Cached from http://www.dm.unito.it/~cerruti/mathnews0504.html
http://chronicle.com
Section: The Chronicle Review
Date: April 23, 2004
Volume 50, Issue 33, Page B10
Mathematics with a moral
By ROBERT OSSERMAN
The past decade has been an exciting one in the world of mathematics
and a fabulous one (in the literal sense) for mathematicians, who saw
themselves transformed from the frogs of fairy tales -- regarded with a
who-would-want-to-kiss-that aversion, when they were noticed at all --
into fascinating royalty, portrayed on stage and screen by such
glamorous stars as Mary-Louise Parker, Matt Damon, and Russell Crowe.
True, the dramatized mathematicians were generally troubled, but they
were geniuses and ultimately sympathetic.
Who bestowed the magic kiss on the mathematical frog? There may have
been two kisses, one from inside and the other from outside the world
of mathematics. The external kiss came first, in the spring of 1993,
with the debut of Tom Stoppard's play Arcadia, which depicted
mathematical genius in the guise of an appealing 13-year-old girl full
of adolescent exuberance, saucy humor, and high spirits. The opening
scene of the play refers to Fermat's Last Theorem, already a famous
problem in 1809, when the scene takes place, and by 1993 considered the
most famous unsolved problem in all of mathematics.
Just two months after the play opened came the kiss from within, as
Andrew Wiles announced that he had finally proved Fermat's Last
Theorem. The formerly obscure realm of mathematical research made the
front pages of newspapers around the world, and the real-life fairy
tale of Wiles's seven-year struggle with the proof was portrayed on
television and in books, reaching what may have been the height of
unreality as he and his wife watched themselves depicted as the lead
characters in a New York musical, Fermat's Last Tango.
Last year the mathematical world was again astir with the announcement
by a Russian mathematician, Grigori Perelman, that he had solved
another of the great open problems in mathematics, the century-old
Poincare conjecture. The story of that problem and the rather
roundabout route to its solution (still to be fully confirmed) has its
own fairy-tale aspects, including a moral or two.
To match the transformation of mathematics, princely sums have been
attached to certain mathematical investigations. In Paris in 2000, the
Clay Mathematics Institute announced its offer of $1-million each for
the solution of seven mathematical problems, among them the Poincare
conjecture.
In addition to Poincare, two other mathematicians play key roles in
our tale: Bernhard Riemann, whose radical rethinking of the foundations
of geometry in 1854 would eventually lead to the solution of the
Poincare conjecture, and William P. Thurston, whose work provided the
essential ingredients for connecting the ideas of Riemann with the
conjecture of Poincare.
What all three have in common is a fertile mathematical imagination --
more specifically, a remarkable geometric vision. In the division
sometimes made between mathematicians who are primarily problem solvers
and those who are theory builders, these three would all fall into the
theory-building category. They also offered solutions to major open
problems, but their solutions were often challenged as incomplete,
while the innovations they presented opened up whole new fields of
research.
In the case of Riemann, one innovation was the notion of a Riemann
surface, a surface that somehow passes through itself without
intersecting itself. The concept turned out to be useful in a variety
of applications, including aerodynamics and string theory.
The specific connection to the Poincare conjecture came from Riemann's
1854 lecture on geometry, in which he swept away the framework of
Euclidean geometry and laid the foundations for what is now known as
Riemannian geometry. For example, rather than start with straight lines
as one of the building blocks of the field, Riemann asked what are the
straightest possible lines -- called "geodesics" -- and what properties
they have. Whether or not they are truly straight depends on a key
property that he called "curvature."
Those notions are not idle speculations or pure abstractions; Riemann
was motivated by a desire to understand the geometric nature of the
space we live in -- nothing short of the shape of the universe. Using
his notion of curvature, he wrote equations for three basic shapes. One
of them is flat space, where the curvature is zero and the geometry is
the familiar one of Euclid. The second has negative curvature and is
now known as "hyperbolic space." It has played an increasingly central
role in various parts of mathematics, as well as being favored in some
circles as the most likely candidate for the present shape of the
universe. The third, with positive curvature, is perhaps the most
important. Called "the hypersphere," it was both Einstein's preferred
choice for the present shape of the universe and the closest model to
what Riemann himself was talking about: the observable universe, or
what we see when we look out in space and back in time toward the Big
Bang.
Riemann's remarkable vision included not only three-dimensional spaces
but also curved spaces of four or more dimensions. As with his
self-penetrating Riemann surfaces, the higher-dimensional curved spaces
seemed like science fiction at the time, but they became the core of
Einstein's general theory of relativity as well as a key component of
modern string theory.
Henri Poincare is sometimes described as the last universal
mathematician -- the last one to make fundamental contributions across
the spectrum of both pure and applied mathematics. Among those
contributions was a new branch of mathematics, one that was to be a
major focus of 20th-century research: algebraic topology. Like Riemann,
Poincare looked at spaces of all dimensions, but he was interested in
their coarse -- or global -- structure, rather than in their fine
structure, described by quantities like curvature.
Topology is a kind of broad-stroke geometry. It is interested in
overall shape, not the fine points. A topologist wants to know if your
house completely encloses an inner courtyard, not whether the courtyard
is rectangular or circular. Although the lack of emphasis on details
might suggest that topology is too imprecise to form the basis of a
mathematical theory, it is important because the first step in
understanding a geometric shape -- whether a strand of DNA or the
entire universe -- is often to determine the overall form.
Poincare devised several ways to assign a set of numbers or algebraic
quantities to geometric figures and spaces, in the hope of making it
possible to describe the overall or global shape of any space in terms
of those quantities. His famous conjecture had to do with a proposed
way to characterize the shape of spherical space.
During the 20th century, as mathematicians studied the possible shapes
of spaces of all dimensions, a curious fact emerged. Two-dimensional
spaces, called "surfaces," could be classified quite well, and spaces
of four or more dimensions turned out in some ways to be more tractable
than the three-dimensional spaces in which we actually live. Perhaps
most surprising, by 1982 Poincare's conjecture had been proved to be
true in all dimensions except three, where it remained an open
question. In the meanwhile, the continuing intensive study of
three-dimensional spaces uncovered a bewildering profusion of possible
shapes.
William Thurston's great contribution was to see a way to systematize
all those shapes -- to provide a kind of periodic table with which to
classify and organize all possibilities, as built up out of components
based on the original positively and negatively shaped geometries of
Riemann, together with a few other basic types. Thurston was able to
prove only a part of that classification; the rest remained perhaps the
most important open problem in geometry and topology: the Thurston
geometrization conjecture. As one measure of its scope, the Poincare
conjecture was subsumed as merely a special a University.
The expression "seems to have succeeded" is shorthand for a rather
complicated and unusual situation. Grigori Perelman announced in 2003
that he had proved the Thurston geometric conjecture and would present
the proof in a series of three papers that he would make available
electronically as they were finished. By late 2003, two of the three
papers had been posted and had become the object of intensive study by
geometers and topologists around the world. In December two
mathematical institutes in the San Francisco Bay area -- the American
Institute of Mathematics, in Palo Alto, and the Mathematical Sciences
Research Institute, in Berkeley -- held weeklong symposia to examine in
detail the components of Perelman's work, as well as additional results
inspired by his methods.
As happened when Wiles proposed his proof of Fermat's Last Theorem, the
experts quickly agreed that whether or not all the parts checked out,
Perelman had made important advances that were bound to be of great
value for further progress. Unlike Wiles, who chose to keep his work
confidential until it was all checked out, Perelman had decided to post
his calculations on the Web as he went along. That allowed many other
mathematicians to try to develop their own proofs of the remaining
pieces. In particular, even without his final paper, which is designed
to give a proof of the complete Thurston conjecture, Perelman has
provided an argument that will resolve the Poincare conjecture -- as
have Tobias Colding, of New York University, and William Minicozzi, of
the Johns Hopkins University, in a paper that uses an alternative
argument together with the work in Perelman's first two papers.
The moral of the story? There are at least two. First, a curious fact
of mathematical life: When faced with a problem that seems intractable,
the best strategy is sometimes to formulate what appears to be an even
harder problem. By expanding one's horizons, one may find an
unanticipated route that leads to the goal.
Second, mathematicians are often thought of as working in isolation,
and that is occasionally the case, as with Andrew Wiles and his
solitary struggle to prove Fermat's Last Theorem. But usually
mathematics is a highly social activity, with collaboration between two
or more individuals the rule rather than the exception.
In fact, institutes like the Mathematical Sciences Research Institute
are based on the premise that fostering collaborative research is one
of the most fruitful ways to advance the discipline. Even when an
individual takes the last step in solving a problem, the solution
invariably depends on elaborate groundwork laid by others -- as is
clearly the case with the solution of the Poincare conjecture.
Robert Osserman, a professor emeritus of mathematics at Stanford
University, is special-projects director at the Mathematical Sciences
Research Institute, in Berkeley, Calif.
http://chronicle.com
Section: The Chronicle Review
Volume 50, Issue 33, Page B10