$1 Million Millennium Problem #5
The Poincaré Conjecture
~ A problem that is over 100 years old ~
This problem is centered on topology. Analogies are helpful. What are the mathematical differences between, let's say, an apple and a doughnut? A rubber band stretched around an apple may be shrunk down to a single point without tearing the apple or leaving its surface. On the other hand, a rubber band stretched around a doughnut, has no way of shrinking to a point without breaking either the rubber band or the doughnut.
The Poincaré Conjecture asserts that the shrinking rubber band idea sets off four-dimensional apples. This quirky reasoning also suggests that mathematically, a doughnut is the same as a coffee cup.
This is a simplified explanation offered strictly to let you visualize the problem.
See Keith Devlin's "The Millennium Problems: The Seven Greatest Unsolved Puzzles of Our Time, Basic Books, 2002.