Spaces of embeddings and representations

The space of embeddings of one manifold N into another manifold M has an extremely rich topology, which can in principle be studied by means of the "embedding calculus" of Goodwillie and Weiss. Such spaces of embeddings are acted upon by the group of symmetries of N and also the group of symmetries of M, and hence their cohomology gives an interesting source of representations of these groups. Letting M denote Euclidean space and acting by symmetries of N, it can often be shown that such representations can be decomposed into relatively simple pieces, so-called algebraic representations, but as representations of the symmetries of N these algebraic pieced can be be combined in interesting, non-algebraic, ways: this makes the study of such groups of symmetries of N richer than the study of their associated arithmetic groups. The goal of this project is to discover whether the cohomology of embedding spaces really can provide interesting representations in this way, or whether they are constrained to actually be algebraic.