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oai:mountainscholar.org:10217/2117672025-12-30T03:39:54Zcom_10217_100532com_10217_100000com_10217_100468com_10217_100303col_10217_182111col_10217_100469

Mirth, Joshua, author Adams, Henry, advisor Peterson, Christopher, committee member Patel, Amit, committee member Eykholt, Richard, committee member 2020-08-31T10:11:49Z 2020-08-31T10:11:49Z 2020 Mirth_colostate_0053A_16090.pdf https://hdl.handle.net/10217/211767 https://doi.org/10.25675/3.02577 If the vertex set, X, of a simplicial complex, K, is a metric space, then K can be interpreted as a subset of the Wasserstein space of probability measures on X. Such spaces are called simplicial metric thickenings, and a prominent example is the Vietoris–Rips metric thickening. In this work we study these spaces from three perspectives: metric geometry, optimal transport, and category theory. Using the geodesic structure of Wasserstein space we give a novel proof of Hausmann's theorem for Vietoris–Rips metric thickenings. We also prove the first Morse lemma in Wasserstein space and relate it to the geodesic perspective. Finally we study the category of simplicial metric thickenings and determine effects of certain limits and colimits on homotopy type. born digital doctoral dissertations English eng Colorado State University. Libraries 2020- Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. Vietoris–Rips metric thickenings and Wasserstein spaces Text Doctor of Philosophy (Ph.D.) Doctoral Mathematics Colorado State University This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). open.access