Mirth, Joshua, authorAdams, Henry, advisorPeterson, Christopher, committee memberPatel, Amit, committee memberEykholt, Richard, committee member2020-08-312020-08-312020https://hdl.handle.net/10217/211767If 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 digitaldoctoral dissertationsengCopyright 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.Text