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，旨在调查的后果，一个新的角度对基础的代数K-理论和理论的跟踪地图拓扑Hochschild和循环同源性（THH和TC）。 人们早就知道，在某种意义上，代数K-理论是模范畴同伦理论的不变量;这一思想可以用动机的观点来精确化，动机的观点认为环和概型的几何是编码在它们的模范畴中的，使这样的模范畴成为研究的中心对象。 PI将发展这一观点，以建立新旧体系，极大地推进我们对代数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伦理论中的新技术：从希尔-霍普金斯-拉文埃尔解决科威尔不变一猜想中产生的无穷大范畴和工具的新兴理论。 PI还建议制定一项计划，以确定对数学感兴趣的本科生，并鼓励他们从事数学科学的研究生工作。 该计划将使参与者接触到一个新的课程，强调通过发现学习。 建议的课程，结合教学技术，从“探究式学习”，是基于一个软件环境，支持通过引导exploration.The拟议的研究将推进我们目前的理解代数和高维几何之间的桥梁初等线性代数和代数拓扑的创新治疗。 这些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。 该教育计划将加强数学训练的本科生的发展，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中招募有才华的本科生。

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.