🍩 Database of Original & NonTheoretical Uses of Topology
(found 13 matches in 0.003709s)


Confinement in NonAbelian Lattice Gauge Theory via Persistent Homology (2022)
Daniel Spitz, Julian M. Urban, Jan M. PawlowskiAbstract
We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instantondyons. Signatures of wellseparated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semiclassical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gaugeinvariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date. 
Topology Identifies Emerging Adaptive Mutations in SARSCoV2 (2021)
Michael Bleher, Lukas Hahn, Juan Angel PatinoGalindo, Mathieu Carriere, Ulrich Bauer, Raul Rabadan, Andreas OttAbstract
The COVID19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARSCoV2. Ideally, one would like to quickly identify new mutations that could confer adaptive advantages (e.g. higher infectivity or immune evasion) by leveraging the large number of genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. However, the large number of currently available genomes precludes the efficient use of phylogenybased techniques. Here, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations based on persistent homology. It identifies convergent events merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the GISAID dataset as of February 2021, comprising 303,651 highquality SARSCoV2 isolates collected since the beginning of the pandemic. We find that topologically salient mutations on the receptorbinding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape, and for many adaptive mutations the topological signal precedes an increase in prevalence. We show that our method effectively identifies emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we extract geotemporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants. 
Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics (2020)
Emerson G. Escolar, Yasuaki Hiraoka, Mitsuru Igami, Yasin OzcanAbstract
Where do ﬁrms innovate? Mapping their locations in technological space is diﬃcult, because it is high dimensional and unstructured. We address this issue by using a method in computational topology called the Mapper algorithm, which combines local clustering with global reconstruction. We apply this method to a panel of 333 major ﬁrms’ patent portfolios in 1976–2005 across 430 technological areas. Results suggest the Mapper graph captures salient patterns in ﬁrms’ patenting histories, and our measures of their uniqueness (the length of “ﬂares”) are correlated with ﬁrms’ ﬁnancial performances in a statistically and economically signiﬁcant manner. We then compare this approach with a widely used clustering method by Jaﬀe (1989) to highlight additional ﬁndings. 
Topological Data Analysis on Simple English Wikipedia Articles (2020)
Matthew Wright, Xiaojun ZhengAbstract
Singleparameter persistent homology, a key tool in topological data analysis, has been widely applied to data problems, with statistical techniques that quantify the significance of the results. In contrast, statistical techniques for twoparameter persistence, while highly desirable for realworld applications, have scarcely been considered. We present three statistical approaches for comparing geometric data using twoparameter persistent homology, and we demonstrate the applicability of these approaches on highdimensional pointcloud data obtained from Simple English Wikipedia articles. These approaches rely on the Hilbert function, matching distance, and barcodes obtained from twoparameter persistence modules computed from the pointcloud data. We demonstrate the applicability of our methods by distinguishing certain subsets of the Wikipedia data, and by comparison with random data. Results include insights into the construction of null distributions and stability of our methods with respect to noisy data. Our statistical methods are broadly applicable for analysis of geometric data indexed by a realvalued parameter. 
Topological Data Analysis of SingleCell HiC Contact Maps (2020)
Mathieu Carrière, Raúl RabadánAbstract
Due to recent breakthroughs in highthroughput sequencing, it is now possible to use chromosome conformation capture (CCC) to understand the three dimensional conformation of DNA at the whole genome level, and to characterize it with the socalled contact maps. This is very useful since many biological processes are correlated with DNA folding, such as DNA transcription. However, the methods for the analysis of such conformations are still lacking mathematical guarantees and statistical power. To handle this issue, we propose to use the Mapper, which is a standard tool of Topological Data Analysis (TDA) that allows one to efficiently encode the inherent continuity and topology of underlying biological processes in data, in the form of a graph with various features such as branches and loops. In this article, we show how recent statistical techniques developed in TDA for the Mapper algorithm can be extended and leveraged to formally define and statistically quantify the presence of topological structures coming from biological phenomena, such as the cell cyle, in datasets of CCC contact maps. 
The Growing Topology of the C. Elegans Connectome (2020)
Alec Helm, Ann S. Blevins, Danielle S. BassettAbstract
Probing the developing neural circuitry in Caenorhabditis elegans has enhanced our understanding of nervous systems. The C. elegans connectome, like those of other species, is characterized by a rich club of densely connected neurons embedded within a smallworld architecture. This organization of neuronal connections, captured by quantitative network statistics, provides insight into the system's capacity to perform integrative computations. Yet these network measures are limited in their ability to detect weakly connected motifs, such as topological cavities, that may support the systems capacity to perform segregated computations. We address this limitation by using persistent homology to track the evolution of topological cavities in the growing C. elegans connectome throughout neural development, and assess the degree to which the growing connectomes topology is resistant to biological noise. We show that the developing connectome topology is both relatively robust to changes in neuron birth times and not captured by similar growth models. Additionally, we quantify the consequence of a neurons specific birth time and ask if this metric tracks other biological properties of neurons. Our results suggest that the connectomes growing topology is a robust feature of the developing connectome that is distinct from other network properties, and that the growing topology is particularly sensitive to the exact birth times of a small set of predominantly motor neurons. By utilizing novel measurements that track biological features, we anticipate that our study will be helpful in the construction of more accurate models of neuronal development in C. elegans 
TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification (2019)
YuMin Chung, ChuanShen Hu, Austin Lawson, Clifford Smyth 
Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background (2019)
Pratyush Pranav, Robert J. Adler, Thomas Buchert, Herbert Edelsbrunner, Bernard J. T. Jones, Armin Schwartzman, Hubert Wagner, Rien van de WeygaertAbstract
We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the \textlessi\textgreaterPlanck\textlessi/\textgreater satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multiscale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊\textlesssup\textgreater2\textlesssup/\textgreater is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric \textlessi\textgreaterχ\textlessi/\textgreater\textlesssup\textgreater2\textlesssup/\textgreatertest shows differences between observations and simulations, yielding \textlessi\textgreaterp\textlessi/\textgreatervalues at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3\textlessi\textgreaterσ\textlessi/\textgreater sporadically at these scales. The highest observed deviation between the observations and simulations for \textlessi\textgreaterb\textlessi/\textgreater\textlesssub\textgreater0\textlesssub/\textgreater and \textlessi\textgreaterb\textlessi/\textgreater\textlesssub\textgreater1\textlesssub/\textgreater is approximately between 3\textlessi\textgreaterσ\textlessi/\textgreater and 4\textlessi\textgreaterσ\textlessi/\textgreater at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by \textlessi\textgreaterPlanck\textlessi/\textgreater’s predecessor, the \textlessi\textgreaterWilkinson\textlessi/\textgreater Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of \textlessi\textgreaterPlanck\textlessi/\textgreater and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Nonparametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on powerspectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to latetime effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the superhorizon scales involved, may motivate the study of primordial nonGaussianity. Alternative scenarios worth exploring may be models with nontrivial topology, including topological defect models. 
Topological Approaches to Skin Disease Image Analysis (2018)
YuMin Chung, ChuanShen Hu, Austin Lawson, Clifford Smyth 
Topology and Geometry for Small Sample Sizes: An Application to Research on the Profoundly Gifted (2018)
Colleen Molloy Farrelly 
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers (2017)
Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J. T. Jones, Mathijs WintraeckenAbstract
Abstract. We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence of 
Statistical Topological Data Analysis  A Kernel Perspective (2015)
Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich BauerAbstract
We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernelbased learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as realworld data.