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We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map,…
When a liquid film lies on a non-wettable substrate, the configuration is unstable and the film then retracts from a solid substrate to form droplets. This phenomenon, known as dewetting, commonly leads to undesirable morphological changes.…
The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical…
Dense granular flows exhibit both surface deformation and secondary flows due to the presence of normal stress differences. Yet, a complete mathematical modelling of these two features is still lacking. This paper focuses on a steady…
Production of gravitational vacuum defects and their contribution to the energy density of our Universe are discussed. These topological microstructures (defects) could be produced in the result of creation of the Universe from "nothing"…
We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution…
The geometry of two-dimensional crystalline membranes dictates their mechanical, electronic and chemical properties. The local geometry of a surface is determined from the two invariants of the metric and the curvature tensors. Here we…
The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from…
Topology is an important determinant of the behavior of a great number of condensed-matter systems, but until recently has played a minor role in elasticity. We develop a theory for the deformations of a class of twisted non-Euclidean…
Determining the positions of lattice defects on elastic surfaces with Gaussian curvature is a non-trivial task of mechanical energy optimization, particularly for surfaces with boundaries. We introduce a simple way to predict the onset of…
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower…
We construct topological defects in two-dimensional classical lattice models and quantum chains. The defects satisfy local commutation relations guaranteeing that the partition function is independent of their path. These relations and…
Topological defects in active liquid crystals can be confined by introducing gradients of activity. Here, we examine the dynamical behavior of two defects confined by a sharp gradient of activity that separates an active circular region and…
Collective cell motions underlie structure formation during embryonic development. Tissues exhibit emergent multicellular characteristics such as jamming, rigidity transitions, and glassy dynamics, but there remain questions about how those…
We investigate topography-driven generation of defects in liquid crystals films coating frozen surfaces of spatially varying Gaussian curvature whose topology does not automatically require defects in the ground state. We study in…
We present a graph theory-based method to characterise flow defects and structural shifts in condensed matter. We explore the connection between dynamical properties, particularly the recently introduced concept of ''softness'', and…
Many scientific and engineering processes produce spatially unstructured data. However, most data-driven models require a feature matrix that enforces both a set number and order of features for each sample. They thus cannot be easily…
We study topological properties of large scale structure in a set of scale free N-body simulations using the genus and percolation curves as topological characteristics. Our results show that as gravitational clustering advances, the…
Recent findings suggest that shell protein distribution and morphology of bacterial microcompartments regulate the chemical fluxes facilitating reactions which dictate their biological function. We explore how the morphology and component…
A continuum model of fracture that describes, in principle, the propagation and interaction of arbitrary distributions of cracks and voids with evolving topology without a fracture criterion is developed. It involves a 'law of motion' for…