We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a $q$-holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ superpolynomial of knots in $3$-space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincaré–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly $q$-holonomic.

We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space $T_{p}X$ of any given point $p$. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of $L$ at $p$. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the growth condition is “equivalent” to the moment polytope. As in the toric case, the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.

One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each ${\mathcal{T}}_g$, the Teichmüller space of a closed surface of genus $g\ge 2$. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential $\phi$. First, we construct a holomorphic embedding $\mathcal{E}:{\mathbb{H}}^k\to {\mathcal{T}}_{g,n}$ corresponding to $\phi$. The criterion says that the two metrics agree on this disk if and only if a certain function $\mathbf{\Phi }:\mathcal{E}\left({\mathbb{H}}^k\right)\to \mathbb{H}$ can be extended to a holomorphic function $\mathbf{\Phi }:{\mathcal{T}}_{g,n}\to \mathbb{H}$. We then show by explicit computation that this is not the case for quadratic differentials arising from $L$-shaped pillowcases.

We characterize groups quasi-isometric to a right-angled Artin group (RAAG) $G$ with finite outer automorphism group. In particular, all such groups admit a geometric action on a $CAT\left(0\right)$ cube complex that has an equivariant “fibering” over the Davis building of $G$. This characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi-isometric to certain RAAGs.