## Devlin's Angle |

The image says it all. When SCIENCE magazine declares that the proof of a theorem in mathematics is the breakthrough of the year in all of science, you know that something special has occurred.

We've had a double wait for this breakthrough. Henri Poincare first formulated his now famous conjecture just over 100 years ago, in 1904, and despite several heroic attempts to prove it, the puzzle remained unsolved until late 2002 and through into 2003, when the somewhat reclusive Russian mathematician Grigori Perelman posted a series of three papers on the Internet that claimed to outline a proof of the Thurston Geometrization Conjecture, which was known to yield the Poincare Conjecture as a consequence.

That was the first wait. But another was in store. Such was the intricacy of Perelman's argument, compounded by the highly compressed manner in which he presented it, that is took over three years of effort by various groups around the world before the consensus agreement came in: he had indeed proved the Poincare Conjecture. (The experts are still unsure about Perelman's proof of the Geometrization Conjecture.)

What made it a major *science* story,
I think, is only in part the magnitude of
the advance within mathematics. Proofs of
major theorems are often of interest to the
entire scientific community, but they rarely
earn the accolade of "the scientific breakthrough
of the year." (Off hand, no previous example
comes to mind, but perhaps some readers have
a better memory than I for that kind of thing.)
I suspect that the editors of SCIENCE gave
their vote to this particular theorem because
it lies at the heart of the domain (if not the
practice) of physics.

Of course, faced with having compared apples with oranges, the editors at SCIENCE did not disclose their criteria. But, in his editorial, editor-in-chief Donald Kennedy does say what appealed to him about the result: "The analysis [described in SCIENCE] struck me as a fascinating exploration, full of metaphors suggesting a multidisciplinary dimension in Perelman's analysis. He first got interested in Ricci flow, a process by which topological regions of high curvature flow into regions of lower curvature. He also identified a quantity, which he called "entropy," that increased during the flow, providing a gradient. Tight spots in spatial connections block the application of these rules to dimensions higher than two, so Perelman dealt with these through "surgical intervention." This story is rich with borrowings: from fluid mechanics, thermodynamics, and even surgery! It's hard to deal with a three-dimensional object in four-dimensional space. Perelman's solution is a stunning triumph of intellect."

Most of the techniques Kennedy alludes to were not developed until long after Poincare had died, but perhaps we should not be surprised that his conjecture required such a rich repertoire of physically-based techniques for its solution. For, although Poincare is generally thought of as one of the greatest mathematicians of all time, his interests were as much in physics as in mathematics, and he came within a whisker of formulating relativity theory before Einstein beat him to the punch. Although usually formulated as a pure mathematics problem about the classification of abstract manifolds, the Poincare conjecture is really about the structure of the space we live in, and the degree to which we humans, as prisoners within that space, unable to break free and observe it from the outside, may nevertheless determine some key features of its structure.

Poincare proposed a procedure which he believed would enable us to make such a determination. It is not a practical procedure, and was never intended to be viewed as such. Thanks to Perelman, we now know that Poincare's proposed method is indeed sufficient to determine the structure of space from the inside - in theory. But viewed from the perspective of physics, it's really an epistemological result, which tells us that the structure of space is in principal knowable, even though the procedure Poincare proposed is not a practical one.

To appreciate what Poincare proposed, consider
the analogy of a two-dimensional ant living on
a surface (actually, it lives *in* the
surface). From the outside, looking down on
the ant's world, we see the creature's universe
as a surface. It could be (the surface of) a
sphere, a torus (the surface of a ring donut),
a two-holed torus, etc. From our viewpoint, we
could see which it is. But would the ant
living in the surface have any way of knowing
the shape? For each of those possible shapes,
the ant has exactly the same freedom of movement,
forwards and backwards or left and right.

Poincare asked the analogous question of our 3-D universe. We can, at least in principle, travel forwards and backwards, left and right, up and down. But as with the ant, this freedom of movement does not tell us the shape. Since we can't step outside our universe and look down on it, is there any way that we could determine the shape of our universe from the inside?

The Poincare conjecture says that the following procedure will do the trick. Imagine we set off on a long spaceship journey. (This is not how Poincare formulated the conjecture; he used the formal mathematical language of topology and geometry.) As we travel, we unreel a long string behind us. After journeying a long while, we decide it is time to head home - not necessarily the same way we went out. When we return to our starting point, we make a slipknot in our string and begin pulling the string through. One of two things can then happen. Either the noose eventually pulls down to a point or else it reaches a stage where, no matter how much we pull, it cannot shrink down any further.

Poincare's conjecture is that if we can always shrink the loop to a point, then the space we live in is the 3D analog of the surface of a sphere; but if we can find a start/end point and a journey where the loop cannot be shrunk to a point, then our universe must have one or more "holes", much like a torus. (Essentially, what is going on is that our string "snags" by going around such a hole.)

In the case of our two-dimensional ant, it is easy to prove that such a procedure does indeed enable the creature to determine from the inside whether its universe is like the surface of a sphere or whether there are holes. But until Perelman came along, nobody had proved the same technique would work for us in our three-dimensional universe and allow us in principle to tell from the inside what kind of universe we live in. We now know that we can. In fact, the more general Geometrization Conjecture that Perelman's argument also established tells us we can in principle determine a great deal more about the shape of the universe.

As I mentioned already, although the Poincare conjecture is about a fundamental question of physics, it is essentially an epistemological matter. There is, as far as I know, no automatic consequence of the conjecture in terms of everyday physics. But the methods Perelman used to finally prove the conjecture seem likely to have major implications in physics. To push through his argument, Perelman had to develop powerful methods to handle singularities. And that may prove to be the start of a whole new chapter in mathematical physics. Stay tuned, scientists.

P.S. *Discover* magazine also listed the
proof of the Poincare conjecture as one of the
top 100 science stories of 2006, but they
ranked it as number 8. Their story led off with
a conjecture of its own: that the proof of the
Poincare conjecture may turn out to be the
number 1 math story of the entire 21st century.
I definitely agree. But then, I would.
I wrote the *Discover* story.

Devlin's Angle is updated at the beginning of each month.