Combinatorial aspects of derived representation theory

卡-丘（CY）三角范畴为最初起源于数学物理的思想提供了一个同调的背景。近年来，他们获得了独立的兴趣，在纯数学，特别是在有关集群倾斜理论和Bridgeland稳定性条件。一个三角范畴是卡-丘范畴，如果它满足一个特别好的塞尔对偶形式：对于整数n，第n个移位函子[n]自然同构于塞尔函子S;这样的范畴称为n-CY。当n >1时，人们对这类范畴的同调结构和组合结构有很多了解，例如（高等）团簇倾斜理论和高等同调代数。当n < 1时的图像发展得不太好，尽管有主流的例子，例如对称代数的稳定模范畴是（-1）-CY，它们的完美导出范畴是0-CY。这个项目的目的是发展负CY范畴的结构理论，例如通过对离散CY三角范畴中的（共）扭对进行分类，并描述稳定模范畴的简单系统。

Calabi-Yau (CY) triangulated categories provide a homological context for ideas that first originated in mathematical physics. In recent years they have acquired independent interest in pure mathematics, particularly in relation to cluster-tilting theory and Bridgeland stability conditions. A triangulated category is Calabi-Yau if it satisfies a particularly nice form of Serre duality: for an integer n, the nth shift functor [n] is naturally isomorphic to the Serre functor S; such a category is called n-CY. When n >1, much is known about the homological and combinatorial structure of such categories, e.g. (higher) cluster-tilting theory and higher homological algebra. The picture when n < 1 is much less well developed despite there being mainstream examples, e.g. stable module categories of symmetric algebras are (-1)-CY and their perfect derived categories are 0-CY. The aim of this project is to develop the structure theory for negative CY categories, for example by classifying (co)torsion pairs in discrete CY triangulated categories, and describing simple-minded systems for stable module categories.