To know a surface, integrate over all the ways of cutting it. The volumes came out polynomial, their coefficients were Witten’s intersection numbers, and the simple closed geodesics turned out to be exponentially rare.
Her 2004 Harvard thesis did three things that had been separate problems, and did them with one idea. It computed the Weil–Petersson volumes of moduli space. It gave a new proof of Witten's conjecture, which Kontsevich had proved in 1992 by an entirely different route. And it counted simple closed geodesics on a hyperbolic surface. The Fields Medal came in 2014, the first to a woman. She died in 2017, forty years old. CITED
The idea is a recursion, and the shape of it is why this chapter belongs in a corpus about unfolding: to know a surface, integrate over all the ways of cutting it. A pair of pants comes off along a simple closed curve, and what is left is a smaller surface of the same kind. The volume of the whole is an integral of the volumes of the pieces over where the cut can be made.
Moduli space $\mathcal{M}_{g,n}$ — hyperbolic surfaces of genus $g$ with $n$ boundary geodesics of lengths $L_1, \ldots, L_n$ — carries the Weil–Petersson symplectic form, and so a volume. Mirzakhani's theorem is that the volume is a polynomial:
V0,3 = 1
V1,1(b) = (b² + 4π²) / 48
V0,4(b1…b4) = 2π² + (b1²+b2²+b3²+b4²)/2
V1,2(b1,b2) = (4π²+b1²+b2²)(12π²+b1²+b2²) / 192
V2,0 = 43π6 / 2160 = 19.138766353582
A polynomial in $b_1^2, \ldots, b_n^2$ of degree $3g - 3 + n$, and its coefficients are intersection numbers on moduli space — the $\psi$-classes of Witten's conjecture. That is the second theorem falling out of the first: compute the volumes and you have computed the intersection numbers, so Witten's conjecture follows.
These polynomials are not independent of each other, and the relation between them is the cleanest check a page like this can carry. Norman Do's identity says that evaluating the derivative at the imaginary boundary length $2\pi i$ drops you one step down the recursion:
$$\frac{\partial V_{g,n+1}}{\partial L_{n+1}}\big(L, 2\pi i\big) = 2\pi i\,(2g-2+n)\,V_{g,n}(L)$$
A boundary of imaginary length is not a boundary. The identity is a statement about the polynomials, and it holds:
V0,4 → V0,3 residual 0
V1,2 → V1,1 residual 1.3 × 10−13
V2,1 → V2,0 recovers 43π6/2160 to 4.5 × 10−13 COMPUTED
$V_{1,1}$ is printed in the literature both as $(b^2+4\pi^2)/24$ and as $(b^2+4\pi^2)/48$, the factor of two being an orbifold convention that different authors absorb in different places. The identity decides it. With $/48$ the residual is $1.3\times10^{-13}$; with $/24$ it is $5.55$ — not a rounding, a wrong answer. SHOWN
A corpus that quotes a constant from a paper without an internal check has no way to notice it picked up the other convention. This is what one looks like.
The third result is the one to state last because it is the one that sounds wrong.
On a closed hyperbolic surface, the number of closed geodesics of length at most $L$ grows like $e^L / L$. Exponentially. That is Huber's theorem and it has been known since 1959. Mirzakhani proved that the number of simple closed geodesics — those that do not cross themselves — grows like
$$s_X(L) \sim c_X \cdot L^{\,6g-6+2n}$$
Polynomial. And $6g-6+2n$ is exactly the dimension of the moduli space the volumes live on — the constant $c_X$ is a Weil–Petersson volume.
The corpus keeps meeting the same structure: a quantity that is not a matter of degree. A conic is not gradually more elliptical; a top is not almost integrable; a plate does not almost have a degenerate mode. Here it is again and in a new form — simple is not a small perturbation of not simple, and the gap between them is the whole difference between a polynomial and an exponential.
And the method is the chain's own: cut, evaluate the pieces, integrate over where the cut could have gone. Kovalevskaya classified by the singularity; Mirzakhani classifies by the decomposition.
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | the pair-of-pants decomposition — a surface cut along simple curves | compression: the constraint that issues the pieces |
| K | integration over where the cut can be made | accumulation toward the recursion's fixed point |
| F | $b = 2\pi i$ — an imaginary boundary, where the polynomial drops a level | the fold SHOWN |
| U | $s_X(L) \sim c_X L^{6g-6+2n}$ — the branch that is polynomially thin | unfolding onto the simple locus CITED |
Every number on this page is produced by book7/ch-mirzakhani-verify.py. It records in its own closing block what it establishes and what it does not.
M. Mirzakhani, “Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces”, Invent. Math. 167, 2007, 179–222.
M. Mirzakhani, “Weil–Petersson volumes and intersection theory on the moduli space of curves”, J. Amer. Math. Soc. 20, 2007, 1–23.
M. Mirzakhani, “Growth of the number of simple closed geodesics on hyperbolic surfaces”, Ann. of Math. 168, 2008, 97–125.
N. Do, “Moduli spaces of hyperbolic surfaces and their Weil–Petersson volumes”, arXiv:1103.4674, 2011 — the identity used above.
M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function”, Comm. Math. Phys. 147, 1992.
H. Huber, “Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II”, Math. Ann. 142, 1961 — the exponential count of all closed geodesics.