This minicourse is an introduction to birational geometry from a local perspective. We will discuss the singularity theory appearing in minimal model program. We start from the definition of singularity of pairs (mld, klt, LC, so on) and what is the minimal model program. The goals of this minicourse are - Shokurov's theorem (ACC conjecture + LSC conjecture imply termination of flips), and - My recent work (a counterexample of PIA conjecture). This is Part 2.