michael_nielsen

Most scientific & mathematical disciplines I know of have results that educated outsiders can appreciate and go "wow" after a brief acquaintance, even without understanding the details. Does anyone know of such a result for category theory?

I've been searching for such a thing for a long time, and it's nice to see the responses here. But I'd ask for something similar for Group Theory. What is your "Brief, nice thing that makes people go Wow!" for Group Theory?

I'm not a group theorist, nor a mathematician, so I'm the wrong person to ask. I do find the Solovay-Kitaev thm wonderful: informally, if you take products of elements in (many) compact Lie groups, they fill in the group exponentially quickly, & surprisingly near to uniformly

A picture is better to explain what this means. And probably best explained concretely using, e.g., rotations and a few similar examples.

I don't quite understand it well enough to make the attempt, but I'll bet there's a mindblowing 5-minute explanation of what Gromov's theorem on groups of polynomial growth says.

Hmm. I'll have to think about that. I certainly don't know enough to be able to see immediately how or why that's interesting, so I'd have to dig into it. One result I did use to impress some 16 year olds was that in a group of even order there is an element a with a^2 = 1.

That's in a completely different league, but it served to show them something that without group theory wasn't obvious, but with group theory was completely obvious. That hinted at the power of the topic, but took about 2 minutes from nothing.

The things you're talking about would not be accessible to the youngsters I deal with, so I'm looking for something they wouldn't be able to show, but which becomes obvious with just basic group theory.