New Frontiers in Homotopy Theory

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、结构的统计集合的涌现特性感兴趣的时候，或者任何对系统状态的描述可能不精确的时候。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，并将其应用于这些以及其他数学和科学领域。这项工作在凝聚态物理、经典代数几何、代数拓扑和范畴逻辑中都有应用。这个项目的范围涉及几个相互关联的研究领域。其中，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。另一个探索了研究给定物理系统的所有模型集合的前景，其思想是物理可测量量的相位作为模型空间的拓扑不变量出现。两位主要研究者将继续他们在色同伦理论中的严格单位的联合研究。这个项目在色同伦理论和高范畴理论之间产生了许多惊人的类比。另外六个项目涉及同伦理论中的新结构及其应用。一个研究了色同伦理论中对偶性的新公式和表达式，另一个研究了首席研究员早期关于“双重性”的工作所表达的极限层次的“泛色”版本，第三个为稳定同伦理论中李代数的普遍存在提供了新的解释。其他项目包括将同伦理论应用于范畴逻辑，应用于数学物理中感兴趣的更高类别的表征，以及应用于数论和代数几何中著名的de Rham-Witt复合体的新构造。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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, emergent properties of a statistical ensemble of structures, 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 these and other areas of mathematics and science. There are applications of this work to condensed matter physics, classical algebraic geometry, algebraic topology and categorical logic.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 explores the prospect of investigating the aggregate of all models of a given physical system, with the idea that the phases of physically measurable quantities occur as topological invariants of the space of models. The two principal investigators will continue their joint work on strict units in chromatic homotopy theory. This project has produced many striking analogies between chromatic homotopy theory and higher category theory. Six further projects involve new stuctures in homotopy theory and its applications. One investigates new formulas and expressions of duality in chromatic homotopy theory, another explores the "transchromatic" version of the hierarchy of limits expressed by the principal investigator's earlier work on "ambidexterity," and a third offers a new explanation for the ubiquity of Lie algebras occurring in stable homotopy theory. Other projects involve applications of homotopy theory to categorical logic, to the characterization of higher categories of interest in mathematical physics, and to a new construction of the famous de Rham-Witt complex used in number theory and algebraic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.