Mathematical Sciences: Quillen-Type Homology and Homotopy   Theory
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In general terms this project is devoted to simplifying and                making more usable several powerful, complicated, and subtle                    algebraic techniques of topology.  In the 1960's, Daniel Quillen                described how to define the homology of simplical objects over                  many different categories, including sets, algebras over a ring,                and unstable algebras over the Steenrod algebra.  Many of the                   calculations in homology theory can be cast in this light, among                them the computation of the E2-term of certain "unstable Adams                  spectral sequences" such as the Bousfield-Kan spectral sequence.                Past experience has indicated the strength of this approach; the                purpose of this project is to explore this idea in depth.  A                    particular ramification would be the exploration of the homology                of function complexes using tools of Jean Lannes and A.K.                       Bousfield.
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