Applications of triangulated categories in representation theory.

在过去的几十年里，三角范畴已经从代数几何中被认为是相当技术性的工具转变为在许多其他代数数学领域中作为研究对象的中心地位：特别是各种表示理论，交换代数和同伦理论。在所有这些领域中，三角分类的语言极大地澄清了现有中心问题背后的思想，同时也提出了重要的新问题。该项目的最初目标是研究有限维代数表示理论中出现的三角化范畴（派生范畴和稳定模范畴）的一些最新主要理论进展和见解，特别是Rouquier关于三角化范畴的维度的工作，以及最近发现的无界派生范畴与长期存在的“同调猜想”之间的联系。这些发展揭示了一系列关于三角分类生成的问题，其中许多问题甚至在看似简单的情况下也是开放的。通过最初尝试回答关于这些相对简单的例子的问题，预计将获得可以应用于更复杂和更不容易理解的例子的洞察力。

Over the last few decades, triangulated categories have moved from being considered rather technical tools in algebraic geometry to assume central prominence as an object of study in many other algebraic areas of mathematics: notably representation theory of various kinds, commutative algebra and homotopy theory. In all of these fields, the language of triangulated categories has immensely clarified the ideas behind existing central problems, as well as raising important new problems. The initial aim of the project will be to study some recent major theoretical advances and insights involving the triangulated categories that arise in the representation theory of finite dimensional algebras (derived categories and stable module categories), particularly the work of Rouquier on dimensions of triangulated categories, and very recent connections discovered between the unbounded derived category and the long-standing "homological conjectures". These developments have opened up a whole range of questions about generation of triangulated categories, many of which are open even in apparently simple cases. By initially trying to answer questions about these relatively simple examples it is expected that insight will be gained that can be applied to far more complicated and less well-understood examples.