Categories in geometry and physics: mirror symmetry, D-branes and Landau–Ginzburg models

Triangulated and derived categories naturally appear in different fields of mathematics: algebra, geometry, topology. Lately they acquired significance in such a branch of physics as string theory where these categories appeared as categories of supersymmetric D-branes in sigma-models and Landau–Ginzburg models. Arising categories are natural and powerful invariants of the corresponding geometric objects that allow to connect apparently incomparable objects from different branches of mathematics and physics. On the one hand in this talk we will attempt to make a survey, on the other hand we are going to present some new results from the theory of triangulated and derived category that have natural applications to geometry and physics.