RUI: Calculus of Functors and Applications in Homotopy Theory

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This project is in the field of topology, which studies the fundamental nature of high-dimensional shapes and the relationships between them. While formerly one of the most abstract areas of mathematics, the usefulness of topology is starting to be recognized in areas such as data analysis and neuroscience where high-dimensional structures have been discovered in a variety of unexpected places. This particular project concerns the ways that different shapes can be related to one another, and applies some of the ideas and intuition of calculus to the study of these connections. Calculus (as taught to undergraduates across the country) is fundamentally about approximation. In this project the PI will study how complicated shapes can be approximated in a useful way by those that are simpler and easier to work with. A systematic understanding of these approximations will help us describe the structure of some of the new shapes that are appearing in modern applications of topology.More technically, the focus of this project is the calculus of homotopy functors developed by Goodwillie. The underlying idea is to approximate an object of interest (say the value of some functor) using other simpler functors that satisfy a polynomial condition. A major component of this project is to understand, in various different situations, how the analogue of the Taylor series (a "Taylor tower" in topology) can be assembled from its components. From this general theory this project aims at several avenues of application. One is to the calculation of the Taylor towers corresponding to various versions of algebraic K-theory, viewed as functors from ring spectra to spectra. Another is to chromatic homotopy theory, specifically to study of the Bousfield-Kuhn functor, and to algebraic K-theory in a K(n)-local setting. This project will also support the PI's efforts to promote mathematics and math education at Amherst College. The PI is engaged with the training of preservice math teachers and has jointly designed and taught a new course on inequality in K-12 math education, expanding the College's support for education studies, and increasing collaboration between mathematics and other academic departments at Amherst.