Investigations in the application of homotopy theory

同伦理论的应用研究

• 批准号: 0905823
• 负责人: Gunnar Carlsson
• 金额: $ 69.01万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905823&HistoricalAwards=false
• 关键词: Investigations; application; homotopy; theory

This project deals with three separate problems in homotopy theory. The first is the study of algebraic K-theory of fields and motivic homotopy theory. We will continue the study of the representation theoretic model for the algebraic K-theory of fields using derived completion constructions, and attempt to use motivic homotopy theory and algebraic K-theory to study the Halperin-Carlsson conjectures on free finite group actions on finite complexes. The second is internal to homotopy theory, and will carry out an analysis of higher topological cyclic homology constructions in the hopes of obtaining good geometric versions of phenomena of higher chromatic levels. The third problem is the study of various aspects of "generalized persistence", which provides a method for studying finite metric spaces, and therefore many data sets coming out of science and engineering. Specifically, we wish to study barcode invariants of certain quivers which will help to assess the "strength" of a topological signal. This project deals with applications of certain geometric techniques, referred to as topological, to other parts of mathematics and eventually even outside of mathematics. Topological techniques capture qualitative features of geometric objects, such as the presence of loops, spheres, and other surfaces within them in a systematic way which allows it to be used as a tool for geometric pattern recognition in a broad sense. Interpreted this way, these techniques have applications in areas as diverse as number theory and the solution of algebraic equations on the one end to problems in data analysis on the other end. In addition, we will pursue an interesting direction in the study of homotopy theory, where we hope to make concrete and geometrical certain constructions which at the moment are less sharply defined.

这个项目涉及同伦理论中的三个独立问题。 第一部分是代数K-场论和动机同伦理论的研究。 我们将继续使用导出的完备构造来研究域的代数K-理论的表示论模型，并尝试使用动机同伦理论和代数K-理论来研究有限复形上自由有限群作用的Halperin-Carlsson定理。 第二个是内部同伦理论，并将进行更高的拓扑循环同调结构的分析，希望获得良好的几何版本的现象，更高的色水平。 第三个问题是研究“广义持久性”的各个方面，它提供了一种研究有限度量空间的方法，因此许多数据集来自科学和工程。 具体来说，我们希望研究条形码不变量的某些颤动，这将有助于评估的“强度”的拓扑信号。这个项目涉及某些几何技术的应用，被称为拓扑，数学的其他部分，最终甚至数学之外。 拓扑技术捕获几何对象的定性特征，例如以系统的方式在其中存在环、球和其他表面，这使得它可以在广义上用作几何模式识别的工具。 这样解释，这些技术在各个领域都有应用，一方面是数论和代数方程的解，另一方面是数据分析问题。 此外，我们将在同伦理论的研究中追求一个有趣的方向，在那里我们希望做出具体的和几何的某些结构，目前还没有那么明确的定义。