Phase 1: Introductory to Abstract Algebra Constructions

• Homological Algebra: Ring and module, exact sequences, Hom and tensor, injective/projective/flat, free resolution, derived functors (Ext and Tor).
• Commutative Algebra: Local rings, Spec,
• Category Theory: Abelian Category, Localization construction, Triangulated Category

Phase 2: Introductory to Algebraic Geometry

• Algebraic Geometry: Varieties, Scheme, Sheaves, and cohomology theory
• Cohomology and base change theorem

Phase 3: Introductory to Derived Category

• Category of sheaves
• Derived Category in Algebraic Geometry
• Semi-orthogonal decomposition
• Fourier-Mukai transforms

Phase 4: Advanced Topics

• Fourier-Mukai transforms on Abelian varieties
• Derived Category of Toric Varieties
• Bridgeland stability conditions