Applications of Chromatic Fixed Point Theory

From its inception the primary concern of algebraic topology has been using algebraic techniques to construct invariants of topological spaces. This pursuit has led to the creation of many important cross-disciplinary tools. One such modern tool is the equivariant Balmer spectrum associated to a finite group G. This object, which lives in the intersection of algebraic geometry and algebraic topology, encodes information about the equivariant stable homotopy category in a systematic yet abstract way. This thesis describes new computational techniques that use knowledge of the Balmer spectrum of the equivariant stable homotopy category to compute explicit topological invariants of spaces. In particular, a new technique for showing certain spectral sequences collapse is presented. This technique is then applied to compute the dimension of the Morava K-theory of a family of finite real Grassmannians. Furthermore, a conjecture for the dimension of the Morava K-theory of all finite real Grassmannians is presented.