We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold$M$induces a prolonged pseudogroup action on the jet spaces$J$^{$n$}of submanifolds of$M$. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on$J$^{$n$}persist under prolongation in the jet order$n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova–Galin equation is given.

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with$d$-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.