Enriched motivic homotopy theory

动机同伦理论及其相关的上同调技巧的多功能性使其成为数学的一个重要分支。最近出现了一些根本性的发展，这些发展已被用来解决一些长期存在的问题。新的具有重要战略意义的发展与V.Voevodsky(Fields Medal 2002)、M.Rost、A.Suslin、I.Panin、M.Levine、F.Morel.的名字有关。这个项目的主要目的是研究丰富的动机同伦理论。希望它的研究能对A^1-拓扑学中的一些经典问题有所帮助。我们建议使用它的方法来构造各种三角动机范畴，这是现代A^1-拓扑学的中心研究对象之一。我们相信，这些构造将具有重要的计算优势。我们发展的同伦理论观点可能在其他数学领域也有用，如代数拓扑学和非交换几何。这项研究将在斯旺西大学数学系进行。

The versatility of motivic homotopy theory and its associated range of cohomological techniques has made it an important branch of mathematics. Recently there have been several fundamental developments which have been used to solve a number of longstanding problems. The new strategically important developments are related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A. Suslin, I. Panin, M. Levine, F. Morel.The principal aim of this project is to investigate enriched motivic homotopy theory. It is hoped that its study will shed light on some classical problems in A^1-topology. We propose to use its methods to construct various triangulated categories of motives, one of the central objects of study in modern A^1-topology. We believe that these constructions will have important computational advantages. The homotopy-theoretic outlook that we develop is likely to be useful in other areas of mathematics such as algebraic topology and non-commutative geometry.The research will be undertaken in the Department of Mathematics, Swansea University.