Abstract: Let f be a function. The two most basic ways to approximate f with a polynomial function are, probably, the interpolation polynomial and the Taylor polynomial. The interpolation polynomial (of degree n) is determined by the n+1 numbers f(0), f(1), ..., f(n). The Taylor polynomial is determined by a different set of n+1 numbers - the first n+1 derivatives of f (at 0 say).

In the talk we will explore the analogues of these two constructions for functors that arise in topology. It turns out that while a polynomial function is determined by a sequence of numbers, a polynomial functor is determined by a (truncated) symmetric sequence with an extra structure. The extra structure can be expressed in terms of operads and their modules. The relationship between the interpolation and the Taylor polynomial can be understood in terms of (a version of) Koszul duality between operads.

A good example to test the theory on is the mapping bi-functor that sends a pair of topological spaces (X, Y) to the space of maps F(X, Y). An equally interesting example is the functor that sends a pair ofsmooth manifolds (M, N) to the space of smooth embeddings Emb(M, N). We will use these functors, and others related to them, to illustrate the general theory.