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In this paper we construct an uncountable union of line segments $T$ which has full intersection with the sets $(\{ 0\} \times [0, 1]) \cup (\{ 1\} \times [0, 1])\subset\mathbb{R}^2$ but has null two-dimensional measure. Further results are…

It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete…

Let $X$ be a closed $3$-manifold, $\mathcal{M}_{R>0}$ the space of metrics on $X$ with positive scalar curvature, and $\text{Diff}(X)$ the group of diffeomorphisms of $X$. Marques proves the fundamental result that $\mathcal{M}_{R>0}/…

In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations…

In this article the algebra and the basis of corresponding analysis in 4-dimensional spaces are constructed, in pseudoeuclidean with signature (1, -1, -1, -1) and pseudo-Riemannian corresponding to the real space-time. In both cases the…

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…

In this paper, we define the non-commuting graph associated to a Lie algebra L and obtain some basic graph properties such as connectivity, diameter, girth, Hamiltonian and Eulerian. Moreover, planarity, outer planarity and isomorphism…

We introduce a new definition of nonpositive curvature in metric spaces and study its relationship to the existing notions of nonpositive curvature in comparison geometry. The main feature of our definition is that it applies to all metric…

Let $\mathfrak{M}(\Sigma)$ be an open and connected subset of the space of hyperbolic metrics on a closed orientable surface, and $\mathfrak{M}(M)$ an open and connected subset of the space of metrics on an orientable manifold of dimension…

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of…

We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of…

A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.

(This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge…

We present the first utterly self-supervised network for dense correspondence mapping between non-isometric shapes. The task of alignment in non-Euclidean domains is one of the most fundamental and crucial problems in computer vision. As 3D…

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any…

We prove that the space $\mathcal{H}_\infty$ of framed infinite volume hyperbolic $3$-manifolds is connected but not path connected. Two proofs of connectivity of this space, which is equipped with the geometric topology, are given, each…

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is…

The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some…

We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are $C^0$ conjugate then the spaces are…