Related papers: Approximately j-dimensional Koch type sets are pot…

We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite…

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the…

We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly…

We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally…

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…

We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n variables over the field of Laurent series with coefficients from a finite field is maximal. This is a analogue of Schmidt's…

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays…

We study isolated quotient singularities by finite and linearly reductive group schemes (lrq singularities for short) and show that they satisfy many, but not all, of the known properties of finite quotient singularities in characteristic…

We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least…

We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension.

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…

We study area-minimizing hypersurfaces in singular ambient manifolds whose tangent cones have nonnegative scalar curvature on their regular parts. We prove that the singular set of the hypersurface has codimension at least 3 in our…

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.

We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is…

In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Kra jewski diagrams [5].…

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1,…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the…

We investigate the correspondence between three theories of deformations of rational surface singularities: de Jong and van Straten's picture deformations, Koll\'ar's P-resolutions, and Pinkham's smoothings of negative weights. We provide…