Counting simple closed geodesics (asymptotics) {#counting-geodesics}

(a) Lay explanation (2–3 sentences)
On a curved surface, a simple closed geodesic is a loop that goes around the surface without crossing itself (think of a rubber band wrapped once around a donut). Mirzakhani answered the question: how many distinct non-self-intersecting loops of length ≤ L exist? She proved that this number grows like a specific polynomial power of L and gave precise asymptotics that depend only on the topology of the surface. Annals of Mathematics

(b) Technical summary (1–2 paragraphs)
Mirzakhani connected the problem of counting simple closed geodesics on a fixed hyperbolic surface to global volume invariants of moduli space — the parameter space of all hyperbolic structures on a surface of given topology. She showed that the number sX(L)s_X(L)sX(L) of simple closed geodesics of length ≤ L on a hyperbolic surface $X$ has polynomial asymptotic growth as L→∞L \to \infty; more precisely sX(L)∼cX Ld, where the exponent d=6g−6+2n depends only on the genus $g$ and number of punctures/boundary components $n$, and the constant cXc_XcX is expressed in terms of Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces. Her approach used integration over moduli space, measured laminations, and careful cutting/gluing arguments to reduce counting on a fixed surface to volume computations on spaces of surfaces with geodesic boundary. The result unified geometric, probabilistic, and topological viewpoints and opened a path to explicit computations of growth constants. Annals of Mathematics+1

(c) Primary/original papers (links)

• Mirzakhani, M. Growth of the number of simple closed geodesics on hyperbolic surfaces. Annals of Mathematics 168 (2008), 97–125. DOI: 10.4007/annals.2008.168.97. Annals of Mathematics+1

(d) Recommended accessible write-ups

• Clay Mathematics Institute interview/profile and commentary that summarizes her counting results and methods. Clay Mathematics Institute

• Expository notes / figures showing the cutting-and-gluing idea (useful visual: Mirzakhani figures in the Annals paper). Semantic Scholar

Weil–Petersson volumes, recursion relations, and Witten’s conjecture {#weil-petersson}

(a) Lay explanation (2–3 sentences)
Mirzakhani found new formulas that compute the volumes (sizes) of moduli spaces measured with the Weil–Petersson metric. These recursion relations allowed her both to compute many volumes explicitly and to give a new geometric proof of a famous result known as Witten’s conjecture, which connects geometry of moduli space to intersection numbers and mathematical physics. arXiv+1

(b) Technical summary (1–2 paragraphs)
The Weil–Petersson metric endows the moduli space of hyperbolic surfaces with a natural symplectic volume form. Mirzakhani derived a recursive formula for these volumes by integrating identities (generalized McShane identities) over moduli space and by analyzing how hyperbolic surfaces decompose along simple closed geodesics. Her recursion expresses higher-genus Weil–Petersson volumes in terms of volumes for simpler surfaces (lower genus and/or fewer boundary components). Mirzakhani then used the volume recursion to give a new proof of Witten’s conjecture (originally proved by Kontsevich), which links intersection numbers of tautological classes on moduli space to the KdV integrable hierarchy. Her approach is geometric and fresh: instead of matrix models or combinatorial ribbon graphs it relies on hyperbolic geometry and the analytic properties of moduli space. This produced a striking bridge between hyperbolic geometry, algebraic geometry, and mathematical physics. arXiv+1

(c) Primary/original papers (links)

• Mirzakhani, M. Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167 (2007), 179–222. (See also the related Invent. Math. corrections/extended notes). arXiv+1

• Wolpert, S. overview of Mirzakhani’s volume recursion and its relation to Witten’s conjecture (useful technical commentary). arXiv

(d) Recommended accessible write-ups

• Survey/exposition: “Mirzakhani’s recursion formula on Weil–Petersson volumes” (overviews available on arXiv and in accessible lecture notes). arXiv+1

• IMU/Fields citation and ICM/AMS expositions summarizing the conceptual impact. International Mathematical Union+1

Dynamics on moduli spaces — the “Magic Wand” work {#magic-wand}

(a) Lay explanation (2–3 sentences)
Mirzakhani (with Alex Eskin and later with Amir Mohammadi) proved a powerful classification theorem about the behavior of orbits under the natural SL(2,ℝ) action on moduli spaces of translation surfaces. Informally called the “Magic Wand” theorem, it says the closure of any such orbit is a nice geometric object — a manifold-like subvariety — which allows many previously intractable dynamical problems to be solved. arXiv+1

(b) Technical summary (1–2 paragraphs)
The SL(2,ℝ) (or GL(2,ℝ)) action on the moduli space of abelian differentials (translation surfaces) encodes the dynamics of billiards in polygons and views translation surfaces as points in a high-dimensional parameter space. Before Mirzakhani and Eskin’s work, the structure of orbit closures for this action was largely mysterious. In a series of papers culminating in the measure-classification and orbit-closure theorems (first by Eskin–Mirzakhani and then Eskin–Mirzakhani–Mohammadi), they proved that ergodic, SL(2,ℝ)-invariant measures are algebraic (affine) and that orbit closures are essentially affine invariant submanifolds in period coordinates. Technically, the proofs adapted and extended methods from homogeneous dynamics (Ratner theory) to the highly non-homogeneous moduli spaces, combining measure rigidity, recurrence properties, and subtle algebro-geometric inputs. The results have sweeping consequences for dynamics, Teichmüller theory, and problems in mathematical physics (for example, wind-tree billiards and counting problems on flat surfaces). arXiv+2

(c) Primary/original papers (links)

• Eskin, A. & Mirzakhani, M., and Eskin–Mirzakhani–Mohammadi: Isolation, equidistribution, and orbit closures for the SL(2,ℝ) action on moduli space. Annals of Mathematics 182 (2015), 673–721. (arXiv:1305.3015). arXiv+1

(d) Recommended accessible write-ups

• Anton Zorich, “The Magic Wand Theorem of A. Eskin and M. Mirzakhani” — a readable popularization explaining motivations and applications (good for advanced undergrads and general readers). arXiv+1

• Lecture notes and expository articles collected in AMS/IMU materials and Zorich’s lecture series (useful for students wanting a guided route into the technical literature). webusers.imj-prg.fr+1

Further topics: random surfaces, large-genus asymptotics, and cross-disciplinary connections {#further-topics}

(a) Lay explanation (2–3 sentences)
Mirzakhani also studied the typical geometry of large or random surfaces and how Weil–Petersson volumes behave as the genus grows. These results let mathematicians say what a “random” high-genus surface typically looks like (e.g., about short geodesics and spectral properties), linking geometry to probability and statistical physics. arXiv

(b) Technical summary (1–2 paragraphs)
In later work Mirzakhani developed asymptotic estimates for Weil–Petersson volumes as the genus $g$ tends to infinity and used these estimates to analyze geometric properties of random hyperbolic surfaces sampled with respect to the Weil–Petersson measure. She proved results about expected values of geometric invariants (e.g., lengths of the shortest geodesics, Cheeger constants, and the distribution of simple closed geodesics) and established large-genus limits that inform probabilistic models of surfaces. These contributions connect to questions in spectral geometry, statistical mechanics, and quantum gravity where ensembles of random surfaces are used as models; analysts and physicists can use her asymptotics to control averages and fluctuations in these models. arXiv+1

(c) Primary/original papers (links)

• Mirzakhani, M. Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus. arXiv:1012.2167; published as J. Differential Geom. (2013/2014). arXiv+1

(d) Recommended accessible write-ups

• Survey articles and lecture notes on Weil–Petersson geometry and large-genus asymptotics (searchable on arXiv and J. Differential Geom. references). arXiv+1

Student help & reading path {#student-help}

• Annotated diagrams: include visuals for (i) simple vs. self-intersecting geodesics on surfaces of genus 0,1,2; (ii) cutting a surface along curves to produce bordered surfaces (illustrating Mirzakhani’s cutting-and-gluing); and (iii) a schematic of moduli space with Weil–Petersson volume regions. (Useful figure sources: Mirzakhani’s Annals paper and Zorich notes). Annals of Mathematics+1

• Prerequisites / what to study next:

  • Beginner: hyperbolic geometry (intro texts), topology of surfaces, and complex analysis refresher.

  • Intermediate: Teichmüller theory (intro lectures), basic ergodic theory/dynamical systems.

  • Advanced: Read Mirzakhani’s Invent. Math. (2007) and Annals (2008) papers, then Eskin–Mirzakhani–Mohammadi (arXiv:1305.3015). arXiv+2Annals of Mathematics+2

• Tutorial links / expositions: Zorich’s “Magic Wand” exposition and Clay/IMU profiles provide excellent narrative routes into the technical work without requiring immediate mastery of all prerequisites. arXiv+1