01 · A life in mathematics
The person behind the work
Shigefumi Mori was born in Nagoya, Japan, and moved to Kyoto University’s Research Institute for Mathematical Sciences in 1990, the year of his Fields Medal. His work reshaped the classification of algebraic varieties, especially three-dimensional ones. Extremal rays identify directions along which certain curves can be contracted, while flips replace problematic contractions by better-behaved birational models. Mori proved the existence of three-dimensional flips, a decisive step in the minimal model programme. Rather than classify every equation separately, the programme seeks simpler representatives related by rational transformations. Its success requires controlled singularities and can lead to a Mori fibre space as well as a minimal model.
The contribution, explained
Different polynomial equations can describe varieties that agree away from lower-dimensional subsets. Birational geometry studies this shared structure instead of insisting that every point remain unchanged. The minimal model programme uses the canonical class to detect curves whose contraction can simplify a variety. Sometimes a direct contraction creates an obstruction; a flip replaces the local configuration with another birational model that lets the process continue. Mori’s existence theorem for three-dimensional flips made a crucial operation available. The outcome can be a minimal model or a Mori fibre space. Controlled singularities are part of the method, not an accidental failure of smoothness.
Why it matters
Mori’s methods turned the classification of threefolds into a systematic programme, providing operations and structural principles that became central to higher-dimensional algebraic geometry.