Related papers: On Topological Defects
When a symmetry gets spontaneously broken in a phase transition, topological defects are typically formed. The theoretical picture of how this happens in a breakdown of a global symmetry, the Kibble-Zurek mechanism, is well established and…
Topology in photonics comes in two distinct flavors: global and local. Global topology considers invariants that are obtained by integrating over the energy band, whereas local topology considers defects, typically vortices, in the…
The homotopical information hidden in a supersymmetric structure is revealed by considering deformations of a configuration manifold. This is in sharp contrast to the usual standpoints such as Connes' programme where a geometrical structure…
In order to apply nonstandard methods to questions of algebraic geometry we continue our investigation from "Enlargements of categories" (Theory Appl. Categ. 14 (2005), No. 16, 357--398) and show how important homotopical constructions…
Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the data into a…
We use a quantitative topological characterization of complex dynamics to measure geometric structures. This approach is used to analyze the weakly turbulent state of spiral defect chaos in experiments on Rayleigh-Benard convection.…
Evaluating the quality of reasoning traces from large language models remains understudied, labor-intensive, and unreliable: current practice relies on expert rubrics, manual annotation, and slow pairwise judgments. Automated efforts are…
Conformal defects -- extended objects in conformal field theories -- carry localised excitations inherited from symmetry currents, known as the displacements and tilts. They capture the linear response of the defect to deformations of its…
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…
This paper considers the problem of detecting topology variations in dynamical networks. We consider a network whose behavior can be represented via a linear dynamical system. The problem of interest is then that of finding conditions under…
This paper explores the topologies of caustics observed in instruments that employ charged particles, such as electron and ion microscopes. These geometrical figures are studied here using catastrophe theory. The application of this…
The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed as a computation moves forward, part-structures for shapes can be…
We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…
Revealing the structural features of a complex system from the observed collective dynamics is a fundamental problem in network science. In order to compute the various topological descriptors commonly used to characterize the structure of…
We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…
Topological defects are fundamental to the collective dynamics of non-equilibrium systems and in active matter, mediating spontaneous flows, dynamic self-organization, and emergent pattern formation. Here, we reveal critical states in…
While the strength of Topological Data Analysis has been explored in many studies on high dimensional numeric data, it is still a challenging task to apply it to text. As the primary goal in topological data analysis is to define and…
We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference…
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…
We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology…