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This volume presents the proceedings from the AMS-IMS-SIAM Summer Research Conference on Homotopy Methods in Algebraic Topology held at the University of Colorado. The conference coincided with the 60th birthday of J. Peter May. An article is included reflecting his wide-ranging and influential contributions to the subject area. Other articles in the book discuss the Adams $EUL2$ term for elliptic cohomology, mapping class groups and function spaces, rational SO(3) equivariant cohomology theories, toral groups and classifying spaces of $p$-compact groups, dual calculus for functors to...