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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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-1: This is not really a question. – Jyotirmoy Bhattacharya Oct 15 '10 at 15:52
It's not phrased with a question mark, but it’s something to which math.SE readers may be able to give illuminating and interesting answers, I think; it’s surely as much a question as most other “big-list” questions are? eg…,…, Or actually, maybe the title would be better with a question mark after all… – Peter LeFanu Lumsdaine Oct 15 '10 at 16:05
@Peter: There should be a check-box to the bottom right just outside of the text box; I'm marking my own response as community wiki in the meantime. – Arturo Magidin Oct 15 '10 at 16:17
@Arturo: that checkbox was what I was expecting (from MO/SE), but I'm darned if I can find it! I guess it could be either a lack of rep, or a browser issue (I'm using the latest Firefox, under Snow Leopard on a MacBook). – Peter LeFanu Lumsdaine Oct 15 '10 at 16:32
@Peter: Hmmm... It was there for my response; it's not there if I try editing your question, and it does not seem to be there if I try to "Ask a new question". (I'm using Firefox under Ubuntu, but as I said, the checkbox was right there when I was editing my own answer). – Arturo Magidin Oct 15 '10 at 16:38

Combinatorics: Counting without your fingers

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There's the old "Topology: Geometry on a rubber sheet."

For both Category Theory and Universal Algebra, I also like The Esperanto of Mathematics/Algebra (except for the negative connotation of "nobody actually uses it"...).

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I must disagree with that. Geometry is distinct from topology. And besides, how will you call geometry of curved manifolds? – Willie Wong Oct 15 '10 at 18:53
@Willie Wong:It's meant to be an "elevator pitch", not a complete and 100% accurate description. And I did not originate it, I am merely reporting it. Yes, of course it is distinct, but then, geometry is not "geometry on a rubber sheet", so "geometry on a rubber sheet" is distinct form geometry. – Arturo Magidin Oct 15 '10 at 20:26

Category Theory: A birds-eye view of mathematics; a framework for investigating how different areas relate to one another.

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Algebra: The mathematics of structures.

Set Theory: The main foundation of mathematics.

Logic: The main foundation of set theory.

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Calculus: the mathematics of motion and change.

(Not original, of course!)

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Geometry: The ticket to Plato's Academy. :-)

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Statistics: Making decisions under uncertainty

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Numerical Analysis: The search for fast, accurate, and stable algorithms for finding numerical solutions.

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Topology: Why you can't turn your shirt inside out while wearing it.

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