🍩 Database of Original & NonTheoretical Uses of Topology
(found 3 matches in 0.001763s)


Loops Abound in the Cosmic Microwave Background: A \$4\sigma\$ Anomaly on SuperHorizon Scales (2021)
Pratyush PranavAbstract
We present a topological analysis of the temperature fluctuation maps from the \emph\Planck 2020\ Data release 4 (DR4) based on the \texttt\NPIPE\ data processing pipeline. For comparison, we also present the topological characteristics of the maps from \emph\Planck 2018\ Data release 3 (DR3). We perform our analysis in terms of the homology characteristics of the maps, invoking relative homology to account for analysis in the presence of masks. We perform our analysis for a range of smoothing scales spanning sub and superhorizon scales corresponding to \$FWHM = 5', 10', 20', 40', 80', 160', 320', 640'\$. Our main result indicates a significantly anomalous behavior of the loops in the observed maps compared to simulations that are modeled as isotopic and homogeneous Gaussian random fields. Specifically, we observe a \$4\sigma\$ deviation between the observation and simulations in the number of loops at \$FWHM = 320'\$ and \$FWHM = 640'\$, corresponding to superhorizon scales of \$5\$ degrees and larger. In addition, we also notice a mildly significant deviation at \$2\sigma\$ for all the topological descriptors for almost all the scales analyzed. Our results show a consistency across different data releases, and therefore, the anomalous behavior deserves a careful consideration regarding its origin and ramifications. Disregarding the unlikely source of the anomaly being instrumental systematics, the origin of the anomaly may be genuinely astrophysical  perhaps due to a yet unresolved foreground, or truly primordial in nature. Given the nature of the topological descriptors, that potentially encodes information of all orders, nonGaussianities, of either primordial or latetype nature, may be potential candidates. Alternate possibilities include the Universe admitting a nontrivial global topology, including effects induced by largescale topological defects. 
Analysis of Kolmogorov Flow and Rayleigh–Bénard Convection Using Persistent Homology (2016)
Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin MischaikowAbstract
We use persistent homology to build a quantitative understanding of large complex systems that are driven farfromequilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatiotemporal behavior.