RUI: Calculus of Functors and Applications in Homotopy Theory

该项目属于拓扑学领域，研究高维形状的基本性质以及它们之间的关系。虽然以前是数学中最抽象的领域之一，但拓扑学的有用性开始在数据分析和神经科学等领域得到认可，这些领域在各种意想不到的地方发现了高维结构。这个特别的项目关注的是不同形状之间的联系，并将微积分的一些想法和直觉应用到这些联系的研究中。微积分（在全国范围内教授给本科生）基本上是关于近似的。在这个项目中，PI将研究如何用更简单、更容易处理的方法来近似复杂的形状。对这些近似的系统理解将帮助我们描述现代拓扑应用中出现的一些新形状的结构。更技术性地说，这个项目的重点是古德威利开发的同伦函子演算。其基本思想是使用其他满足多项式条件的简单函子来近似感兴趣的对象（例如某个函子的值）。该项目的一个主要组成部分是了解，在各种不同的情况下，如何模拟泰勒级数（拓扑学中的“泰勒塔”）可以从其组件组装。从这一一般理论出发，本项目的目标是几种应用途径。一个是计算泰勒塔对应的各种版本的代数K理论，视为函子从环谱谱。另一个是色同伦理论，特别是对Bousfield-Kuhn函子的研究，以及在K（n）-局部设置下的代数K-理论。 该项目还将支持PI促进阿默斯特学院数学和数学教育的努力。PI参与了业余数学教师的培训，并共同设计和教授了一门关于K-12数学教育不平等的新课程，扩大了学院对教育研究的支持，并增加了数学与阿默斯特其他学术部门之间的合作。

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.