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The first Millennium Problem I will be exploring is the Poincaré Conjecture. In the early 1900s, a mathematician by the name of Henri Poincaré stated a hypothesis regarding a field called topology. Although many agreed with him, no one could actually prove that his conjecture was true for many decades. Therefore, one of the Millennium Problems became proving or disproving the truth of what became known as the Poincaré Conjecture.

Now, before you go and try solving this yourself, the conjecture was actually proven true by a mathematician named Grigori Perelman. Interestingly, Perelman actually declined the million-dollar prize, stating that it was unfair for he alone to be awarded for the collaborative work of him and the contributors before him, especially the work of a certain Richard Hamilton. In any case, he felt the recognition of the validity of his proof was reward enough, and his work has left a definite time-stamp in mathematical history.

Even though this Millennium Problem is already solved, I think it is valuable to explore what it is and why it became a Millennium Problem in the first place. My goal is to express the conjecture in easy-to-read terms so anyone can understand it, and with that, let's dive in.

What is this conjecture?

The Poincaré Conjecture itself is actually very nice and concise. Simply, it is the following:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

But, as simple as this definition is, the mathematical jargon can make it very difficult to actually understand or envision. Just to begin to make sense of the conjecture, you have to know what "simply connected", "closed", "manifold", and "homeomorphic" all mean. I personally prefer the more colloquial definition below:

Every finite 4D surface with no holes in it can be transformed into a 4D sphere.

Oh dear, 4D? The fourth dimension? Yep, that's right. This problem is in the fourth dimension. That's what was originally meant by a "3-manifold" or a "3-sphere". So if the problem is in the fourth dimension, how can we ever hope to comprehend the meaning of the conjecture? After all, our brains weren't really designed to think about anything higher than the three dimensions we've come to know and love.

Thankfully, we can make sense of the Poincaré Conjecture by making an analogy to a lower dimension, namely the third dimension. By looking at how the properties of "simply connected", "closed", "manifold", and "homeomorphic" all work in the third dimension, perhaps we can get a sense of how things work in a dimension higher. And we're going to do it with cookie dough.

It's just like cookie dough

Using cookie dough, we are going to illustrate what each of the four mathy terms in Poincaré's conjecture means.

A simply connected surface is essentially any surface without a hole in it. Below, the first batch of dough is simply connected, but the second is not.

Simply Connected

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Since this has no holes, it is simply connected.

Not Connected

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The hole means this cookie dough is not simply connected.

A closed surface is, for our intents and purposes, finite, meaning it does not go on forever and ever. All the cookie dough balls on the left are closed surfaces since each one has an end. On the other hand, the cookie sheet on the right just keeps going and going, and therefore its surface is not considered closed.

Granted, I wouldn't mind living in a world where cookie dough sheets extended infinitely.

Closed

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Finite in size, therefore closed.

Open

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Even though this has no holes, the sheet goes infinitely, and hence it is not closed.

A surface is a manifold if it is, roughly speaking, smooth. That is, if you zoom in onto a curved surface enough, it should eventually appear flat with no bumps. In general, fractal surfaces are not manifold since a fractal, by definition, always looks like itself when you zoom in.

Now, there is no such thing as a fractal cookie, but we can pretend like the cookie on the right is too rough to be a manifold surface. In other words, no matter how much you zoom in, it will appear rough.

Manifold

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If you were to keep zooming in, the surface will appear more and more smooth.

Not Manifold

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We can pretend that if you keep zooming in that the surface will always look rough. Therefore it is not manifold.

Finally, a surface is homeomorphic to another surface if one can be transformed into the other through squishing, pulling, and so forth. You only cannot merge surfaces, meaning you cannot close up holes. Regarding cookie dough, we can easily knead the cube on the left into the ball on the right; therefore, these two pieces of dough are homeomorphic.

From a Cube

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To a Sphere

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The Poincaré Conjecture says that if a surface is simply connected, closed, and manifold, then it is homeomorphic to a sphere. That is, we can transform any piece of cookie dough that matches the three stated characteristics into a ball of dough. If the dough fails to be simply connected, closed, or manifold, then it cannot be turned into a ball.

The Millennium Problem, then, was to somehow prove that this property holds for four-dimensional objects. That is, if you had 4D cookie dough, you could knead it into a 4D ball if and only if it has no holes, is finite, and is smooth.

How was it solved?

So far, this is the only Millennium Problem to be solved, and it was solved by a mathematician named Grigori Perelman in 2002. As one might expect, the math used to prove the conjecture is ludicrously complex. However, I will try to give a simple explanation here.

Essentially, Perelman used a process called Ricci Flow, a geometric procedure invented by another mathematician named Richard Hamilton. This procedure is an official formulation that, over time, smooths out a surface into a more uniform shape. Put more simply, it turns a blob into a sphere over a long period of time.

There were some issues with Ricci Flow, however, on specific kinds of surfaces, and so Perelman invented techniques for circumventing those exceptions, among other things. You can find out more about Ricci Flow and its relationship to the Poincaré Conjecture on Numberphile.

Significance

While the proof of the Poincaré Conjecture will not lead to a cure for cancer or what have you, its truth does say a great deal about the mathematical field of topography and can lead to a number of other discoveries in the field. This conjecture is basically an identity statement, essentially another way of saying 1+1 = 3-2 = 1, but in topography.

Hopefully you enjoyed this article. Be sure to leave a message below in the comments section if you have any lingering questions! More Millennium Problems will be coming soon.

References

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