Representation of Galois groups and descent in algebraic K-theory

代数 K 理论中伽罗瓦群的表示和下降

• 批准号: 0104162
• 负责人: Gunnar Carlsson
• 金额: $ 30.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104162&HistoricalAwards=false
• 关键词: Representation; Galois; groups; descent; algebraic

DMS-0104162Gunnar CarlssonThis project includes several different directions of research. Carlsson will study a homotopy theoretic model he has constructed for the algebraic K-theory spectrum of a field, which is built out of the representation theory of the absolute Galois group of the field. He will try to prove that the model is indeed equivalent to the K-theory of the field, as well as to work out the consequences of this result for the Quillen-Lichtenbaum conjectures and the relationship between this result and the Bloch-Kato conjecture. He will also investigate potential applications of algebraic topology in high dimensional data analysis. He expects to improve software which he and V. De Silva have developed for homology computation, with the goal of identifying features such as singular points as well as global topology for data sets of dimension greater than three. He also plans to study homotopy theoretic issues which arise in computational questions, relating to sampling subcomplexes from large simplicial complexes. Kiem plans to compute intersection numbers for singular moduli spaces of curves. This is an important problem since the intersection numbers in this case are related to well-known geometric invariants, such as the Casson invariant.One of the great themes in mathematics over the last two centuries is the strong relationship between arithmetic and geometry. The study of this theme was initiated by Abel and Galois in the early 19th century, and in this century its further development has resulted in our obtaining very precise information concerning sets of integer or rational solutions to systems of equations. The goal of this project is to explore another manifestation of this theme, in the form of the so-called algebraic K-theory of fields. Algebraic K-theory is a geometric construction attached to arithmetic objects, called fields. Fields are arithmetic objects in which one can add, multiply, and divide. Algebraic K-theory attaches geometric objects to these fields, and in such a way that already well understood invariants of fields can be extracted easily. Algebraic K-theory also contains many less well understood invariants as well, and the goal of understanding these invariants has been one toward which topologists have been striving since the early 1970's, when Quillen defined higher algebraic K-theory. Two important conjectures have been formulated concerning algebraic K-theory, the Quillen-Lichtenbaum and Bloch-Kato conjectures. They would relate the algebraic K-theory of a field to properties of its "absolute Galois groups", an object which incorporates all possible symmetries of sets of solutions of sets of equations over the field in question. This project aims to understand very clearly how algebraic K-theory is built out of information of this large group of symmetries, and to use this understanding to approach the two central conjectures mentioned above.

DMS-0104162Gunnar CarlssonThis project includes several different directions of research. 卡尔松将研究同伦理论模型，他已经建立了代数K-理论频谱的领域，这是建立了代表性理论的绝对伽罗瓦群的领域。 他将试图证明该模型确实等价于该领域的K理论，以及计算出这一结果对Quillen-Lichtenbaum猜想的影响以及这一结果与布洛赫-加藤猜想之间的关系。他还将研究代数拓扑在高维数据分析中的潜在应用。 他希望改进他和V. De Silva为同源计算开发的软件，目标是识别特征，如奇点以及大于3维的数据集的全局拓扑结构。他还计划研究同伦理论的问题出现在计算问题，有关采样subcomplex从大型单纯复杂。 Kiem计划计算曲线的奇异模空间的相交数。 这是一个重要的问题，因为在这种情况下的交集数与著名的几何不变量，如卡森不变量。在过去的两个世纪中，数学中的一个伟大主题是算术和几何之间的密切关系。 这一主题的研究是由阿贝尔和伽罗瓦在19世纪初世纪，并在本世纪的进一步发展，导致我们获得非常精确的信息有关的整数或合理的解决方案系统的方程组。 这个项目的目标是探索这一主题的另一种表现形式，即所谓的代数K-场论。 代数K-理论是一个几何结构，它依附于算术对象，称为域。字段是一种算术对象，可以在其中进行加、乘和除。 代数K理论重视几何对象这些领域，并在这样一种方式，已经很好地理解不变量的领域可以很容易地提取。 代数K-理论也包含许多不太好理解的不变量，而理解这些不变量的目标一直是拓扑学家自1970年代初以来一直在努力的目标之一，当时奎伦定义了高等代数K-理论。 关于代数K-理论，有两个重要的定理，Quillen-Lichtenbaum定理和Bloch-Kato定理。 他们将把一个域的代数K-理论与它的“绝对伽罗瓦群”的性质联系起来，绝对伽罗瓦群是一个包含了所有可能的对称性的域上的方程组的解的集合。 该项目旨在非常清楚地了解代数K理论是如何从这一大群对称性的信息中构建出来的，并利用这种理解来接近上述两个中心结构。