János Kollár:
Consider
          $x^{4} + 2x^{2}y^{2} + y^{4} = x^{2} - y^{2}$.
It looks like a nice figure of eight. It goes around and comes back here and back, OK.




What happens if I add to it a plus zero point one?

          $x^{4} + 2x^{2}y^{2} + y^{4} = x^{2} - y^{2} + 0.1$

Turns out I get something that hugs the figure eight from the inside. So I get something like this. And I also get the same thing at the other side.
Instead of being made out of just one piece of matter, its suddenly made out of two pieces.



Chris Lincoln: And the blue lines don't intersect.



János Kollár: What happens if I subtract zero point one from it?

          $x^{4} + 2x^{2}y^{2} + y^{4} = x^{2} - y^{2} - 0.1$

Well then again I get something that will hug the figure 8 but this time it will hug it from the outside. It doesn't have this part where it intersects.


You can think of it as a circle where you're trying to bring the top and the bottom together.