chakravyuha

Poincaré conjecture is a mathematics problem that (as I understood it) deals with the structure, nature and properties of the three dimensional space that we live in. This region of three dimensional space is referred to as a 3-manifold. The fundamental structure of the shape we live in is a difficult thing to determine and comprehend because we are situated in the space. Henry Poincaré was a French mathematician who, nearly a 100 years ago, proposed a way to understand how space would look from the outside, looking in this three dimensional space. The problem can be restated as trying to determine the shape of very large object in relation to a human, say trying to determine whether or not earth is flat or to trying to define what a hole is. The absence of matter surrounded by matter would probably be an inadequate mathematical answer for the latter!

His approach, as explained by Keith Devlin in a recent broadcast of Forum[1], was as follows. Assume that you start from some arbitrary point in the universe and travel for a very long time taking arbitrary turns, leaving a string in your wake all this while. After this rather long travel, where you were leaving a snake trail of string all the while, you proceed to return to the point you started from. You then proceed to pull the string together tight, holding both the ends of the string, trying to close the loop formed by holding both ends of the string in your hand. If you are able to pull this string till there is no loop and end up with a point then we can determine that the space that covered during the course of the travel is really, well, as unremarkable as it looks from the inside. It essentially has no definable characteristic shape in three dimensions. If, however, we are not able to tie the string to get a point, it’s whole different story altogether and the space you covered would look different from the outside and would have a definable shape. It’s almost like if were living in doughnut shaped planet, had traveled over the ring of the doughnut and then trying to close the loop. We would not be able to tie the string to a point in that case. This was the idea that Poincaré thought would help determine how space looks from the outside. So Poincarés conjecture would help answer question on the shape and nature of this space. Questions like what is the simplest 3-dimensional space, what are their properties, among others. Poincaré’s conjecture was in the news recently because a Russian mathematician named Grigory Perelman recently got awarded a very prestigious award in mathematics called the Fields Medal for providing a proof for this conjecture. A proof that essentially says that Poincaré’s idea was indeed correct and would work to determine the nature of 3-dimensional spaces.

What made this story very interesting was the human drama leading to Perelman getting the award. It involves a whole cast actors, among whom is a prominent scientist of Chinese origin, Shing-Tung Yau. Proving such a theorem is not a discrete step. There are a lot of intermediate steps involved, one research playing into the other, one insight being built upon and used by the other to come to the final conclusion etc. The latest edition of the New Yorker magazine has a lengthy feature on the events and the people involved in this story. Yau make a claim of playing a much larger role in the development of the solution, if not the claim of being the sole person solving it. His almost militant interest to get credit stems in large part due to his desire to cement his status as the leading scientist in his native China. Perelman himself is a rather singular character. He fits the stereotype of the obsessed scientist to the tee. He is ultra reclusive, lives in Russia with his mother and refuses to accept the award because he does not find any significance in it, purity of purpose being the only ideal. There is, however, near universal recognition of Perelman’s contribution in developing new and profound mathematical constructs that form the essential tools to construct and understand the proof.

Keith Devlin mentions that often its these mathematical constructs and insights that are probably as important, if not more so, as solving the actual problem itself. Consequences resulting from Poincaré conjecture have already been studied from all points so finally knowing that the theory does indeed hold, is not all that important. He says that the constructs developed by Perelman will prove to be invaluable in solving equations in physics that “develop singularities” when applied locally to small areas. (I cannot claim to understand what these are so here’s a link that talks about them). His constructs will help physicists in controlling these singularities. The steps involved in solving a problem of this nature are as important as the result itself. They lead to development of ideas, constructs and tools which themselves reveal hidden structures and truths that not only help evaluate a theory, but also lead to significant and possibly hidden truths and relationships.

To elaborate on this point, Douglas Hofstadter (who was the other guest on the program with Keith Devlin and who wrote that mind-bending book Gödel, Escher, Bach: An Eternal Golden Braid which is a potential subject of a post in itself) gave the example of prime numbers. Prime numbers are an oddity because they do not seem to fit any particular pattern. The existence of primes has been known for thousands of years. You can divide prime number in different classes in many different ways. Besides 2, all the prime numbers are odd numbers. Now, there are two types of odd primes. Those that are 1 greater than a multiple of 4 e.g. 41 which is 40 + 1, 40 being a multiple of 4, and those that are 1 below a multiple of 4 e.g. 43 which is 44-1, 44 again being a multiple of 4. Although there are infinitely many numbers in both classes, these two classes of primes have a fundamentally different nature. There is a very interesting connection that can be drawn between these two classes of primes and squares. It turns out that primes of the former class can always be represented as the sum of two squares. Taking the example of 41, the square would be 25 (5 squared) and 16 (4 squared). What’s more, they will always be exactly one pair of squares that will add up to this prime! And what about the other latter class? Well, for that class there is no pair of squares that will add up to the primes in that class. So there turns out to be this unexpected connection between squares, which are easy to think about and are predictable, and prime numbers which are so very irregular. In nature too there are so many patterns that we come across everyday, but rarely realize how they are so closely related with mathematical symmetries and relationships. The petals of some flowers are arranged in fibonacci symmetry, earth is the perfect shape for minimising the pull of gravity on its outer edges – a sphere (although centrifugal force from its spin actually makes it an oblate spheroid, flattened at top and bottom)[2], bees construct the beehive made up Hexagons because a hexagon fits most closely together without any gaps which maximise the use of space[2] and so on.

[1]. KQED Forum Fri, Aug 25, 2006. What Mathematicians Do
[2]. abc.net.au. Maths in Nature; Photos.

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