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，同伦理论起源于代数拓扑学，代数拓扑学是通过代数不变量研究拓扑空间（粗略地说，几何学中有一个接近但没有距离的概念）;然而，它逐渐在数学的许多领域，特别是代数和微分几何中找到了更广泛的应用，并且与物理学有许多有趣的联系。部分原因是，专注于数学对象的变形不变属性使分类和计算更容易处理。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有力的应用，并且最近的许多研究关注同伦对象的代数几何和更高范畴的性质，阐明了许多结构方面，并激发了这些不同的数学学科之间的进一步互动。这个项目的目的是采用代数几何和更高，代数拓扑学和同伦理论研究中的范畴技术。第一个目标是研究代数K-理论和相关理论，如拓扑Hochschild同调，以及代数几何中产生的更一般的动机，重点是通过识别本地化序列来开发计算方法。第二个目的是分类厚子类的某些稳定的更高的范畴，特别是那些出现在几何中的形成完美的复杂的各种类别的衍生计划。第三个涉及椭圆上同调，并关注各种orbifolds的椭圆上同调的明确描述，是尚未完全理解的orbifolds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，另一个最近备受关注的主题。最终的目标是了解单位的衍生计划使用新兴工具的衍生代数几何，特别是皮卡德和布劳尔集团和他们的更高的范畴类似物和可能的连接拓扑场理论。

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.