Bush, Johnathan E., authorAdams, Henry, advisorPatel, Amit, committee memberPeterson, Chris, committee memberLuong, Gloria, committee member2021-09-062021-09-062021https://hdl.handle.net/10217/233794The geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combinatorial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the trigonometric moment curve. Topological information about metric thickenings allows us to give new generalizations of the Borsuk–Ulam theorem and a selection of its corollaries. Finally, we prove a centrally-symmetric analog of a result of Gilbert and Smyth about gaps between zeros of homogeneous trigonometric polynomials.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