Related papers: Transitional geometry
The present paper gives an account and quantifies the change in topology induced by small and type II geometric transitions, by introducing the notion of the \emph{homological type} of a geometric transition. The obtained results agree…
We summarize an explicit construction of a duality cycle for geometric transitions in type II and heterotic theories. We emphasize that the manifolds with torsion constructed with this duality cycle are crucial for understanding different…
In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real…
General hierarchical lattices of coupled maps are considered as dynamical systems. These models may describe many processes occurring in heterogeneous media with tree-like structures. The transition to turbulence via spatiotemporal…
A corner in a map is an edge-vertex-edge triple consisting of two distinct edges incident to the same vertex. A corneration is a set of corners that covers every arc of the map exactly once. Cornerations in a dart-transitive map generalize…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal…
Fluid interfaces, such as soap films, liquid droplets or lipid membranes, are known to give rise to several special geometries, whose complexity and beauty continue to fascinate us, as observers of the natural world, and challenge us as…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications.Periodic structures such as solid crystalline materials or textiles were previously classified in…
Many physical situations are characterized by interfaces with a non trivial shape so that relevant geometric features, such as interfacial area, curvature or unit normal vector, can be used as main indicators of the topology of the…
This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential…
In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and…
We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
The geometry of antisymmetric fields with nontrivial transitions over a base manifold is described in terms of exact sequences of cohomology groups. This formulation leads naturally to the appearance of nontrivial topological charges…
The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are…
The tenfold classification provides a powerful framework for organizing topological phases of matter based on symmetry and spatial dimension. However, it does not offer a systematic method for transitioning between classes or engineering…
We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary 'critical' region containing a…
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an…