Poincaré’s Conjecture’s been in the news quite a bit recently, but I’ve yet to read a description of the problem that was clear enough for me to understand. As near as I can figure, Heni Poincaré theorized (in layman’s terms) that any object without a hole in it can be distorted into a sphere that has the same surface area and volume. Am I close?
You’re pretty close, actually. To clarify a couple of things:
First, “without a hole in it” has a more precise mathematical sense, namely that any closed path in the space can be continuously deformed to a point. As an example, let’s look at two two-dimensional manifolds, the surface of a sphere (usually called the two-sphere) and the surface of a donut (usually called the torus.) If we had a rubber band stretched out on the surface of the two-sphere, we could move it around, keeping it in contact with the surface, in such a way that we could scrunch the rubber band down to a single point. However, this is not the case for the torus: If we put the rubber band on the surface so that it wrapped entirely around the torus, or went through the hole at the center of the torus, then there would be no way to contract the rubber band to a point without breaking it or lifting it off the surface.
You might be asking why we want to deal with this “non-contractability” stuff, when it seems easier to just count the holes — the two concepts seem intimately related, after all. The answer is that when you deal with more complicated spaces (see below), we might not be even able to visualize them in three dimensions in order to count the holes, whereas the notion of closed loops is more straightforward to generalize.
Second, in the above example, the space we were dealing with were two-dimensional. The Poincaré conjecture, however, deals with three-dimensional spaces, which are pretty darn hard to wrap your head around — in some real sense, they would have to “bend around” in a fourth dimension, but our primate brains are pretty bad at visualizing four dimensions, having evolved to do silly things like recognize food sources and flee from predators instead. Suffice it to say that there is an analogue of the two-sphere for any number of dimensions, and that the original Poincaré conjecture deals with the three-dimensional version. (As an aside, the Poincaré conjecture in even higher dimensions turns out to be even easier to prove: it was done for five dimensions and higher in the '60s, and for four dimensions in the '80s.)
Hope this helps!
This page talks pretty generally about Perelman’s work, and links to a number of explanations of Poincaré, including one in the form of a short story.
To add to the previous explanations: the surface area and volume are not considered when comparing surfaces topologically.
Also, you cannot conserve both surface area and volume. The sphere you end up with can have either the same surface area or the same volume, but not both (unless you started with a sphere).
Is it necessary that either be preserved?
No. I was just addressing the OP’s implication that they were.