What does the psychological examination in police academy screen for?

The Poincaré conjecture depends on the almost mind-numbing problem of understanding the shapes of spaces: mathematicians call it topology. Think about it this way-

It might help us to understand the space we are currently living in, but how do you understand a 3-sphere as being a manifold of a 4-sphere? What does a four-dimensional sphere look like? It's too difficult to imagine. The surface of the earth looks like a flat plane anywhere we stand on it because the earth is a large 3-sphere and relative to its size we are standing in an infinitesimal neighbourhood that looks flat to us. The earth might be a tiny surface on some weird object that is a 4-dimensional sphere. This is the super layman terms of what the conjecture says. So it may help to understand physics better.

We did not know that earth was spherical with our eyes until NASA sent spacecraft and took photos of planet earth.

But,

If you can go back to ancient Greece, the Greeks lived on what seemed to be like flat earth but by using mathematics they were able to figure out that earth is, in fact, spherical and they even calculated the diameter of the earth with impressive accuracy. So, using mathematics you can step out of the world you live in and see what it would look like.

Poincaré conjecture theorem helps us, as creatures living in the 3D world, figure out and understand the shape of the universe by using mathematics. Maybe the shape of the universe is like an inside of a sphere or a like a tube where we can keep going on and on. Poincaré came up with a method that can be understood in those terms.

The surface of an apple is simply connected. But the surface of a doughnut is not. How do you start with the idea of simple connectivity and then characterise space in three dimensions?

That’s about the Poincaré conjecture.