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