Mathematical Sciences: Research in K-Theory

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This research is concerned with three projects in algebraic K-theory. The first is a continuation of previous work on the K-theory of twisted forms of reductive group schemes, with the goal of computing a part of the K-cohomology of such schemes. The second project is to use this computation of the K-theory of group schemes to get a good description of the K-theory of homogeneous spaces for reductive groups, and their twisted forms. The third project is an investigation of some constructions in motivic cohomology. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns the evolution of concepts from linear algebra such as basis and vector space. This work has significant implications for number theory and algebraic geometry, and promises to make exciting connections between a number of different areas in mathematics.