RUI: Calculus of Functors and Applications in Homotopy Theory

这个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。虽然拓扑学以前是数学中最抽象的领域之一，但在数据分析和神经科学等领域，人们开始认识到拓扑学的有用性，在这些领域，人们在各种意想不到的地方发现了高维结构。这个特别的项目涉及不同形状之间相互联系的方式，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(就像全国各地的本科生一样)从根本上讲是关于近似的。在这个项目中，PI将研究如何以一种有用的方式用更简单和更容易处理的形状来逼近复杂的形状。对这些近似的系统理解将有助于我们描述现代拓扑学应用中出现的一些新形状的结构。更严格地说，这个项目的重点是Goodwillie发展的同伦函子演算。其基本思想是使用满足多项式条件的其他更简单的函数值来逼近感兴趣的对象(比如某个函数值)。这个项目的一个主要组成部分是，在各种不同的情况下，了解如何将泰勒级数(拓扑学中的“泰勒塔”)的模拟从其组件组装起来。从这个一般理论出发，这个项目的目标是几个应用途径。一个是与各种形式的代数K-理论相对应的泰勒塔的计算，它被视为从环谱到谱的函子。二是色同伦理论，特别是Bousfield-Kuhn函子的研究，以及K(N)-局部环境下的代数K-理论。该项目还将支持国际数学联合会促进阿默斯特学院数学和数学教育的努力。国际数学学院致力于职前数学教师的培训，并共同设计和教授了一门关于K-12数学教育中不平等的新课程，扩大了学院对教育研究的支持，并加强了数学与阿默斯特大学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.