Motives, geometry, and higher category theory

动机、几何和更高范畴论

• 批准号: 1406529
• 负责人: David Gepner
• 金额: $ 15.03万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406529&HistoricalAwards=false
• 关键词: Motives; geometry; higher; category; theory

Homotopy theory is the study of properties of mathematical objects which remain invariant under deformation. Historically, homotopy theory emerged from algebraic topology, which is the study of topological spaces (roughly, geometry in which there is a notion of closeness but not distance) through algebraic invariants; gradually, however, it has found applications much more broadly throughout many areas of mathematics, especially algebraic and differential geometry, and has many interesting connections to physics. Part of the reason for this is that focusing on deformation-invariant properties of mathematical objects makes both classifications and calculations much more tractable. The interaction goes both ways; recently algebraic and geometric methods have found powerful applications in homotopy theory, and much recent research has been concerned with the algebro-geometric and higher-categorical nature of homotopical objects, illuminating many structural aspects and inspiring further interactions between these various mathematical disciplines.The purpose of this project is to employ algebro-geometric and higher-categorical techniques in the study of algebraic topology and homotopy theory. The first goal is to study algebraic K-theory and related theories such as topological Hochschild homology, as well as more general motives arising in algebraic geometry, with emphasis on developing computational methods through identification of localization sequences. The second aims to classify thick subcategories of certain stable higher categories, especially those which arise in geometry by formation of perfect complexes over various classes of derived schemes. The third involves elliptic cohomology and is concerned with the explicit description of the elliptic cohomology of various orbifolds and is an important example of the not yet fully understood stable homotopy theory of orbifolds, also known as global stable homotopy theory, another subject of much recent attention. The final goal is to understand the units of a derived scheme using the emerging tools of derived algebraic geometry, in particular the Picard and Brauer groups and their higher categorical analogues and possible connections to topological field theories.

同伦理论是对在变形下保持不变的数学对象的性质的研究。历史上，同伦理论起源于代数拓扑，代数拓扑是通过代数不变量研究拓扑空间（大致是几何学，其中有接近的概念，但没有距离的概念）；然而，它逐渐在数学的许多领域（尤其是代数和微分几何）中得到了更广泛的应用，并且与物理学有许多有趣的联系。造成这种情况的部分原因是，关注数学对象的变形不变特性使得分类和计算变得更加容易处理。互动是双向的；最近，代数和几何方法在同伦理论中得到了强大的应用，并且最近的许多研究都关注同伦对象的代数几何和高范畴性质，阐明了许多结构方面并激发了这些不同数学学科之间的进一步相互作用。该项目的目的是在代数拓扑和高范畴研究中采用代数几何和高范畴技术 同伦论。第一个目标是研究代数 K 理论和相关理论，例如拓扑 Hochschild 同调，以及代数几何中出现的更一般的动机，重点是通过识别定位序列来开发计算方法。第二个目的是对某些稳定的较高类别的厚子类别进行分类，特别是那些在几何中通过在各种派生方案上形成完美复合体而出现的子类别。第三个涉及椭圆上同调，涉及各种轨道折叠的椭圆上同调的显式描述，是尚未完全理解的轨道折叠稳定同伦理论的一个重要例子，也称为全局稳定同伦理论，是最近备受关注的另一个主题。最终目标是使用派生代数几何的新兴工具来理解派生方案的单位，特别是皮卡德群和布劳尔群及其更高的分类类似物以及与拓扑场论的可能联系。