When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of mathematics. It turned out not to be… so here’s the question I thought it was!
There are lots of times when one wants a quick way to explain to someone (family, students, teachers, colleagues) what some field is about, and (hopefully) catch their interest as well — perhaps to come and learn about it themselves, perhaps just to understand why you find it interesting, perhaps to convince them why they should give you funding to work on it…
The question linked nicely describes how a good such pitch should work:
This isn't as easy as it sounds. Imagine the user who will never read your FAQ and you have two seconds to grab their attention. It should be catchy but descriptive. It should be thoroughly clear but painfully concise. Make every... word... count.
A couple of extra thoughts: the level of technicality of the pitch should probably depend on how specialised the field is. You don't need to describe “algebra” or “calculus” to a professional mathematician; dually, you probably aren't trying to explain “homotopy theory of CDGA’s” to your friend who never took any maths courses (at least, not in one sentence). This isn’t about any level of maths in particular — whether in recreational maths or cutting-edge research, everyone can benefit from a good slogan sometimes. Posting a few examples myself, to get the ball rolling
I’m not sure to what extent guidelines/conventions for big-list community-wiki questions have solidified on this site, but a couple which work well on MO and SE are:
just one example (tagline) per posted answer, so that they can be voted up/down individually;
don't be shy of making near-duplicates if you think an idea could have been executed a bit better, nor of suggesting improvements to answers in their comments.
Related questions: Best intuitive metaphors for math concepts; Cocktail party math (at MO).
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