Homotopy-Theoretic Aspects of the Theory of Motives

这个建议的目的是发展一个类似的，在同伦范畴，类别的混合泰特动机已被证明是如此重要的算术几何的最近发展。其中心思想是用Waldhausen的谱的K-理论来代替后一主题中的群环的代数K-理论。这将解释zeta-valuesat奇数正整数的外观在理论的动机，并作为valuesof Igusa的Reidemeister不变量在微分拓扑。黎曼zeta函数的某些值被广泛地证明是超越的。这些数字最近被证明[通过工作的康纳斯，Kreimer，和Marcolli]发挥核心和意想不到的作用在理论基础的重整化技术fundamental微扰量子场论，但为什么他们应该是如此重要的是相当神秘的。然而，这些相同的数字是fundamentalimportance在其他部分的数学，如算术代数几何和微分拓扑。他们的作用在代数geometryhas被合理化的发展理论的“动机”，它似乎有可能，一些版本的这些想法可以扩展到微分拓扑以及。拓扑量子场论和微分拓扑学之间的深刻联系表明，这将导致对物理学几何基础的理解达到一个新的水平。

The goal of this proposal is to develop an analog, in the homotopycategory, of the category of mixed Tate motives which have proved tobe so important in recent developments in arithmetic geometry. Thecentral idea is to replace the algebraic K-theory of the ring ofintegers in the latter subject with Waldhausen's K-theory of thesphere spectrum. This would explain the appearance of zeta-valuesat odd positive integers in both the theory of motives, and as valuesof 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 ofConnes, Kreimer, and Marcolli] to play a central and unexpected rolein the theoretical basis for the renormalization techniques fundamentalto perturbative quantum field theory, but WHY they should be so importantis quite mysterious. However, these same numbers are of fundamentalimportance in other parts of mathematics, eg in arithmetic algebraicgeometry and in differential topology. Their role in algebraic geometryhas been rationalized by the development of a theory of `motives', and itseems likely that some version of these ideas can be extended todifferential topology as well. The deep links between topological quantumfield theories and differential topology suggest that thiswill lead to a new level of understanding of the geometric foundationsof phyics.