New Frontiers in Homotopy Theory

同伦理论的领域是研究对变形不敏感的数学不变量。 它适用于任何有兴趣研究系统的定性方面、统计结构系综的涌现性质，或者任何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这些和其他领域。 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等方面都有应用。这个项目的范围涉及几个相互关联的研究领域。 其中之一，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。 另一个探讨了研究给定物理系统的所有模型的集合的前景，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。 两位主要研究人员将继续他们在色同伦理论中严格单位的联合工作。 这个项目在色同伦理论和更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及其应用的新结构。 一个研究新的公式和表达的对偶色同伦理论，另一个探讨了“transchromatic”版本的层次结构的限制表示的主要研究者的早期工作“双元性”，第三个提供了一个新的解释无处不在的李代数发生在稳定同伦理论。 其他项目包括同伦理论在分类逻辑中的应用，数学物理中更高类别兴趣的表征，以及在数论和代数几何中使用的著名的de Rham-Witt复形的新构造。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.