Math and non-specialists:

I got a question at one of the interviews that went something like this:

"So what is algebraic topology, anyway? [I hesitated for a second] In less then a hundred words."

How should I have answered that? My response was something like: "Well, topology takes a set and looks at open subsets in it. So, like, in the real numbers, you could look at open intervals. An algebraic topology then is like looking at these open sets on surfaces [pause] like on a sphere."

Is there some better way of explaining this to someone whose background is probably in engineering? Nothing I said is exactly wrong, though I don't think that that's how an algebraic topologist would describe what he does either.

This brings up a problem that I've had before. How do you explain math to a non-specialist? This is hard for every field, I think, but particularly so for math for two reasons. One is that math is so interconnected and there are so many terms defined (and these definitions are so precise) that it is difficult to explain something small to a nonspecialist. That is part of the problem with expaining algebraic topology, right? "Well, topology is the study of topological spaces, which is a set plus a topology, which is a collection of subsets with the following properties..." No one wants to hear that. Another example, I think I know a fair amount of algebra for an undergrad. I took a pretty advanced undergrad course here, and worked in roughly that field for a summer. But when I look at a textbook for a second-year (say) grad class in algebra, I have no idea what many of the terms mean. To read this, I have to go look up all of these things. There are of course ways of getting around this, though often those ways aren't mathematically precise, and this is definitely something that people trying to explain math have to worry about.

Another problem with explaining math is that many people believe that they aren't good at math and don't believe that they can understand what you're saying. Even engineers, who are comfortable with numbers, often spend a lot of time talking about how they hate math/ didn't understand it/didn't do well in it. If you believe that you can't understand something, you won't.

As I said, I know that the first problem at least is something that people in many fields face and I may just believe that it's worse in math because that's where I know the most of these terms. Does anyone have any thoughts on this?