Mathematical Sciences: Loop Spaces and Classical Homotopy Theory

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The principal investigator is studying several related areas. A number of problems are connected to the classical distributivity formula obtained from the Hilton-Milnor theorem. Problem one here is a reformulation of the strong form of the Kervaire invariant, the first step being the analysis of zero divisors in Dickson's analogues of the Cayley numbers. The second problem is the analysis of a simpler, stable distributivity law. One direction is a certain stable analogue of Barratt's finite exponent conjecture. In a related project, he continues some of his past work on the homology of configuration spaces. These investigations are mainly attempts to understand and perfect algebraic techniques. Specifically, the techniques permit calculations concerning qualitative properties of geometric manifolds, e.g. connectedness, knottedness, the fixed- point property, and so forth.