Related papers: Duality of moduli and quasiconformal mappings in m…
F. Gehring and W. Ziemer proved that the p-modulus of the family of paths connecting two continua is dual to the p^*-modulus of the corresponding family of separating hypersurfaces. In this paper we show that a similar result holds in…
We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincar\'e…
A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…
In the present paper, homeomorphisms in metric spaces which generalize quasiconformal mappings, are investigated. It is proved that, at some conditions on metric spaces and boundaries of corresponding domains, families of above mappings are…
The article is devoted to the study of mappings with finite distortion in metric spaces. Analogues of results relating to equicontinuity and normality of families of quasiregular mappings are obtained. It is proved that the indicated…
A paper is devoted to study of topological properties of some class of space mappings. It is showed that, sense preserving mappings $f:D\rightarrow \overline{{\Bbb R}^n}$ of a domain $D\subset{\Bbb R}^n,$ $n\geqslant 2,$ satisfying some…
This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then…
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincar\'e inequalities. Our…
For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light…
In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincar\'e inequality we show that quasiopen and…
We establish a sharp reciprocity inequality for modulus in compact metric spaces $X$ with finite Hausdorff measure. In particular, when $X$ is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M.…
We provide a general framework to study convergence properties of families of maps. For manifolds $M$ and $N$ where $M$ is equipped with a volume form $\mathcal{V}$ we consider families of maps in the collection $\{(\phi, B) : B \subset M,…
We discuss analogs of Faber's conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincare…
In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion…
We study geometric characterizations of the Poincar\'{e} inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of…
In this paper we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings $\Omega\rightarrow\mathbb{R}^m$ from some domain $\Omega\subset\mathbb{R}^n$ to $\mathbb{R}^m$, where $n\leq m$, which belong in…
We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a…
We develop a theory of quasimaps to a moduli space of sheaves $M$ on a surface $S$. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic…
We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there…
We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use…