Applications of homological category theory to algebraic geometry and representation theory

(I)三角化范畴上的同调结构：三角化范畴上的挠对同时推广了“t-结构”，一个在代数几何中也很重要的经典概念，以及最近在代数表示论中提出的“簇倾斜子范畴”。在研究扭对上的代数结构时，我们给出了一个将阿贝尔范畴与扭对联系起来的构造，它推广了t-结构的心和簇倾斜商。(II)与有限群相关的双变函子：我们正在研究“麦基函子”-一个在有限群上的函子双变，特别是“坦巴拉函子”，它在麦基函子理论中扮演着交换的角色，从范畴的角度来看。作为交换环理论的双变模拟，我们制定了Tambara函子的基本运算，对应于理想商、分数、多项式和素谱。

(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.