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In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…
Dynamic graph theory is a novel, growing area that deals with graphs that change over time and is of great utility in modelling modern wireless, mobile and dynamic environments. As a graph evolves, possibly arbitrarily, it is challenging to…
Framework materials and their deformations provide a compelling relation between materials science and algebraic geometry. Physical distance constraints within the material transform into polynomial constraints, making algebraic geometry…
In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…
We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe…
Over the past two decades, complex network theory provided the ideal framework for investigating the intimate relationships between the topological properties characterizing the wiring of connections among a system's unitary components and…
We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in $\mathbb{R}^d$ in the cases when the lattice is either fixed, fully flexible, or…
I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random…
Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or…
We present necessary and sufficient conditions for the generic rigidity of body-bar frameworks on the three-dimensional fixed torus. These frameworks correspond to infinite periodic body-bar frameworks in $\mathbb{R}^3$ with a fixed…
Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the…
This paper introduces the notion of weak rigidity to characterize a framework by pairwise inner products of inter-agent displacements. Compared to distance-based rigidity, weak rigidity requires fewer constrained edges in the graph to…
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental…
This paper investigates self-maps T from X to X which satisfy a distance constraint in a metric space which mixed point-dependent non-expansive properties, or in particular contractive ones, and potentially expansive properties related to…
We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in…
This paper develops a mirror symmetry theory of Spencer cohomology within the geometric framework of constrained systems on principal bundles, revealing deep symmetric structures in constraint geometry. Based on compatible pairs…
The graph layouts used for complex network studies have been mainly been developed to improve visualization. If we interpret the layouts in metric spaces such as Euclidean ones, however, the embedded spatial information can be a valuable…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
In this paper the notion of the rigid frame of reference within special relativity is analysed. Three definitions of rigidity are formulated. By using several examples of non-inertial frames, it is shown that these definitions are not…