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

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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.