Collery, Brian, authorPeterson, Chris, advisorShonkwiler, Clayton, advisorCavalieri, Renzo, committee memberKirby, Michael, committee memberPouchet, Louis-Nöel, committee member2024-09-092024-09-092024https://hdl.handle.net/10217/239223The Grassmann manifold Gr(k, n) is a geometric object whose points parameterize k dimensional subspaces of Rn. The flag manifold is a generalization in that its points parameterize flags of vector spaces in Rn. This thesis concerns applications of the geometry of the Grassmann and flag manifolds, with an emphasis on image compression. As a motivating example, the discrete versions of Daubechies wavelets generate distinguished n-dimensional subspaces of R2n that can be considered as distinguished points on Gr(n, 2n). We show that geodesic paths between "Daubechies points" parameterize families of "good" image compression matrices. Furthermore, we show that these paths lie on a distinguished Schubert cell in the Grassmannian. Inspired by the structure of Daubechies wavelets, we define and explore sparse matrix varieties as a generalization. Keeping in that theme, we are interested in understanding geometric considerations that constrain the "good" compression region of a Grassmann manifold.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.Grassmann manifoldDaubechies waveletsText