CAREER: Algebraic K-theory, trace methods, and non-commutative geometry

职业：代数 K 理论、迹方法和非交换几何

• 批准号: 1151577
• 负责人: Andrew Blumberg
• 金额: $ 42.59万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1151577&HistoricalAwards=false
• 关键词: CAREER; Algebraic; theory; trace; methods

This proposal describes a broad research program aimed at investigating the consequences of a new perspective on the foundations of algebraic K-theory and on the theory of trace maps to topological Hochschild and cyclic homology (THH and TC). It has long been known that in some sense algebraic K-theory is an invariant of the homotopy theory of the category of modules; this idea can be made precise in terms of a motivic perspective, which views the geometry of rings and schemes as encoded in their categories of modules, making such module categories the central object of study. The PI will develop this perspective to establish conjectures old and new, dramatically advancing our understanding of algebraic K-theory and its deep role in topology, geometry, and arithmetic. The work relies in part on new technology in homotopy theory: the emerging theory of infinity categories and tools arising from the Hill-Hopkins-Ravenel solution of the Kervaire invariant one conjecture. The PI also proposes to develop a program to identify undergraduates interested in mathematics and encourage them to pursue graduate work in the mathematical sciences. The program will expose participants to a novel curriculum emphasizing learning by discovery. The proposed curriculum, incorporating pedagogical techniques from "inquiry-based learning", is based on a software environment supporting an innovative treatment of elementary linear algebra and algebraic topology via guided exploration.The proposed research will advance our current understanding of the bridge between algebra and high-dimensional geometry. Some aspects of the proposed research will have impact on mathematical physics, particularly the study of topological field theories and string theory. The educational program will enhance the development of mathematically trained undergraduates and will leverage the University of Texas' existing strengths in recruiting talented undergraduates from traditionally under-represented groups.

该提案描述了一项广泛的研究计划，旨在调查代数 K 理论基础以及拓扑 Hochschild 和循环同调（THH 和 TC）迹图理论的新视角的后果。 人们早就知道，在某种意义上，代数K理论是模范畴同伦论的不变量；这个想法可以从动机的角度来表达，即将环和方案的几何形状视为编码在其模块类别中，使这些模块类别成为研究的中心对象。 PI 将发展这一视角来建立新旧猜想，极大地增进我们对代数 K 理论及其在拓扑、几何和算术中的深层作用的理解。 这项工作部分依赖于同伦理论中的新技术：无穷范畴的新兴理论和工具，源自 Kervaire 不变一猜想的 Hill-Hopkins-Ravenel 解。 PI 还建议制定一项计划，以确定对数学感兴趣的本科生，并鼓励他们攻读数学科学领域的研究生工作。 该计划将使参与者接触到强调通过发现学习的新颖课程。 拟议的课程结合了“基于探究的学习”的教学技术，基于一个软件环境，支持通过引导探索对初等线性代数和代数拓扑进行创新处理。拟议的研究将增进我们目前对代数和代数之间桥梁的理解。高维几何。 所提出的研究的某些方面将对数学物理产生影响，特别是拓扑场论和弦理论的研究。 该教育计划将促进受过数学训练的本科生的发展，并将利用德克萨斯大学现有的优势，从传统上代表性不足的群体中招募有才华的本科生。