Explanation for Poincare's Conjecture

GoodOmens

A student of mine in AP Calculus mentioned to me that Grigori Perelman is going to be awarded one of the Clay Mathematics Prizes for his work on the Poincare Conjecture. I had missed that news and was happy to hear about it. Some of my other students are interested in finding out more about it as well.

I'm trying to figure out how to explain it to them. These are very advanced high-school math students, but none have ever experienced any sort of topology before (at least, not that I know of). I've barely done any topology myself, and that was more than a decade ago so I've forgotten all of it, so I'm quite baffled about the conjecture and its proof.

Anyone have any ideas about how I can present this to my class in a way that they're likely to understand without totally watering it down?

scrivenerjones

Awesome idea. I have no stomach for topology so I can't really offer much advice, though. I would maybe hit them with the Science writeup, and maybe some highlights from the New Yorker profile on Perelman.

shadydentist

Explain it in fewer dimensions first. i.e.:

"Imagine a sphere. Any closed loop on the surface of a sphere can be continuously tightened into a point. Now imagine some arbitrary 3-D shape. If all closed loops on the surface of this shape can be drawn into a point, then the shape can be represented as a sphere without losing any points on the shape, or any of the relationships between the points."

The wiki article does a pretty good job.

GoodOmens

I suppose that's the part, shadydentist, that I don't get myself. My topology-fu is weak, so bear with me a second. Think of a barbell shape, essentially 2 spheres connected by a bar. Put the closed loop on one of the spheres, so that it surrounds the connecting with the bar (so it would have to be on the "inside" surface of the sphere, facing the other one).

If that loop closes, it will eventually squeeze onto the bar, but then it would stop, right? Or does it, instead, slide along the bar to the other spheres and finish its contraction to a point there? My explanation is not great, but if I'm understanding things correctly the second option would apply, meaning that the barbell and the sphere are, topologically, the same. Am I good so far?

If, on the other hand, you drill a hole through the sphere and put the loop on the inside surface of the hole, it can never contract to a point, so that's not a sphere-equivalent anymore.

I never "got" topology on a deep level, so I'm afraid I'm coming off as a blithering idiot here.

shadydentist

GoodOmens wrote: »

I suppose that's the part, shadydentist, that I don't get myself. My topology-fu is weak, so bear with me a second. Think of a barbell shape, essentially 2 spheres connected by a bar. Put the closed loop on one of the spheres, so that it surrounds the connecting with the bar (so it would have to be on the "inside" surface of the sphere, facing the other one).

If that loop closes, it will eventually squeeze onto the bar, but then it would stop, right? Or does it, instead, slide along the bar to the other spheres and finish its contraction to a point there? My explanation is not great, but if I'm understanding things correctly the second option would apply, meaning that the barbell and the sphere are, topologically, the same. Am I good so far?

If, on the other hand, you drill a hole through the sphere and put the loop on the inside surface of the hole, it can never contract to a point, so that's not a sphere-equivalent anymore.

I never "got" topology on a deep level, so I'm afraid I'm coming off as a blithering idiot here.

You're right. For the barbell example, any closed loop can be "slid" onto the ends of the spheres, closing to a point. With the hole drilled through the sphere, there are some loops that cannot, so it is not sphere-equivalent.