Homotopy-Theoretic Aspects of the Theory of Motives

本建议的目标是在同伦范畴中发展一种混合泰特动机范畴的类比，这种混合泰特动机在算术几何的最新发展中已被证明是如此重要。其中心思想是用瓦尔德豪森的球谱k理论取代后一学科中指环的代数k理论。这将解释奇正整数的ζ值在动机理论中的出现，以及作为微分拓扑中Igusa的Reidemeister不变量的值。黎曼ζ函数的某些值被广泛推测为超越。这些数字最近被证明[通过connes， Kreimer和Marcolli的工作]在微扰量子场论的基本重整化技术的理论基础中发挥了核心和意想不到的作用，但是为什么它们应该如此重要却相当神秘。然而，这些相同的数字在数学的其他部分，如算术、代数几何和微分拓扑中是至关重要的。它们在代数几何中的作用已经被“动机”理论的发展合理化了，而且这些思想的某些版本似乎也可以扩展到微分拓扑中。拓扑量子场理论和微分拓扑之间的深刻联系表明，这将导致对物理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.