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dc.contributor.advisorSeshadri, Harish
dc.contributor.authorMaitra, Sayantan
dc.date.accessioned2018-05-22T16:18:20Z
dc.date.accessioned2018-07-31T06:08:45Z
dc.date.available2018-05-22T16:18:20Z
dc.date.available2018-07-31T06:08:45Z
dc.date.issued2018-05-22
dc.date.submitted2017
dc.identifier.urihttp://etd.iisc.ac.in/handle/2005/3588
dc.identifier.abstracthttp://etd.iisc.ac.in/static/etd/abstracts/4456/G28199-Abs.pdfen_US
dc.description.abstractThis thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis. The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that, Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties. On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved. Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.en_US
dc.language.isoen_USen_US
dc.relation.ispartofseriesG28199en_US
dc.subjectMetric Measure Spaceen_US
dc.subjectMetric Spacesen_US
dc.subjectNon-positive Alexandrov Curvatureen_US
dc.subjectGauged Measure Spacesen_US
dc.subjectLipschitz Distanceen_US
dc.subjectHausdorff Distanceen_US
dc.subjectGromov-Hausdorff Distanceen_US
dc.subject.classificationMathematicsen_US
dc.titleThe Space of Metric Measure Spacesen_US
dc.typeThesisen_US
dc.degree.nameMSen_US
dc.degree.levelMastersen_US
dc.degree.disciplineFaculty of Scienceen_US


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