🍩 Database of Original & NonTheoretical Uses of Topology
(found 5 matches in 0.0016s)


Finding Universal Structures in Quantum ManyBody Dynamics via Persistent Homology (2020)
Daniel Spitz, Jürgen Berges, Markus K. Oberthaler, Anna WienhardAbstract
Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider simulated data of a twodimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium universal phenomena. A possible explanation of the underlying processes is provided in terms of mixing wave turbulence and vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum manybody dynamics in terms of robust topological structures beyond standard field theoretic techniques. 
Abnormal Hole Detection in Brain Connectivity by Kernel Density of Persistence Diagram and Hodge Laplacian (2018)
Hyekyoung Lee, Moo K. Chung, Hyejin Kang, Hongyoon Choi, Yu Kyeong Kim, Dong Soo Lee 
Felix: A Topology Based Framework for Visual Exploration of Cosmic Filaments (2016)
Nithin Shivshankar, Pratyush Pranav, Vijay Natarajan, Rien van de Weygaert, E. G. Patrick Bos, Steven RiederAbstract
The largescale structure of the universe is comprised of virialized bloblike clusters, linear filaments, sheetlike walls and huge near empty threedimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, walls and voids are relatively well separated, when compared to filaments, which span a relatively larger range. The large scale filamentary network thus forms an intricate part of the cosmic web. In this paper, we describe Felix, a topology based framework for visual exploration of filaments in the cosmic web. The filamentary structure is represented by the ascending manifold geometry of the 2saddles in the MorseSmale complex of the density field. We generate a hierarchy of MorseSmale complexes and query for filaments based on the density ranges at the end points of the filaments. The query is processed efficiently over the entire hierarchical MorseSmale complex, allowing for interactive visualization. We apply Felix to computer simulations based on the heuristic Voronoi kinematic model and the standard \$\Lambda\$CDM cosmology, and demonstrate its usefulness through two case studies. First, we extract cosmic filaments within and across cluster like regions in Voronoi kinematic simulation datasets. We demonstrate that we produce similar results to existing structure finders. Filaments that form the spine of the cosmic web, which exist in high density regions in the current epoch, are isolated using Felix. Also, filaments present in voidlike regions are isolated and visualized. These filamentary structures are often over shadowed by higher density range filaments and are not easily characterizable and extractable using other filament extraction methodologies. 
Cosmic Web Reconstruction Through Density Ridges: Method and Algorithm (2015)
YenChi Chen, Shirley Ho, Peter E. Freeman, Christopher R. Genovese, Larry WassermanAbstract
The detection and characterization of filamentary structures in the cosmic web allows cosmologists to constrain parameters that dictate the evolution of the Universe. While many filament estimators have been proposed, they generally lack estimates of uncertainty, reducing their inferential power. In this paper, we demonstrate how one may apply the subspace constrained mean shift (SCMS) algorithm (Ozertem & Erdogmus 2011; Genovese et al. 2014) to uncover filamentary structure in galaxydata. The SCMS algorithm is a gradient ascent method that models filaments as density ridges, onedimensional smooth curves that trace highdensity regions within the point cloud. We also demonstrate how augmenting the SCMS algorithm with bootstrapbased methods of uncertainty estimation allows one to place uncertainty bands around putative filaments. We apply the SCMS first to the data set generated from the Voronoi model. The density ridges show strong agreement with the filaments from Voronoi method. We then apply the SCMS method data sets sampled from a P3M Nbody simulation, with galaxy number densities consistent with SDSS and WFIRSTAFTA, and to LOWZ and CMASS data from the Baryon Oscillation Spectroscopic Survey (BOSS). To further assess the efficacy of SCMS, we compare the relative locations of BOSS filaments with galaxy clusters in the redMaPPer catalogue, and find that redMaPPer clusters are significantly closer (with pvalues \textless10−9) to SCMSdetected filaments than to randomly selected galaxies.