New Algebraic Structures in Topology

同伦理论的领域是研究对变形不敏感的数学不变量。当人们对研究系统的定性方面感兴趣时，或者在系统状态的说明中可能存在不精确的时候，它都适用。近年来，同伦理论的方法在凝聚态物理和数学基础等领域得到了广泛的应用。这个项目旨在用代数拓扑学中的新工具来支持这些关系，并将它们应用到数学和科学的其他领域。这项工作在凝聚态物理、经典代数几何以及长期的教育中都有应用。这个项目的范围涉及几个相互关联的研究领域。其中之一，在代数向量丛上，刻画了复分析与代数拓扑学之间的一种新的接口，旨在克服具有代数结构的拓扑向量丛的障碍。另一种是将为研究微分流形而开发的工具组织到更高的类别中，以便为拓扑量子场论中的“高斯积分”的计算提供一个通用的拓扑表达式。两位主要研究人员将共同致力于一个项目，该项目旨在为代数拓扑学中出现的“单元组”建立一个引人注目的、猜想的、结构性的结果。这一结果对拓扑量子场论的构建具有重要意义，并为理解“球面同伦群”提供了一种潜在的新途径。进一步的三个项目涉及同伦理论、代数K-理论、跨色同伦理论以及Goodwillie演算与无穷范畴稳定化中“粉碎”积的构造之间的关系的新的推广。

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.