Poincare’s conjecture (1904) is related to the nature of three-dimensional manifolds, or spaces. It is in many ways deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
(The conjecture is fundamental to topology, and incidentally one of the $1 million Millenium Prize Problems awarded by The Clay Mathematics Institute of Cambridge, Massachusetts).
In three dimensions, it is hard to discern the overall shape of something; so for example, when we envision the surface of a sphere, we are really seeing a two-dimensional object embedded in three dimensions. To understand the problem it is therefore easier to start with two dimensions.
The topology of two-dimensional manifolds or surfaces has been well understood since the 19th century. In fact, there is a simple list of all possible smooth compact orientable surfaces. Any such surface has a well-defined genus g ≥ 0, which can be described intuitively as the number of holes; and two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have
the same genus.
This means that all such two-dimensional surfaces are classified by a single integer, g. A surface of genus zero is topologically a sphere (an apple), a surface of genus one is a torus (a doughnut) and a surface of genus two is, well a dougnut with two handles (or whatever you want to call it), and so on.
In the case of two dimensions, it is easy to see how the conjecture should be understood: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
The corresponding question in higher dimensions is much more difficult. The most basic example of such a manifold is the three-dimensional unit sphere, that is, the set of all points (x,y,z,w) in four-dimensional Euclidean space which have distance exactly 1 from the origin: x^2 + y^2 + z^2 + w^2 = 1.
A distinguishing feature of the two-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere, or that “you can’t lasso a two-sphere”.
In mathematical terms, the Poincare Conjecture can be phrased as follows: If a compact three-dimensional manifold M has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M is homeomorphic to the sphere S^3 ?
From the first, the apparently extremely simple nature of this statement has led mathematicians to overdo it. For about 100 years, several mathematicians have tried to prove the conjecture. But they all failed. Well, except from one, it seems.
Grigori Perelman’s proof is based on a study of the geometry of the infinite dimensional space consisting of all Riemannian metrics on a given smooth three-dimensional manifold.
The metric g_ij on a Riemannian manifold determines the length between points according to ds^2 = gij dx^i dx^j. (From the first and second derivatives of this metric tensor, one can compute the Ricci curvature tensor R_ij , and the scalar curvature R; for example, Euclidean flat space has R_ij = R = 0, while a usual three-dimensional sphere of radius r, has Ricci curvature R_ij = 2g_ij /r^2 and scalar curvature R = 6/r^2).
Perelman’s proof is based on the study of the Ricci flow, i.e. the solutions to the differential equation:
dg_ij/dt = −2R_ij ,
where t is like an external time-parameter. In other words, the metric is required to change with time so that distances decrease in directions of positive curvature.
Perelman’s succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process; however, the fates of more complicated objects were previously thought to be problematic. As the Ricci flow progressed, kinks and neck pinches – or singularities – could appear, pinch off and even shrink away. Perelman has showed that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started (his papers at arXiv.org).
For his work, Dr. Perelman has just been awarded the Fields Medal (an analog of the Nobel Prize) and as said before, also the Millenium Prize; but Perelman seems to have refused to accept the award (news.google.com). What a coolheaded guy 😉
In the context of string theory, the Ricci flow represent the world-sheet renormalization group (RG) flow; Perelman’s proof might have important implications for understanding space-time singularities.
To be continued…