Derived categories of sheaves over finite partially ordered sets and their homological properties

三角化范畴和派生范畴已经成功地用于联系不同数学起源的对象（例如孔采维奇的同调镜像对称猜想）以及相同性质的对象（例如里卡德的森田理论，布鲁猜想）。在这个项目中，我们研究派生类别产生的组合对象，如偏序集（偏序集），箭图与潜在的和其他箭图的关系。我们的主要目标是了解这些对象的组合属性是如何反映在表示理论和相关的派生类别的同调属性。其中一个主要问题是关于是否存在一个算法，该算法给定两个这样的对象，决定它们的派生类别是否等价。

Triangulated and derived categories have been successfully used to relate objects of different mathematical origins (e.g. Kontsevich’s Homological mirror symmetry conjecture) as well as objects of the same nature (e.g. Rickard’s Morita theory, Broue’s conjecture). In this project we investigate derived categories arising from combinatorial objects, such as partially ordered sets (posets), quivers with potential and other quivers with relations. Our main goal is to understand how the combinatorial properties of these objects are reflected in representation theoretic and homological properties of the associated derived categories. One of the main questions concerns the existence of an algorithm that given two such objects decides whether their derived categories are equivalent, or not.