Applications of triangulated categories in representation theory.

在过去的几十年里，三角范畴已经从被认为是代数几何中的技术工具，转变为在许多其他数学代数领域的研究对象，特别是各种表示论、交换代数和同伦理论。在所有这些领域中，三角范畴的语言极大地澄清了现有中心问题背后的思想，并提出了重要的新问题。该项目的最初目的将是研究一些最近的主要理论进展和见解，涉及有限维代数表示理论中出现的三角范畴（导出范畴和稳定模范畴），特别是Rouquier关于三角范畴维度的工作，以及最近发现的无界导出范畴和长期存在的“同调代数”之间的联系。这些发展揭示了关于三角范畴生成的一系列问题，其中许多问题甚至在表面上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.