Quiver representations, singularity categories, and monoidal structures

从不同的角度研究了代数的表示，包括单面结构和三角结构。一个目标是用生成元和关系来描述表示图的投影函子的生成元、表示环和么半群。另一个目标是对三角奇点范畴的粗和局部化子范畴进行分类。这涉及到连续和离散的参数化，反映了现有的么一或组合结构。选择奇点类别的动机是与有限群的表示理论、矩阵因式分解的研究以及在代数几何中的应用，包括加权投影线。

Representations of algebras are studied from various directions, involving monoidal and triangulated structures. One goal is to describe in terms of generators and relations representation rings and monoids of projections functors for representations of quivers. Another goal is to classify thick and localising subcategories of triangulated singularity categories. This involves continuous and discrete parametrisations, reflecting the existing monoidal or combinatorial structures. The choice of singularity categories is motivated by connections with the representation theory of finite groups, the study of matrix factorisations, and applications in algebraic geometry, including weighted projective lines.