dc.contributor.advisor Seshadri, Harish dc.contributor.author Maitra, Sayantan dc.date.accessioned 2018-05-22T16:18:20Z dc.date.accessioned 2018-07-31T06:08:45Z dc.date.available 2018-05-22T16:18:20Z dc.date.available 2018-07-31T06:08:45Z dc.date.issued 2018-05-22 dc.date.submitted 2017 dc.identifier.uri http://etd.iisc.ac.in/handle/2005/3588 dc.identifier.abstract http://etd.iisc.ac.in/static/etd/abstracts/4456/G28199-Abs.pdf en_US dc.description.abstract This 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.iso en_US en_US dc.relation.ispartofseries G28199 en_US dc.subject Metric Measure Space en_US dc.subject Metric Spaces en_US dc.subject Non-positive Alexandrov Curvature en_US dc.subject Gauged Measure Spaces en_US dc.subject Lipschitz Distance en_US dc.subject Hausdorff Distance en_US dc.subject Gromov-Hausdorff Distance en_US dc.subject.classification Mathematics en_US dc.title The Space of Metric Measure Spaces en_US dc.type Thesis en_US dc.degree.name MS en_US dc.degree.level Masters en_US dc.degree.discipline Faculty of Science en_US
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