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

这项建议描述了一个广泛的研究计划，旨在调查一个新的观点的结果，在代数K理论的基础上，以及到拓扑Hochschild和循环同调(THH和TC)的迹映射理论上。人们早就知道，在某种意义上，代数K-理论是模范畴的同伦理论的一个不变量；这一思想可以根据动机的观点而变得精确，该观点将环和模式的几何视为编码在它们的模范畴中，使这样的模范畴成为研究的中心对象。PI将发展这一视角来建立新旧猜想，极大地促进我们对代数K理论及其在拓扑学、几何学和算术中的深层作用的理解。这项工作在一定程度上依赖于同伦理论中的新技术：无限范畴的新兴理论和工具，源于Kervaire不变猜想的Hill-Hopkins-Ravenel解。国际数学联合会还建议制定一项计划，以确定对数学感兴趣的本科生，并鼓励他们从事数学科学方面的研究生工作。该计划将让参与者接触到一个强调从发现中学习的新颖课程。建议的课程结合了“探究式学习”的教学技巧，以软件环境为基础，透过引导式探索，创新地处理初等线性代数和代数拓扑学。建议的研究将加深我们目前对代数和高维几何之间的桥梁的理解。拟议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培训的本科生的发展，并将利用德克萨斯大学在从传统上代表不足的群体中招收有才华的本科生方面的现有优势。

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.