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。 它适用于任何对研究系统的定性方面感兴趣的时候，或者任何对系统状态的规范可能不精确的时候。 近年来，同伦理论的方法在凝聚态物理和数学基础等不同领域得到了应用。 该项目旨在用代数拓扑学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。 这项工作的应用凝聚态物理，经典代数几何，并从长远来看，教育。这个项目的范围涉及几个相互关联的研究领域。 其中之一，代数向量丛，描绘了一个新的接口之间的复杂的分析和代数拓扑，并打算在障碍，拓扑向量丛具有代数结构。 另一个组织工具开发的研究微分流形到更高的类别，在服务提供一个一般的拓扑表达式的评估“高斯积分”的拓扑量子场论。 两位主要研究者将共同致力于一个项目，旨在为代数拓扑中出现的“单位群”建立一个引人注目的、结构化的结果。 这一结果对拓扑量子场论的构建具有启示意义，并为理解“球面同伦群"提供了一种潜在的新方法。三个进一步的项目涉及同伦理论的新的概括，在代数K-理论，“transchromatic同伦理论”的背景下，并在古德威利演算和“粉碎”产品的建设之间的关系，在稳定的无穷大范畴。

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.