Let $X$ be a compact toric Kähler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya, Oh, Ohta, and Ono defined open Gromov–Witten (GW) invariants of $X$ as virtual counts of holomorphic disks with Lagrangian boundary condition $L$. We prove a formula that equates such open GW invariants with closed GW invariants of certain $X$-bundles over ${\mathbb{P}}^1$ used by Seidel and McDuff earlier to construct Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disk potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya, Oh, Ohta, and Ono.

The moduli space of Riemann surfaces of genus $g\ge 2$ is (up to a finite étale cover) a complex manifold, so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$. This expectation is verified in low genus and is supported by Harer’s computation of its de Rham cohomological dimension and by vanishing results in the tautological intersection ring. In this article, we prove that such a dimension is at most $2g-2$. We also prove an analogous bound for the moduli space of Riemann surfaces with marked points. The key step is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most $g$. In order to do that, we produce an exhaustion function whose complex Hessian has controlled index: the construction of such a function relies on some basic geometric properties of translation surfaces.

Suppose that $\mathcal{X}$ is either the mapping class group equipped with the word metric or Teichmüller space equipped with either the Teichmüller metric or the Weil–Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that for any large box in ${\mathbb{R}}^n$ there is a standard model of a flat in $\mathcal{X}$ such that the quasi-Lipschitz image of a large sub-box is near the standard flat. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.

We say that permutations ${\pi }_1,\dots ,{\pi }_r\in {\mathcal{S}}_n$ invariably generate ${\mathcal{S}}_n$ if, no matter how one chooses conjugates $\pi {\text{'}}_1,\dots ,\pi {\text{'}}_r$ of these permutations, the $\pi {\text{'}}_1,\dots ,\pi {\text{'}}_r$ permutations generate ${\mathcal{S}}_n$. We show that if ${\pi }_1,{\pi }_2$, and ${\pi }_3$ are chosen randomly from ${\mathcal{S}}_n$, then, with probability tending to $1$ as $n\to \infty$, they do not invariably generate ${\mathcal{S}}_n$. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate ${\mathcal{S}}_n$ with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.