

Mirzakhani considered analogues of these well-studied questions for Weil–Petersson random Riemann surfaces Reference Mir13Reference MZ15Reference MP17 and devoted her 2010 talk at the International Congress of Mathematicians to this topic Reference Mir10.

In this section we discuss this work. We will leave out the background on graphs, but many readers will wish to keep in mind the comparison between a random graph, with -regular fixed and a large number of vertices, and a random surface with large genus.

### 10.1. Understanding the volume polynomials

We begin with the constant term of the polynomial which is the volume , of Improving on previous results of Mirzakhani and others, Mirzakhani and Zograf proved the following .Reference Mir13Reference MZ15.

#### Theorem 10.1.

There exists a universal constant such that for any fixed , is asymptotic to

as .

This largely verified a previous conjecture of Zograf, except that his prediction that is still open Reference Zog08. Mirzakhani and Zograf also gave a more detailed asymptotic expansion. The proof uses the recursions satisfied by discussed in Section 6.

Previous results gave asymptotics as for fixed Reference MZ00. See Reference Mir13, Section 1.4 for open questions concerning asymptotics as both and go to infinity.

Also by studying recursions, Mirzakhani proved results in Reference Mir13 that imply

Mirzakhani and Petri showed this bound is asymptotically sharp for fixed and bounded as Reference MP17, Proposition 3.1. The proof of the inequality actually gives a bound with replaced with one of its Taylor polynomials.

### 10.2. An example

To illustrate Mirzakhani’s techniques, we will give an upper bound for the probability that a random surface in has a nonseparating simple closed geodesic of length at most some small .

We begin by studying the average over of the number of simple, nonseparating geodesics of length at most on If . is a simple nonseparating curve, we can express this as

where is the characteristic function of the interval Mirzakhani’s Integration Formula gives that this is equal to a constant times .

Since is small, inequality Equation 10.1.1 gives that is approximately equal to the constant term of the volume polynomial, so the average is approximately a constant times

The asymptotics in Theorem 10.1 imply that converges to 1 as so we get that the average number of simple, nonseparating geodesics of length at most , is asymptotic, as to a constant times , In particular, this implies that the probability that a random surface in . has such a geodesic is bounded above by a constant times .

A similar lower bound is possible by giving upper bounds for the average number of pairs of nonseparating simple closed curves.

### 10.3. Results

Here is an overview of results from Reference Mir13, which concern random as .

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The probability that has a geodesic of length at most is bounded above and below by a constant times .

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The probability that has a separating geodesic of length at most goes to 0.

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The probability that has Cheeger constant less than goes to 0.

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The probability that the first eigenvalue of the Laplacian, is less than , goes to 0.

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The probability that the diameter of is greater than goes to 0.

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The probability that has an embedded ball of radius at least goes to 1.

The first two results are proven using the techniques in the example. The Cheeger constant is defined as

where the infimum is over all smooth multi-curves that cut into two subsurfaces Mirzakhani defines the geodesic Cheeger constant . to be the same quantity where is required to be a geodesic multi-curve, so obviously She proves that .

and is then able to study using the techniques in the example. The result on follows from the Cheeger inequality .

We conclude with a special case of the main result of Mirzakhani and Petri Reference MP17.

#### Theorem 10.2.

For any the number of primitive closed geodesics of length in , viewed as a random variable on , converges to a Poisson distribution as , .

What is fascinating about this result of Mirzakhani and Petri is that it concerns all primitive closed geodesics, not just the simple ones. The proof uses that a geodesic of length at most a constant on a surface of very large genus is contained in a subsurface of bounded genus and with a bounded number of boundary components (depending on The boundary of that subsurface is a simple multi-curve ). associated to By showing that, as . most , do not have a separating multi-curve of bounded length, they are able to show that on most most primitive geodesics are simple and, hence, use the techniques illustrated in the example.

### 10.4. Open problems

For some problems, we list an easier version followed by a harder version.

#### Problem 10.3.

Does there exist a sequence of Riemann surfaces