I see that a lot of people have responded to the first question about modern math savyiness, so that makes me nervous about responding to the second question about what mathy idea interests me.
I think fractals are super cool, as well as repeating, never ending patterns like that.
I know that fractals aren't super mathy, but technically they are. Fractals are pretty, and it's difficult to imagine that they go on forever and ever! It reminds me of trying to find the smallest/biggest number. If you take one, and divide it by two, it is one half. If you half a half, it's one fourth. If you half one fourth, it's one eighth. If you half one eighth, you get one sixteenth. And it continues to become a smaller and smaller number, just like a fractal. Cool stuff :)