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Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an…
We construct a family of non-PCF dendrites $K$ in a plane, such that in each of them all subarcs have the same Hausdorff dimension $s$, while the set of $s$-dimensional Hausdorff measures of subarcs connecting the given point and a…
Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up…
We prove linear upper and lower bounds for the Hausdorff dimension set of minimal interval exchange transformations with flips (in particular without periodic points), and a linear lower bound for the Hausdorff dimension of the set of…
This is a "software upgrade" to a paper originally published in 1976, with cleaner statements and improved proofs. The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is…
This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $Rc_{M^n_i}\geq -(n-1)$ as…
We revisit the global structure of the extended Kerr spacetime and of a broader class of Kerr-type spacetimes possessing ring singularities. By working with the elementary analytic extension (the union of the interior and exterior regions…
H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced the Spanier group of a based space $(X,x)$ which is denoted by $\psp$. By a Spanier space we mean a space $X$ such that $\psp=\pi_1(X,x)$, for every $x\in X$.…
Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the…
For a bounded closed convex set $K$, in this note, we study the FPP for $\alpha$-H\"older nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^\alpha$ for all $x, y\in K$, $\alpha\in (0,1)$. First, we note…
A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with…
In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of…
A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard…
We show that non-collapsed $\text{RCD}(K,3)$ spaces without boundary are orbifolds whose topological singularities are locally finite and locally homeomorphic to cones over $\mathbb{RP}^2$, and that the topology of such spaces is stable…
The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via…
A central objective of topological data analysis is to identify topologically significant features in data represented as a finite point cloud. We consider the setting where the ambient space of the point sample is a compact Riemannian…
When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a…
The Theorem on Invariance of Domain due to L.E.J. Brouwer states that one connected, compact (Hausdorff) m-dimensional manifold embedded into another actually realizes a homeomorphism. This fundamental result is relevant to Functional…
We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do…
Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor…