Motives, geometry, and higher category theory

同伦理论是研究数学对象在变形下保持不变的性质。从历史上看，同伦理论是从代数拓扑中产生的，代数拓扑是通过代数不变量研究拓扑空间（粗略地说，几何中有接近但没有距离的概念）；然而，渐渐地，它在数学的许多领域得到了更广泛的应用，特别是代数和微分几何，并且与物理学有许多有趣的联系。部分原因是，关注数学对象的形变不变特性使分类和计算更加容易处理。这种互动是双向的；近年来，代数和几何方法在同伦理论中得到了强有力的应用，最近的许多研究都关注于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。本项目的目的是在代数拓扑和同伦理论的研究中运用代数几何和高范畴技术。第一个目标是研究代数k理论和相关理论，如拓扑Hochschild同调，以及代数几何中出现的更一般的动机，重点是通过识别局部序列来开发计算方法。第二个目的是对某些稳定的高范畴的粗子范畴进行分类，特别是那些在几何中通过在各种派生方案上形成完美复形而产生的子范畴。第三种方法涉及椭圆上同，是对各种轨道的椭圆上同的显式描述，是目前尚未完全理解的轨道稳定同伦理论的一个重要例子，也被称为全局稳定同伦理论，是最近备受关注的另一个主题。最终目标是利用派生代数几何的新兴工具，特别是Picard和Brauer群及其更高范畴的类似物，以及与拓扑场理论的可能联系，来理解派生方案的单位。

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.