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Metric thickenings and complexes on metric spaces

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dc.date.accessioned 2021-11-29T08:53:13Z
dc.date.available 2021-11-29T08:53:13Z
dc.date.issued 2021-11-29
dc.identifier.uri http://hdl.handle.net/123456789/38623
dc.title Metric thickenings and complexes on metric spaces en
ethesis.faculty Matemaattis-luonnontieteellinen tiedekunta fi
ethesis.faculty Faculty of Science en
ethesis.faculty Matematisk-naturvetenskapliga fakulteten sv
ethesis.faculty.URI http://data.hulib.helsinki.fi/id/8d59209f-6614-4edd-9744-1ebdaf1d13ca
ethesis.university.URI http://data.hulib.helsinki.fi/id/50ae46d8-7ba9-4821-877c-c994c78b0d97
ethesis.university Helsingin yliopisto fi
ethesis.university University of Helsinki en
ethesis.university Helsingfors universitet sv
dct.creator Vuorenmaa, Elmo
dct.issued 2021 xx
dct.abstract In topology, one often wishes to find ways to extract new spaces out of existing spaces. For example, the suspension of a space is a fundamental technique in homotopy theory. However, in recent years there has been a growing interest in extracting topological information out of discrete structures. In the field of topological data-analysis one often considers point clouds, which are finite sets of points embedded in some R^m. The topology of these sets is trivial, however, often these sets have more structure. For example, one might consider a uniformly randomly sampled set of points from a circle S1. Clearly, the resulting set of points has some geometry associated to it, namely the geometry of S1. The use of certain types of topological spaces called Vietoris-Rips and Cech complexes allows one to study the "underlying topology" of point clouds by standard topological means. This in turn enables the application of tools from algebraic topology, such as homology and cohomology, to be applied to point clouds. Vietoris-Rips and Cech complexes are often not metrizable, even though they are defined on metric spaces. The purpose of this thesis is to introduce a homotopy result of Adams and Mirth concerning Vietoris-Rips metric thickenings. In the first chapter, we introduce the necessary measure theory for the main result of the thesis. We construct the 1-Wasserstein distance, and prove that it defines a metric on Polish spaces. We also note, that the 1-Wasserstein distance is a metric on general metric spaces. In the sequel, we introduce various complexes on spaces. We study simplicial complexes on R^n and introduce the concept of a realization. We then prove a theorem on the metrizability of a realization of a simplicial complex. We generalize simplicial complexes to abstract simplicial complexes and study the geometric realization of some complexes. We prove a theorem on the existence of geometric realizations for abstract simplicial complexes. Finally, we define Vietoris-Rips and Cech complexes, which are complexes that are formed on metric spaces. We introduce the nerve lemma for Cech complexes, and prove a version of it for finite CW-complexes. The third chapter introduces the concept of reach, which in a way measures the curvature of the boundary of a subset of R^n. We prove a theorem that characterizes convex, closed sets of R^n by their reach. We also introduce the nearest point projection map π, and prove its continuity. In the final chapter, we present some more measure theory, which leads to the definitions of Vietoris-Rips and Cech metric thickenings. The chapter culminates in constructing an explicit homotopy equivalence between a metric space X of positive reach and its Vietoris-Rips metric thickening. en
dct.subject Topology
dct.subject Vietoris-Rips complex
dct.subject simplicial complex
dct.subject metric thickening
dct.subject 1-Wasserstein metric
ethesis.isPublicationLicenseAccepted true
ethesis.language.URI http://data.hulib.helsinki.fi/id/languages/eng
ethesis.language englanti fi
ethesis.language English en
ethesis.language engelska sv
ethesis.thesistype pro gradu -tutkielmat fi
ethesis.thesistype master's thesis en
ethesis.thesistype pro gradu-avhandlingar sv
ethesis.thesistype.URI http://data.hulib.helsinki.fi/id/thesistypes/mastersthesis
dct.identifier.ethesis E-thesisID:02d755fa-39cc-469c-a235-a97684aef35b
dct.identifier.urn URN:NBN:fi:hulib-202111294111
ethesis.facultystudyline Matematiikka fi
ethesis.facultystudyline Mathematics en
ethesis.facultystudyline Matematik sv
ethesis.facultystudyline.URI http://data.hulib.helsinki.fi/id/SH50_050
ethesis.mastersdegreeprogram Matematiikan ja tilastotieteen maisteriohjelma fi
ethesis.mastersdegreeprogram Master's Programme in Mathematics and Statistics en
ethesis.mastersdegreeprogram Magisterprogrammet i matematik och statistik sv
ethesis.mastersdegreeprogram.URI http://data.hulib.helsinki.fi/id/MH50_001

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