Mathematical Sciences: Periodic Homotopy and Unstable Localization

9401166 Langsetmo This career advancement award supports mathematical research in the field of topology. The work continues studied in homotopy theory involving the relationship between periodic homotopy and higher Morava K-theories. In particular, work will be done in extending the construction of Dyer-Lashof operations and calculating the Morava K-theory of infinite loop spaces. The calculation of the Morava K-theory of various finite loop spaces and determining the when the cobar (base to fiber) spectral sequence converges will also be carried out. Further study of the relationship between stable and unstable localization in the case of non-connective homology theories and the relationship between unstable periodic homotopy and periodic homology theories via unstable localization and mapping telescopes are planned. Finally, applications to related fields such as representation theory will be considered. Topology is the study of spaces in which shape and distance become less important while qualities of connectedness and deformability. Homotopy, in particular, focuses on the development of a mathematical structure which measures the extent to which curves may or may not be deformed into one another while constrained to stay on a particular surface or within a given space. ***