I've heard that at least one mathematician answers this by saying that they are a tennis instructor. It sets things up to avoid the regular 'I'm terrible at math' and 'I can't even balance my checkbook' replies. Somehow, I seem to have stumbled into a section of math that I think is easy to explain to people:

So, I do topology. You know that in geometry you study shapes. And you have some notion of when two shapes are "the same", that is, if you can rotate them and shift them around and maybe flip them over, and they line up. But that's all you are allowed to do, these few rigid motions. In topology, we allow more things to be the same. The standard analogy goes: if you made those circles and squares and triangles in geometry using a piece of stretchy string or something, you could deform the string until you got all three of those shapes. So now we decide that they are really all the same. The question now becomes, "how can you tell when two things are the same?", and this is the question topologists try to answer.

If I want to go further, and the person is still listening, and I feel like they remember what polynomials are (look like), I might mention that within topology is the field of algebraic topology. Here, the way you distinguish shapes is you look at them, and assign some sort of more rigid (algebraic) object to them. The idea being that whenever two shapes are the same, in the loose sense of topology, then there is some correspondence between their associated algebraic objects. For example, in the study of knots, you look at a knot and write down a polynomial (by following some particular rules). Two pictures of a knot could look different, but if you go through the process of writing down the associated polynomials, you'll get the same answer.

Finally, if the person is still listening, I might even introduce the calculus of functors (without saying functor). I make sure they remember Taylor polynomials from calculus. It was this process of getting the best approximation to a given function by using polynomials. The idea being that somehow polynomials are easier to understand than your average function, so if you can get a good enough approximation to your crazy function using polynomials, you're in good shape. Well, it turns out that you can define what it means to be 'polynomial' and 'the best polynomial approximation' in other contexts. And so I'm supposed to be actually computing the best polynomial approximations for this one particular example. Heavy on the supposed to be.

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