When someone who doesn't know much about math asks me what I study, this is what I tell them. Of course, in person there are a lot of hand gestures.

I study a kind of math called topology. Topology is like geometry, because we study shapes, not numbers.

In topology, unlike geometry, we don't care about the lengths or sizes of things. It's as if the shapes we study are made of clay: they can be mushed around, bent, stretched, and that's all the "same thing" to a topologist.

For example, a bowl is the same as a plate, because you can just bend the walls down. And a plate may as well just be a solid ball.

If you have a coffee mug, the "mug" part is just like a bowl, so you can mush that down, and you're left with the handle. But if you tear things apart or glue things together, that's changing things, from the view of topology. So you can't break open the handle or seal it up.

So the classis math joke is that a topologist thinks that a coffee mug is the same as a donut: they both have one hole.

That might be surprising: that a coffee mug and a donut are the same. And mathematicians don't like to be surprised. So we come up with tools to tell things apart. Ways to say "No matter how much you bend and stretch Thing 1, as long as you don't break it apart or glue things together, you will never get Thing 2."

In fact, the tools I'm referring to are all just fancy ways of counting the number of holes in things. So that's what I study: fancy ways of counting the number of holes in things.