Calculus of Functors and Homotopy Theory

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9971855Arone Arone is interested in applying the theory of calculus of functors(developed by T. Goodwillie and M. Weiss) to study the unstablehomotopy of such spaces as spheres and classifying spaces of compactLie groups. Calculus of functors allows one to introduce certain``derivatives'' of homotopy groups. If one understands thesederivatives well, one can use them to recover a lot of informationabout unstable homotopy theory (this is analogous to the fact that ifone knows all the derivatives of an ordinary function at a point, onecan often recover a lot of information about the function by means ofTaylor series). The derivatives are really objects of stable homotopytheory, and they normally come equipped with interesting symmetries ofcertain classical groups, such as the symmetric or orthogonal groups(division by the action of the symmetric group corresponds to divisionby n! in ordinary Taylor series). Thus calculus allows one to reducequestions about unstable homotopy theory to questions aboutequivariant stable homotopy theory. This does not yield an effectivemethod for calculating homotopy groups (there probably is no suchmethod), but it does allow one to make interesting qualitativestatements about the global structure of homotopy theory. Analyzingthe derivatives of different objects, together with the accompanyingsymmetries, has been the focus of much of Arone's work. Thesederivatives seem to be very interesting objects on their own, andstudying them involves combining methods from stable homotopy theory,equivariant topology, and group cohomology in fresh and unexpectedways. Many basic questions in mathematics, and in topology, in particular,are of the following form: given two objects, describe the possiblefunctions between them. For instance, one can ask to describe thespace of continuous maps between two topological spaces, or the spaceof embeddings of a circle into some physical space (this is informallyknown as the space of ``knots''). These questions are usually verydifficult. This is so because the dependence of the space offunctions on the source and the target is very complicated.Informally, one could say that the space of functions is a complicatedfunction of two variables. It turns out that, in analogy to ordinarydifferential calculus, one can develop a theory of ``Taylor series''of sorts for this kind of function and use it to approximatecomplicated spaces such as function spaces by simpler spaces, inmuch the same way as an ordinary function can be approximated by itsTaylor polynomials. This very striking idea was conceived byT. Goodwillie. Much of Arone's work has focused on understanding the``Taylor polynomials'' and the ``Derivatives'' in this context. Thishas proved to be a fascinating and rewarding journey, which hasalready led to interesting new results in (algebraic) topology. Someof the results are of such a universal nature that there is at least aslight hope that this work will have an impact on mathematics outsidehardcore algebraic topology.***