Applications of homological category theory to algebraic geometry and representation theory

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(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.