I think it's nice to be able to give some knowledge of further courses in math, of a very "who's who in the zoo" sort. By which I mean descriptive, not axiomatic, and not even a detailed or working description. Just some extremely light and vague concept to attach to a word. And then let it go.
Not sure why you are pushing this guy, Richard Borcherds. I've never heard of him. I trust you that he's a big wheel in some way and interesting to people further advanced in their studies. But I think it's "A Bridge Too Far" for high school. Be a little wary of sharing your passion, what interests YOU, with what is good for your students. I think you're better off showing them the Horizon video on Andrew Wiles.
https://www.dailymotion.com/video/x223gx8 (probably copyright violating, but you can either buy a physical copy of the video or license or...just decide to steal it.)
In terms of awareness of math topics fields, I actually think what's closer to interest to them is something that talks about the terms and how they connect to school courses, further down the line. Not so much, where's active research going on. (Yeah, I get that you probably are interested in coal face of research...but remember what I said about pushing your interests versus thinking about audience.) We get questions here all the time about "what comes after Calculus BC" (US). Showing a lack of understanding even of 3rd semester calculus and ODEs, that most engineers go on to take a year later.
I think for AA, just knowing that it's like high school algebra but more...uh...abstract is kind of enough. Tell them that it's a core course that math majors take in undergrad, but engineers and scientists almost always don't...don't need it. And that it's related to how they proved that 5th level and higher polynomials don't have a "quadratic formula"-style solution. [This connects to high school algebra and teases that the topic actually did something.] I include GT within AA. You really don't need to get complicated enough to differentiate.
And then, let it go. Teach them the pre-calc they need to know. Honest, even just knowing the (names of, not really even the content) basic future math courses in undergrad for science/engineering (who are much more numerous than math majors) and math majors, is an above and beyond enrichment. Then get back to teaching the actual topic you're supposed to.