{"id":{"repo_id":"colostate","oai_identifier":"oai:mountainscholar.org:10217/211767"},"canonical_url":"https://search.dev.ndltd.org/etd/colostate/oai:mountainscholar.org:10217/211767","repository":{"repo_id":"colostate","name":"Colorado State University","base_url":"https://api.mountainscholar.org/server/oai/request"},"display":{"title":"Vietoris–Rips metric thickenings and Wasserstein spaces","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Mirth, Joshua, author","Adams, Henry, advisor","Peterson, Christopher, committee member","Patel, Amit, committee member","Eykholt, Richard, committee member"],"institution":"Colorado State University. Libraries","degree_name":"Doctor of Philosophy (Ph.D.)","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-27T19:13:23Z","subjects":[],"languages":["eng","English"],"rights":["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."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://doi.org/10.25675/3.02577"],"render_values":[{"text":"https://doi.org/10.25675/3.02577","href":"https://doi.org/10.25675/3.02577","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10217/211767","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Mirth, Joshua, author","Adams, Henry, advisor","Peterson, Christopher, committee member","Patel, Amit, committee member","Eykholt, Richard, committee member"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-08-31T10:11:49Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-08-31T10:11:49Z"]},{"key":"dc:date.issued","label":"Date","values":["2020"]},{"key":"dc:publisher","label":"Institution","values":["Colorado State University. Libraries"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (Ph.D.)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Colorado State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["Mirth_colostate_0053A_16090.pdf"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10217/211767","https://doi.org/10.25675/3.02577"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["born digital","doctoral dissertations"]},{"key":"dc:title","label":"Title","values":["Vietoris–Rips metric thickenings and Wasserstein spaces"]}]}],"canonical_facts":{"dc:creator":["Mirth, Joshua, author","Adams, Henry, advisor","Peterson, Christopher, committee member","Patel, Amit, committee member","Eykholt, Richard, committee member"],"dc:date.accessioned":["2020-08-31T10:11:49Z"],"dc:date.available":["2020-08-31T10:11:49Z"],"dc:date.issued":["2020"],"dc:description.abstract":["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."],"dc:format.medium":["born digital","doctoral dissertations"],"dc:identifier":["Mirth_colostate_0053A_16090.pdf"],"dc:identifier.uri":["https://hdl.handle.net/10217/211767","https://doi.org/10.25675/3.02577"],"dc:language":["English"],"dc:language.iso":["eng"],"dc:publisher":["Colorado State University. Libraries"],"dc:rights":["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."],"dc:title":["Vietoris–Rips metric thickenings and Wasserstein spaces"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy (Ph.D.)"],"thesis:institution_name":["Colorado State University"]},"updated_at":"2026-07-27T19:13:23Z"}