Algebraic Homotopy Theory and Algebraic Cycles

Morel-Voevodsky对A^1-同伦理论的建立，使得在代数几何的背景下系统地研究了同伦理论，并将同伦理论的技术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：继续研究a1 -同伦理论的基本结构问题，并利用该理论引入的新观点来阐明代数几何中的基本问题，特别是代数循环理论中的基本问题。具体项目包括：1；理解Voevodsky对自然上同调理论的切片塔，并从这些上同调理论构造有趣的循环理论。这包括由ko理论产生的无取向环，以及由等变束的k理论产生的代数Bredon理论。2 .极限动机和极限动机上同调的构造及其在几何基本群上的伽罗瓦群作用的应用，以及混合塔特动机周期的研究。利用n点属零曲线模空间的代数协坐标给出张量范畴的有趣变形。给出了研究光滑曲线模中稳定现象的一个代数集。在代数和拓扑学这两个看似无关的领域之间建立类比，往往是解决这两个领域难题的有效方法。最近，Morel和Voevodsky将拓扑学的一个分支——稳定同伦理论——转移到代数集合中，使稳定同伦理论的思想适用于代数和数论问题。我的研究是尝试从稳定同伦理论中提取一些特定的结构，并使它们适应这个新的环境。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产生的结果。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整数上的混合泰特动机周期积分时，总是会出现某些特殊的数字，即所谓的多重泽塔值。

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.