> Hironaka solves a major problem concerning the resolution of singularities on an algebraic variety.
Essentially, sometimes the varieties (which are geometric objects like curves, surfaces, etc.) studied in algebraic geometry are singular: they might have singularities, like nasty self-crossings or sharp edges. Hironaka's result lets you take a "bad" variety and "resolve its singularities", giving you a good (nonsingular) variety which you can work with instead.
This is in "characteristic zero", i.e. over fields like the real or complex numbers. We also have fields of positive characteristic, e.g. the integers modulo any prime number. Over such fields, I understand that this is a much harder problem to solve.
The aforesaid Heisuke Hironaka is 86 now.
In March of this year, he published a (purported) proof of resolution of singularities in positive characteristic.
The /r/math thread has some good explanations.
https://www.reddit.com/r/math/comments/6aqwbo/hironaka_publi...