01 · A life in mathematics
The person behind the work
Heisuke Hironaka was born in Yū, now part of Iwakuni in Japan, in 1931. His 1970 Fields Medal recognised the resolution of singularities of algebraic varieties in characteristic zero, proved in arbitrary dimension. He was then a professor at Harvard. The theorem showed that singular geometric objects could be related systematically to nonsingular ones through carefully chosen modifications. Hironaka’s achievement became a foundational tool in algebraic geometry and complex geometry, enabling arguments that replace difficult singular spaces with smoother spaces while retaining a controlled relationship to the original object.
The contribution, explained
A polynomial equation may define a space with a crossing, cusp or more complicated singularity. A blow-up replaces a chosen centre with extra geometric information that separates directions previously merged there. One modification need not finish the job: the challenge is choosing centres and proving that a controlled sequence eventually produces a nonsingular model. Hironaka established this in characteristic zero and arbitrary dimension. Away from the singular region, the construction preserves the intended birational relationship, providing a smooth space mapped back to the original rather than simply erasing the troublesome points.
Why it matters
Resolution of singularities became a basic reduction tool in algebraic and complex geometry, allowing smooth methods to address spaces originally defined with singularities.