Mathematical Sciences: Stable Homotopy

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The research projects to be undertaken are in the following two areas of algebraic topology: the stable homotopy groups of spheres, and the homotopy limit problem. In the first problem, the investigator will study certain third-order periodic phenomena in the stable homotopy groups of spheres. This is a continuation of a general program for studying the global behavior of stable homotopy groups of spheres, as formulated by Miller, Revenel and Wilson. Results obtained from this research will enhance our understanding of the underlying periodicity of stable homotopy. In the second problem, the author will study the so called "homotopy limit problem." The importance of this problem lies in the fact that various well-known conjectures and theorems can be formulated as particular homotopy limit problems. One of the goals is to extend known affirmative solutions to the homotopy limit problem to include other important spaces from algebraic K-theory. The result can them be applied to the study of important homotopy theoretic questions. Bascially, homotopy theory is a tool for investigating the topology of spheres in all dimensions and other such natural geometric objects. It is an algebraic tool, useful because algebra permits computation.