Related papers: Algorithms and topological invariants for dynamic …

Traditionally, robots are regarded as universal motion generation machines. They are designed mainly by kinematics considerations while the desired dynamics is imposed by strong actuators and high-rate control loops. As an alternative, one…

Topology is a powerful tool for categorizing magnetization textures by defining a topological index in both two-dimensional (2D) systems, such as thin films or curved surfaces, and in 3D bulk systems. In the emerging field of 3D…

A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of…

This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components,…

Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…

We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we show that the (generalized) third…

Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…

We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe''…

We propose and investigate a new algorithm for quantifying the topological properties of cosmological density fluctuations. We first motivate this algorithm by drawing a formal distinction between two definitions of relevant topological…

The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are…

This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with…

Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…

This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as the background notes to a series of seminars given at NTNU and subsequently at Sheffield. I am…

We show how to define a dynamical topological invariant for general one-dimensional topological systems after a quantum quench. Focusing on two-band topological insulators, we demonstrate that the reduced momentum-time manifold can be…

One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor…

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…

We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary two-dimensional faceted surfaces $z = h(x,y)$. Our method uses a three-component facet/edge/junction storage model,…

Using digital topology approach, we construct digital models of closed surfaces as the intersection graphs of LCL covers of the surfaces. It is proved that digital models of closed surfaces are digital 2-dimensional surfaces preserving the…

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful…