Combinatorial aspects of derived representation theory

Calabi-Yau(CY)三角范畴为最初起源于数学物理的思想提供了一个同源上下文。近年来，他们对纯数学产生了独立的兴趣，特别是与星团倾斜理论和布里奇兰稳定性条件有关的数学。三角范畴是Calabi-Yau，如果它满足一种特别好的Serre对偶形式：对于整数n，第n个移位函子[n]自然同构于Serre函子S；这样的范畴称为n-CY。当n&gt；1时，关于这类范畴的同调和组合结构，例如(更高的)簇倾斜理论和更高的同调代数，人们已经知道了很多。尽管有主流的例子，但当n&lt；1时，图的发展要差得多，例如对称代数的稳定模范畴是(-1)-CY，而它们的完美派生范畴是0-CY。这个项目的目的是发展负CY范畴的结构理论，例如通过分类离散CY三角范畴中的(余)扭偶，以及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.