New Algebraic Structures in Topology

同伦理论是研究对变形不敏感的数学不变量的领域。它适用于任何对研究系统的定性方面感兴趣的时候，或者任何在系统状态的说明中可能存在不精确的时候。近年来，同伦理论的方法在凝聚态物理和数学基础等各个领域都得到了应用。该项目旨在利用代数拓扑中的新工具来加强这些关系，并将其应用于数学和科学的其他领域。这项工作可以应用于凝聚态物理、经典代数几何，从长远来看，还可以应用于教育。这个项目的范围涉及几个相互关联的研究领域。其中，在代数向量束上，描述了复杂分析和代数拓扑之间的一个新的接口，并旨在解决拓扑向量束具有代数结构的障碍。另一篇文章将研究微分流形的工具组织到更高的类别中，为拓扑量子场论中“高斯积分”的计算提供一般的拓扑表达式。两位主要研究人员将共同致力于一个项目，旨在为代数拓扑中的“单元群”建立一个引人注目的、推测性的结构结果。这一结果对拓扑量子场论的构建具有启示意义，并为理解“球的同伦群”提供了一种潜在的新方法。进一步的三个项目涉及到同伦理论中思想的新推广，在代数k理论的背景下，“转色同伦理论”，以及在无穷范畴稳定中Goodwillie微积分与“smash”积构造之间的关系。

The field of homotopy theory is the study of mathematical invariants that are insensitive to deformations. It is applicable whenever one is interested in studying qualitative aspects of a system, or whenever there might be imprecision in the specification of the state of a system. In recent years the methods of homotopy theory have found use in fields as diverse as condensed matter physics and the foundations of mathematics. This project aims to bolster these relationships with new tools from algebraic topology, and to apply them to other areas of mathematics and science. There are applications of this work to condensed matter physics, classical algebraic geometry, and, in the long term, to education.The scope of this project involves several interrelated areas of study. One of these, on algebraic vector bundles, depicts a new interface between complex analysis and algebraic topology, and is intended to get at the obstruction to topological vector bundles having algebraic structures. Another organizes tools developed for the study of differential manifolds into higher categories, in service of providing a general topological expression for the evaluation of "Gaussian integrals" in topological quantum field theories. The two principal investigators will work jointly on a project designed to establish a striking, conjectured, structural result for "groups of units" occurring in algebraic topology. This result has implications for the construction of topological quantum field theories and offers a potentially new approach to understanding the "homotopy groups of spheres." Three further projects involve new generalizations of ideas in homotopy theory, in the contexts of algebraic K-theory, "transchromatic homotopy theory" and in a relationship between Goodwillie calculus and the construction of "smash" products in the stabilizatons of infinity categories.