A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation that can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be regarded as the Schwarzian derivative of a ${\mathbb{CP}}^1$-structure, to which one can naturally associate another measured geodesic lamination using grafting.

We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism $ML(S)\to Q(X)$ for each conformal structure $X$ on a compact surface $S$. We show that these maps are nearly identical, differing by a multiplicative factor of $-2$ and an error term of lower order than the maps themselves (which we bound explicitly).

As an application, we show that the Schwarzian derivative of a ${\mathbb{CP}}^1$-structure with Fuchsian holonomy is close to a $2\pi$-integral Jenkins-Strebel differential. We also study two compactifications of the space of ${\mathbb{CP}}^1$-structures, one of which uses the Schwarzian derivative and another of which uses grafting coordinates. The natural map between these two compactifications is shown to extend to the boundary of each fiber over Teichmüller space, and we describe that extension

Let $\Omega$ be some domain in the hyperbolic space ${\mathbb{H}}^n$ (with $n\ge 2$), and let $S_1$ be a geodesic ball that has the same first Dirichlet eigenvalue as $\Omega$. We prove the Payne-Pólya-Weinberger (PPW) conjecture for ${\mathbb{H}}^n$, namely, that the second Dirichlet eigenvalue on $\Omega$ is smaller than or equal to the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius

This article investigates the relationship between the topology of hyperbolizable $3$-manifolds $M$ with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to $M$. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold $\mathrm{DM}$, and this inequality is sharp. This article proves that the inequality is, in fact, sharp in every pleating variety of AH$(M)$

We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety $X=G/P$. Using this formula, we give an effective algorithm to compute the $3$-point, genus zero, equivariant Gromov-Witten invariants on $X$, which are the structure constants of its equivariant quantum cohomology algebra

Let $F$ be a global field of positive characteristic, and let $D$ be a maximal order in a central division algebra over $F$. We give a precise properness criterion for moduli spaces of $D$-shtukas with general modifications, which includes the statement that for some division algebras, the space of ordinary $D$-shtukas is not proper. The proof is based on a result on the smoothness of a suitable stack of complete homomorphisms of $D$-modules of rank $1$