Keenan, Liam2025-01-282025-01-282024-05https://hdl.handle.net/11299/269657University of Minnesota Ph.D. dissertation. May 2024. Major: Mathematics. Advisor: Tyler Lawson. 1 computer file (PDF); iv, 63 pages.This thesis is divided into two parts. In the first part, we prove that topological Hochschild homology satisfies both 1-connected and flat descent for connective E₂₋ rings. Along the way, we provide an alternative construction of the ``May filtration'' on topological Hochschild homology, which was originally considered by Angelini--Knoll and Salch. In the second part of this thesis, which is based on joint work with Jonas McCandless, we show that topological restriction homology has a chromatic vanishing property similar to that of algebraic K-theory and topological cyclic homology. This is based on a careful analysis of the interaction of algebraic K-theory with infinite products.enChromatic homotopy theoryTopological Hochschild homologyTopological restriction homologyTrace methodsThesis or Dissertation