Hironaka's resolution of singularities

Hironaka's resolution of singularities is a mathematical process that transforms complex geometric shapes with "rough" or "irregular" points into smooth, well-behaved forms. Imagine carefully "smoothing out" kinks, folds, or sharp corners on a surface without changing its essential shape. This method applies to abstract algebraic objects called varieties, turning them into nicer, smooth versions. The key achievement is proving that such a smoothing process always exists in characteristic zero (like real or complex numbers), allowing mathematicians to better understand and study the fundamental structure of these shapes in a systematic and consistent way.