The goal of this proposal is to develop an analog, in the homotopy category, of the category of mixed Tate motives which have proved to be so important in recent developments in arithmetic geometry. The central idea is to replace the algebraic K-theory of the ring of integers in the latter subject with Waldhausen's K-theory of the sphere spectrum. This would explain the appearance of zeta-values at odd positive integers in both the theory of motives, and as values of Igusa's Reidemeister invariants in differential topology.
Certain values of Riemann's zeta function are widely conjectured to be transcendental. These numbers have recently been shown [through work of Connes, Kreimer, and Marcolli] to play a central and unexpected role in the theoretical basis for the renormalization techniques fundamental to perturbative quantum field theory, but WHY they should be so important is quite mysterious. However, these same numbers are of fundamental importance in other parts of mathematics, eg in arithmetic algebraic geometry and in differential topology. Their role in algebraic geometry has been rationalized by the development of a theory of `motives', and it seems likely that some version of these ideas can be extended to differential topology as well. The deep links between topological quantum field theories and differential topology suggest that this will lead to a new level of understanding of the geometric foundations of phyics.
