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，旨在研究代数K理论基础的新观点的后果以及拓扑图与拓扑图和环状同源性（THH和TC）的痕量地图理论。 早就知道，在某种意义上，代数K理论是模块类别的同型理论的不变。可以从动机的角度来精确地做出这个想法，该观点将戒指和方案的几何形状视为其模块类别中编码的几何形状，从而使此类模块类别成为研究的核心对象。 PI将开发这种观点，以建立新老式的猜想，从而极大地推进了我们对代数K理论及其在拓扑，几何和算术中的深刻作用的理解。 这项工作部分依赖于同义理论中的新技术：无穷大类别的新兴理论和工具是由kervaire不变的一个猜想引起的山丘 - 霍普金斯 - 雷诺尔解决方案。 PI还建议制定一个计划，以识别对数学感兴趣的大学生，并鼓励他们从事数学科学的研究生工作。 该计划将使参与者接受新的课程，以通过发现来强调学习。 拟议的课程纳入了“基于查询的学习”中的教学技术，它基于一个软件环境，该软件环境通过指导探索来支持基本线性代数和代数拓扑的创新处理。拟议的研究将促进我们目前对桥梁之间的桥梁和高维度的理解。 拟议的研究的某些方面将对数学物理学有影响，尤其是对拓扑领域理论和弦理论的研究。 该教育计划将增强经过数学培训的大学生的发展，并将利用得克萨斯大学现有的优势从传统上代表性不足的群体招募有才华的本科生。