Algebraic Homotopy Theory and Algebraic Cycles

由于Morel-Voevodsky构造了A^1-同伦理论，使得在代数几何的背景下系统地研究同伦理论，以及将同伦理论的技巧和方法应用于代数几何的基本问题。我们计划在这一新理论的两个方面进行研究：继续我们对A1-同伦理论的基本结构问题的研究，并用这一理论引入的新观点来阐明代数几何中的基本问题，特别是代数圈理论中的基本问题。具体项目包括：1.理解沃沃茨基切片塔中自然产生的上同调理论，并从这些上同调理论中构建有趣的循环理论。其中包括由KO-理论产生的无向圈，以及由等变丛的K-理论产生的代数Bredon理论。极限动机和极限动机上同调的构造，并应用于q的Galois群在几何基本群上的作用，以及混合Tate动机的周期的研究。利用n点亏格零曲线的模空间的代数余边给出了张量范畴的有趣变形。为研究光滑曲线的模中的稳定现象提供了一个代数环境。在代数和拓扑学这两个看似不相关的领域之间建立类比，往往是解决这两个领域的难题的有效方法。最近，Morel和Voevodsky将拓扑学的一个完整分支，称为稳定同伦理论，转移到了代数环境中，使稳定同伦理论的思想适用于代数和数论中的问题。我的研究是试图从稳定同伦理论中提取一些具体的结构，并使它们适应这种新的环境。此外，我计划进行具体的计算，以确定这些新结构在应用于熟悉的代数对象时会产生什么结果。最后，我希望回答伽罗瓦理论中的基本问题，并解释为什么在计算被称为对整数的混合动机周期的积分时，总是出现某些特殊的数字，即所谓的多重Zeta值。

The construction by Morel-Voevodsky of A^1-homotopy theory has enabled the systematic study of homotopy theory in the setting of algebraic geometry, as well as the application of techniques and approaches of homotopy theory to basic problems in algebraic geometry. We plan to pursue a research program in both aspects of this new theory: to continue our investigation of the basic structural issues of A1-homotopy theory and to use the new viewpoint introduced by this theory to shed light on basic problems in algebraic geometry, especially in the theory of algebraic cycles. Specific projects include:1. Understanding Voevodsky's slice tower for naturally occuring cohomology theories, and constructing interesting cycle theories from these cohomology theories. These include non-oriented cycles arising from KO-theory, and an algebraic Bredon theory, arising from the K-theory of equivariant bundles.2. Constructions of limit motives and limit motivic cohomology, with applications to actions of the Galois group of Q on geometric fundamental groups, and the study of periods of mixed Tate motives.3. Using the algebraic cobordism of the moduli space of n-pointed genus zero curves to give interesting deformations of tensor categories.4. Giving an algebraic setting for the study of stable phenomena in the moduli of smooth curves.Creating analogies between the seemingly unrelated fields of algebra and topology has often been a fruitful approach to solving difficult problems in both fields. Recently, Morel and Voevodsky have transferred an entire branch of topology, called stable homotopy theory, to the algebraic setting, making ideas from stable homotopy theory applicable to problems in algebra and number theory. My research is an attempt to take a number of specific constructions from stable homotopy theory and adapt them to this new setting. In addition, I plan to make specific computations to determine what these new constructions yield when applied to familiar algebraic objects. Finally, I hope to answer basic questions in Galois theory and to explain why certain special numbers, the so-called multiple zeta values, always turn up when one evaluates the integrals known as periods of mixed Tate motives over the integers.