Now What

My favourite is the one about elliptic curves. In short, an elliptic curve is a curve in a plane that looks like this one:

While most of the points on this curve will typically have coordinates in real numbers (eg. square root of two, which is not nice at all) there are usually a few special points with (very nice) rational coordinates that we are interested in.

Furthermore, there is a way to add these points: given two rational points on the curve, consider the line through them. The line intersects the curve in a third point which is assigned to be the sum of the previous two (well, in fact it isn’t - because we also need to reflect the point in the horizontal axis for things to work well).

It turns out that this addition, despite defined by means of geometry, satisfies some standard algebraic properties and the resulting structure contains a lot of valuable information (at least from mathematical point of view). Indeed, elliptic curves have significant applications in (public key) cryptography and can be used to establish a secure connection in a browser (eg. for purposes of online banking).

More generally, in case of elliptic curves we managed to solve two consecutive problems: understanding both an abstract object and its use in real world. This is a major success, since finding an application of a mathematical theory can apparently take up to a hundred years.

On a similar note, I can confirm that choosing how to apply mathematics in real life is not at all trivial. There are many options - some very obvious, some less obvious (and potentially the more interesting). In any case there are way too many to easily choose one. I only hope it will take me less than a century to do so.