Great explanation. I went ahead and got your book. But I got lost right around where you explain what an exotic sphere is.
What do you mean my jerks and jumps? I asked my math major friend and couldn't quite understand the explanation. He said smooth means that "all of it's functions have derivatives in all orders." How does that relate to jumps and jerks?
What does "not diffeomorphic" mean?

Man I wish more Math was explained like this!! It sounds so interesting.