9704761 Arone The investigator's general area of specialization is algebraic topology. His recent work has been concerned with achieving a better understanding of the global structure of homotopy theory. This work combines two hitherto disjoint approaches, the "chromatic approach," and the "calculus of functors" approach. In joint work with M. Mahowald, the investigator examined the Goodwillie derivatives of unstable homotopy theory and used them to prove a new result about unstable homotopy of spheres. In other work, the investigator studied the Taylor polynomials (as opposed to derivatives) of unstable homotopy theory and provided an explicit space-level description of these. One of the key ingredients of the investigator's contribution to his joint work with Mahowald was a combinatorial analysis of the poset of partitions of a finite set, which turned out to have consequences for the chromatic filtration of spheres. Further investigation of the combinatorics involved has led the investigator to discover seemingly new and mysterious recurrence patterns. He believes that some of the deep periodic phenomena in homotopy theory can be studied via this combinatorial periodicity. In very recent work, the investigator utilized this combinatorial periodicity to construct certain "self maps" on the derivatives of unstable homotopy theory, which imply, among other things, the existence of finite complexes whose cohomology realizes certain finite subalgebras of the Steenrod algebra. The investigator believes that these finite complexes will be useful for localization and periodicity. In the investigator's area of Algebraic Topology, he is interested in how to analyze the space of continuous functions between two topological spaces. It turns out that an analogue of the theory of Taylor series can be developed, and one can study the space of continuous maps via its "polynomial approximations" (an idea due to T. Goodwillie). The investigato r combined this theory with more classical methods in topology, and he used it (in joint work with M. Mahowald) to prove a new result on the global structure of an important algebraic tool of the trade, the so-called unstable homotopy of spheres. He now intends to use the aformentioned polynomial approximations to construct certain spaces with interesting (cohomological) properties, spaces whose mere existence should lead to further insights in topology. His work has a strong combinatorial flavor (specifically, one of the main gadgets in his work is the lattice of partitions of a finite set), and he hopes that some of his results will be of interest to combinatorialists. ***
