Solving a Conjecture

Print edition : September 08, 2006

The Poincare Conjecture has baffled mathematicians for over a century.


Perelman displays an ascetic adherence to the purity of mathematical pursuit.


WHAT is the Poincare Conjecture whose solution by the maverick Russian mathematician Grigori `Grisha' Perelman has created such a stir in the mathematics world? This conjecture, formulated a little over 100 years ago by the famous French mathematician and physicist, Henri Poincare, is currently the most outstanding one in the branch of mathematics known as topology.

Topology deals with the fundamental properties of abstract spaces. Whereas classical geometry is concerned with measurable quantities such as angle, distance and area, topology is concerned with notions of continuity and relative position. It is the study of properties of spaces that remain unchanged, when the spaces are bent, stretched, shrunk, or twisted, subject only to the condition that nearby points in one space correspond to nearby points in the transformed version of that space. Since allowed deformations are like manipulating a rubber sheet, topology is often called "rubber-sheet geometry". In contrast, cutting and then gluing together parts of a space are bound to fuse two or more points and separate points once close together.

The Poincare Conjecture has to do with the nature of the three-dimensional space and classification of certain types of three-dimensional geometric objects. It may be thought of as part of a more far-reaching conjecture, known as the Thurston Geometrisation Conjecture, which hypothesises a complete classification of all three-dimensional geometric objects.

As is the case in all sciences, classification is one of the first questions that is sought to be asked and answered in mathematics too. Of course even framing the question `how many different kinds of things are there', though apparently simple, is never quite as easy as it sounds. To group `things' - in this case geometric objects - together, one needs something of a precise notion of what we mean by similarity. Or, as mathematicians put it, we need a notion of equivalence. In the case of geometric objects, it turns out that we may ask this question at different levels.

The most basic kind of similarity is what is known as topological equivalence. Loops provide a simple example of such equivalence for one-dimensional geometric objects. One may lay the loop, made by a piece of string for instance, on a flat surface and arrange it, with some care, so that it forms a perfect circle. On the other hand, one may carelessly let it take any shape one chooses, while ensuring that the string does not cross itself. Since any such shape can be deformed to a circle, without having to cut the loop or take it off the surface, we may declare that all such loops and the circle are indeed all `topologically equivalent'.

By the end of the 19th century, relying above all on the work of the great German mathematician Berhard Riemann, the classification of two-dimensional objects, namely surfaces, was well understood. The notion of equivalence in this case is similar to the simple example given above, except that we must now allow for deforming the whole surface, as if it were made from a rubber sheet, and not just some lines on it.

A famous theorem, originally due to Riemann and put in its final form by the German mathematician Paul Koebe, proved that all two-dimensional geometric objects were essentially of three kinds. They could be equivalent to a sphere, like the surface of an orange or a bonda. Or they could be equivalent to a torus, a shape like that of the surface of a doughnut or a medu vada. For that matter, an equivalent shape would be a coffee cup with the handle or a person with a hand on the waist like a dancing girl. And thirdly, they could be like several doughnuts or tori (or vadas, to persisist with our analogy) stuck together, objects with several holes, as long as the holes were finite in number. In fact, the number of holes or genus, in mathematical parlance, provides a classification scheme for this third class.

So in contrast to the one-dimensional example, there is more than one `topological type' in the classification of two-dimensional surfaces. These topological types cannot be transformed into one another without drastic actions such as cutting and gluing or pasting. But it would certainly be preferable if there were an easier way to characterise the difference between these `topological types' based on some intrinsic properties without attempting to cut and glue one shape into another. As it turns out, there is more than one way to characterise precisely these distinctions but there is one that is of particular relevance to the story of the Poincare Conjecture.

AT THE FUNCTION in Madrid, where the Fields Medals for 2006 were awarded.-

A distinguishing characteristic is the behaviour of loops on the surface as they are shrunk. In the case of a bonda any such loop inscribed on it can be continuously shrunk to a point. But in the case of a vada, loops that thread through the hole cannot be shrunk to a point. This is a fundamental distinction between the sphere and the other two topological types. In fact, all two-dimensional surfaces on which loops can be shrunk to a point are topologically the same as the familiar sphere. The Poincare Conjecture is an extension of this classification story to three-dimensional geometric objects. Poincare merely asked whether an analogous classification using the idea of shrinking loops on them holds for abstract three-dimensional objects as well.

At the beginning of the 20th century, Poincare was working on the basic foundations of topology. He had devised some methods (known as homology in modern mathematics) of characterising the intrinsic properties of geometric objects and claimed that using these he could determine when any three-dimensional object was indeed the same as a three-dimensional sphere. The claim turned out to be incorrect as Poincare himself established later. However it was in the course of these studies that Poincare formulated his celebrated conjecture: Any three-dimensional space, on which any loop can be continuously shrunk to a point, is indeed topologically equivalent to a `three-dimensional sphere'.

Wisely, as it turned out, Poincare did not attempt to prove this in his time.

The first major attack on proving the conjecture was in the work of the English mathematician Alfred Whitehead. And with this effort also began a long history of wrong proofs of the conjecture. Just as with the celebrated Fermat's theorem before it, the effort to prove the Poincare Conjecture yielded many dividends in the form of an improved understanding of low-dimensional topology, particularly the topology of three dimensions. Analogues of the Poincare Conjecture were eventually proved for higher dimensions, in two separate efforts. The first by Stephen Smale was for dimensions greater than or equal to five. Subsequently Michael Freedman proved it for the four-dimensional case. Both efforts earned their authors the Fields Medal, the first being awarded in 1966, and the second in 1986.

But the great leap forward in three-dimensional topology arose from the work of William Thurston in the late 1970s. Thurston extended the Poincare Conjecture to another conjecture regarding the complete classification of all three-dimensional spaces, providing the analogue of the Riemann-Koebe theorem for the classification of two-dimensional surfaces. According to this `Geometrisation Conjecture', as it is known in the world of mathematics, all three-dimensional topologies can be built out of eight basic geometries. From these eight types, all three-dimensional geometric objects can be produced, by certain operations like rotation, reflection and translation. Poincare's original conjecture could now be conceived as a subset of this larger conjecture.

An important feature of Thurston's work that is of special interest in the light of subsequent developments, is that, in developing the analogue of Riemann's result for three-dimensional spaces, he also tied topology to geometry. In topology, shapes, sizes and distances do not matter. On the other hand the notion of geometry arises in the mathematical study of different spaces when the concept of a distance (and hence also size and volume) is introduced. What Thurston's work did was to relate the two together. The eight basic types were produced as geometries for which the curvature was constant. Each topological type was described in a particular geometry in Thurston's classification. Or to put it differently, the classification was easiest described in a framework where three-dimensional objects had particular geometries ascribed to them.

The next advance came from the work of Richard Hamilton, whose work provided the first real possibility that the Poincare Conjecture and perhaps even the Geometrisation Conjecture of Thurston could eventually be proved. Indeed mathematician John Milnor, in his introduction to the Poincare Conjecture on the website of the Clay Mathematical Institute (CMI), which listed Poincare Conjecture as one of the seven Millennium Prize Problems with a prize-tag of $1 million, explicitly mentioned the possibility that the work of Hamilton could perhaps be used to prove the conjecture, reflecting the expert consensus on the subject.

Hamilton's work crucially utilised this description in terms of geometries of constant curvature. In this description, these geometries emerged as the final equilibrium state of evolution over a very long time beginning from some initial geometry, just as the distribution of temperature in a heated body would eventually result in a uniform temperature across the object. In mathematics literature, this is referred to as `Ricci flow' and the equations indeed resemble the `heat flow equation' in physics. Thus the hope was that if indeed there were unique geometries that were the essence of the classification then these would automatically emerge from any arbitrary initial geometry that one started with. It soon emerged from Hamilton's own work that the description of this evolution in the general case was extremely complicated, involving interrupting the flow to cut and paste the geometry and then letting the evolution continue further.

But the crucial step that was needed then was to deal with the possibility that in the evolution process, the geometries would tend to limits other than the desired ones of the type that Thurston had described, like `neck pinches' or other unwanted strange features. The essence of the problem then was to describe what happened along the flow and how these `singularities' in the flow had to be tamed. Indeed, Hamilton himself described as recently as 1995 the nature of these puzzles. Perelman's critical contribution was to bridge this last gap, or chasm, as it might be more appropriately described, given the state of the art of the subject a decade ago.

To any reader who was tuned into the buzz surrounding the Thurston and Poincare Conjectures, and the role of Hamilton's Ricci flow method in their proof, the abstract of Perelman's first paper itself would have set off alarm bells, even if Perelman was fairly cautious in the wording of his claims in the first paper. Perelman's work on the Ricci flow indeed took him beyond what was strictly needed to prove the Thurston and Poincare Conjectures. In the third of the series of three preprints that he posted on the Internet, he demonstrated a shortcut showing that the proofs of the conjectures required less than the full strength of his results.

Perelman's telegraphic style, "frugal with the details" as some mathematicians describe it, has taken nearly three years to be understood and digested, in major part, by experts who have dedicated themselves to this task. But given the focus on the Geometrisation and Poincare Conjectures, the part relating to them in Perelman's work has been the primary target of scrutiny. Current expert wisdom (with one voice of dissent coming from Chinese-American mathematician S.T. Yau) has it that Perelman's work is complete and sufficient in this regard, with no gaps in the proof, even if there may be some in exposition. But already Perelman's work has opened new directions and the methods that he has established are thought to be of considerable significance for the future of this sub-discipline of mathematics.

Perelman himself has spurned the awards and honours that have begun to flow his way, most notably the Fields Medal for 2006, the mathematics equivalent of the Nobel Prize. In his brief comments and indications as reported in the international media, he reveals an almost ascetic adherence to the purity of mathematical, scientific and intellectual pursuit, unwilling, it appears, to have it sullied by mundane matters such as monetary awards and official recognition. He has not even interacted very much with those studying his work, confident in his correctness and patient with the time taken by others to accept it.

Given the strong and almost complete evidence that points to the grandeur of his achievement, accomplished in its crucial final steps in solitary splendour, Grisha Perelman's asceticism appears to be less the eccentricity of a somewhat arrogant genius and more the quiet satisfaction of an intellect that has laboured to win the ultimate prize of a lasting and honoured place in the history of mathematics.

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