[maximilian]

Wow, 16 is sooo old. I'm sure plenty of great mathematicians didn't know what they wanted at 16 either.

25 is pretty young still I think. Thats only 3 years into a PhD, so really I wouldn't expect anyone (except super geniuses) to make many achievements by that age. I'd say 30 between 27 and 30 seems like a pretty solid age, as you've reached a peak with your PhD and you're still young and inquisitive and hopefully humble.

[mark-t]

Ask around. 16 is quite late to start taking math seriously and end up at the very top. He wasn't doing math for fun. He wasn't asking himself mathematical questions. He wasn't doing math competitions. He had no knowledge outside of what was taught in school to students at his grade level. This is freakishly uncommon, but he shows it's possible.

25 is young. He is just a few years into his PhD, and he hasn't made many major achievements (he's been a grader for the IMO, written one paper that's still being refereed for publication). The point is that he's noticeably better than I am at almost every branch of mathematics except combinatorics, and he had no interest until he was 16.