Applications of homological category theory to algebraic geometry and representation theory

(I)三角范畴上的同调结构：三角范畴上的扭偶同时推广了代数几何中一个著名的经典概念“t-结构”和最近在代数表示论中提出的“簇倾斜子范畴”。在研究扭转对上的代数结构时，我们给出了一个构造，它将一个交换范畴与每个扭转对联系起来，推广了t-结构的核心和簇的倾斜商。(Ii)与有限群相关的双变函子：我们从范畴的角度研究了有限群上的一个函子双变量，特别是在Mackey函子理论中起交换作用的‘Tambara函子’。作为交换环理论的二元类比，我们建立了Tambara函子的基本运算，对应于理想商、分式、多项式和本谱。

(I) Homological structures on triangulated categories: Torsion pair on a triangulated category generalizes simultaneously ‘t-structure’, a well-known classical notion which is also important in algebraic geometry, and ‘cluster tilting subcategory’, which is formulated recently in representation theory of algebras. While investigating algebraic structures ontorsion pairs, we have given a construction which associates an abelian category to each torsion pair, which generalizes the heart of a t-structure and the cluster tilting quotient.(II) Bivariant functors associated to finite groups: We are investigating ‘Mackey functor’ ‐a functor bivariant on afinite group, and especially ‘Tambara functor’, which plays a role of commutativering in Mackey functor theory, from a categorical point of view. As a bivariant analog of commutative ring theory, we have formulated fundamental operations on Tambara functors, corresponding to ideal quotient, fraction, polynomial and primespectrum.