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Speaking
by e-mail, Lafforgue said: "When non-mathematicians ask me
what I work on, I don't try to explain it to them because I believe
that this is nearly impossible. The same with mathematicians who
work in other fields." Voevodsky is traveling and could not
be reached. But Sudan has been successful, he said, in explaining
his work to his 3-year old daughter. If she can get it, so can we.
Let us recognize that mathematicians
are not like you or me. We the many can detect some beauty in
the paintings of Titian, feel a certain sad hope in a Chopin sonata,
recognize the grace in Frank Lloyd Wright's Fallingwater. But,
most likely, the isomorphism between a modified motivic cohomology
of an algebraic variety and the modified singular cohomology of
its natural topological space does little for us.
Which is, really, a shame.
For there is a beauty in mathematics, which you may have glimpsed
that day in first grade when it struck you how peculiar zero was:
that you could add it to any other number -- any number at all!
-- and the number would stay the same. Or maybe you've encountered
a slick little thing called the square root of -1. There are men
who have this number engraved on their tombstones.
This wonder, of course, gets
beaten out of us by dull teachers, media stereotypes, the massacre
of our attention span, and the manufacturers of standardized college
entrance exams. But it hasn't been beaten out of these guys. "There
exist in mathematics things extremely beautiful," said Lafforgue.
"One thing that's always astonishing is that, occasionally,
one realizes that in mathematics the truth is beautiful."
Over the millennia the tree
of mathematics has branched in dozens of different directions,
arching out from a base that looks like a high school transcript:
geometry, algebra, analytic geometry, calculus. In some ways mathematicians
have been working there ever since, extending those basic concepts
to more and more sophisticated ideas, building mathematical objects
(like the set of all positive integers, 1, 2, 3, ...), and constructing
ever more complicated beasts (like the set of all fractions, to
give a trivial example,). Mathematicians often find this beauty,
this truth, in their efforts to unite previously disparate areas
of their field, to tame the unruly beasts they have unleashed.
Lafforgue made his mark in
such unification. Decades ago a young Princeton mathematician
named Robert Langlands conjectured that two very different animals
are intimately connected. Roughly speaking, as Charles Seife described
in Science magazine, these animals are mathematical objects that
can be distorted in certain ways and still retain their original
shape [such as the fundamental equivalence between a rubber coffee
cup and a doughnut -- one can be stretched into the other, as
long as you respect the hole] and objects that reveal the relations
between solutions of equations."
Langlands' conjecture, described
as a "Rosetta stone" of mathematics, was formalized
into the Langlands Program, a quest that has happily occupied
scores of mathematicians for more than 30 years. Andrew Wiles,
the Princeton mathematician who a few years ago announced a heralded
proof of Fermat's Last Theorem, established a important ingredient
of the conjecture in his own work.
In general, mathematicians
believed that Langlands' conjecture was true, but proving it was
extremely difficult. Parts of the proof had already garnered two
Fields Medals, and in 1999 Lafforgue made his mark with a 300-page
handwritten proof of the conjecture in the case of what are called
"function fields."
This is where the writer,
fearing that he is losing the reader, must bring the discussion
back to earth. He'll begin by pointing out that, the foregoing
remarks notwithstanding, mathematicians are as human as you or
me, even if they often have funny-looking hair or peculiar habits.
The genius Paul Erdos called little children "epsilon,"
which is humorous if you're a mathematician (the Greek letter
epsilon is often used as a symbol to express the concept of something
approaching zero) but probably irritating if you are not.
Sometimes mathematicians
make mistakes. After his proof of Fermat's Last Theorem was announced
in the New York Times ("At Last, Shout of 'Eureka!' in Age-Old
Math Mystery") Wiles discovered a mistake in his work and
presumably just about had a bird. It took him a year to fix the
problem, and Fermat was put to bed at last.
In 2000 Lafforgue was awarded
the Clay Research Award by the Clay Mathematical Institute in
Massachusetts. Just five days later he found a mistake in his
own work. Lafforgue contacted Arthur Jaffe, the Clay Institute's
president and a mathematics professor at Harvard, and offered
to return his prize.
"Andrew Wiles and I
convinced him that the award would be, under the circumstances,
even more valuable to him," said Jaffe. "He could travel
or collaborate however he liked in order to repair his proof."
Lafforgue said he worked
day and night and fixed the flaw a few months later, proving that
two very different-looking things are the same. At the same time
it gave mathematicians confidence that the Langlands Program would
succeed in other areas where work proceeds on the conjecture.
Why does Lafforgue's proof
matter to me or to you? For a moment let's set aside the part
about beauty.
In 1960 the physicist Eugene
Wigner spoke of "the unreasonable effectiveness of mathematics."
Mathematics is useful, and not just in the spreadsheet that is
going to be part of the report you have due in three hours. Scientists
and engineers have constructed our world on it, from Newton's
calculus -- which he invented to describe the laws of motion --
to the quantum mechanics that describe the workings of the chip
inside your personal computer.
Mathematics, amazingly enough,
works; that is why it has been called the queen of the sciences.
It works in the real world, and not just in the airy heights of
the mathematician's imagination. It's an ugly kind of thing, in
a way, in the minds of some pure mathematicians. "Real mathematics
has no effect on war," wrote Hardy in "A Mathematician's
Apology," a book intended to justify his existence as a pure
mathematician. "No one has yet discovered any warlike purpose
to be served by the theory of numbers or relativity."
Hardy wrote this in 1940,
five years before the nuclear bomb was built on Einstein's fundamental
ideas in relativity and well before today's cryptography built
on prime numbers. These days no one is clean. That is one thing
of which we can all be sure. Voevodsky also solved a major mathematical
problem, called the Milnor conjecture. Milnor, one of the best
mathematicians of the past half century and a close friend of
John Nash, the mathematician portrayed in the movie "A Beautiful
Mind," believed there was an equivalence between different
ways of describing the properties of different kinds of surfaces.
(This is a vast oversimplification, but I'm worried again that
I'm losing you.) Voevodsky created new mathematical tools that,
in 1996, enabled him to solve the problem.
Voevodsky is also a Clay
Institute Prize fellow, and the institute sponsors his visits
to Russia to lecture and inspire the current generation of young
students there. Russia has a proud tradition of mathematicians
and mathematical physicists (built in part, it has been speculated,
because the country was unable to mount major efforts in the experimental
sciences), which like other areas has suffered with the collapse
of the Soviet Union.
How do mathematicians like
Voevodsky work? This is, in the end, difficult to say. Hardy spent
most days at the side of a cricket field, drinking tea. The great
Grothendieck, who as much as anyone else is responsible for a
vision of the unity of all mathematics, spent 18 hours a day creating
mathematics that has astonished and inspired mathematicians ever
since. (Grothendieck won his own Fields Medal in 1966, Milnor
in 1962.) "There was far more imagination in the head of
Archimedes than in that of Homer," Voltaire said, which English
majors might doubt. But they would be wrong.
Your tax dollars pay for
about $300 million dollars of mathematical research each year,
and we are finally going to get to something you can understand.
As you read this article,
you trust that the words your computer retrieved from Salon's
Web servers are faithful to their original. But how do you know?
You know because mathematicians
and engineers have thoughtfully included error-correcting codes
in the computers that talk to one another across the Internet,
because sometimes bits get scrambled, dropped, or mutilated. Madhu
Sudan, winner of the 2002 Nevanlinna Prize, explained it to his
daughter as follows: "When my 3-year-old daughter, Roshni,
asked me what I do, I told her I correct errors. She asked me
what is an error and what does it mean to correct them. So I wrote
'Rothni' on a piece of paper and asked her to circle any mistakes
(without explaining what I intended to write). She circled the
't.' I told her that's what I do for a living. She understood
what I did, but not why it was a big deal."
It's a big deal because Sudan
has shown that certain codes can correct many more errors than
was previously thought possible.
Sudan is a theoretical computer
scientist. He does not, he said, use a computer in his work. He
was recognized for his breakthroughs in error-correcting codes,
probabilistically checkable
proofs, and the non-approximability of optimization problems.
Given a proposed proof of
a mathematical statement -- say, the statement that there is an
infinite number of prime numbers (numbers, such as 7 or 17, evenly
divisible only by 1 and themselves) -- the theory of probabilistically
checkable proofs recasts the proof so that its fundamental logic
is encoded as a sequence of bits that can be stored in a computer.
Checking only some of these bits, Sudan and others have shown,
can determine with high probability whether the proof is correct.
Amazingly, the number of bits one must examine can be made extremely
small.
How small? Let's just leave
it as "small," because this article must soon come to
an end, and because ... well, you know.
Consider two sets: all towns
in your state, and all states whose names end in the letter "A."
Given a finite collection of finite sets such as this, what is
the largest size of a subcollection such that every two sets in
the subcollection have no overlap? I forgot to ask Sudan about
this particular problem, but he probably doesn't know anyway --
hey, it's a tough problem. You might propose a solution, which
could be easily checked, but in general there may be no known
algorithm that will easily produce a solution from scratch.
What Sudan and others showed
is that, for many such problems, approximating an optimal solution
is just as hard as finding an optimal solution. Now, this could
obviously be useful to scientists and engineers. It also has implications
for a fundamental mathematical problem called P=NP (if you solve
it, the Clay Institute will give you a million dollars).
Why, in the end, does all
of this mathematics matter? Why have Lafforgue, Voevodsky, and
Sudan been culled from the set of all mathematicians for the highest
honors?
Yes, mathematics is wonderfully
useful for calculating the properties of superstrings and the
path of the next comet that will collide with earth. Yada yada.
But, really, does a poem
matter only because it can be read at a memorial? Isn't a busker's
real worth the gleam in his eye when he performs? Is whatever
art you may have created more important than the way you felt
when you created it, or the way others felt when it was received?
"The case for my life,
then," Hardy wrote in his "Apology," "or for
that of any one else who has been a mathematician in the same
sense in which I have been one, is this: that I have added something
to knowledge, and helped others to add more; and that these somethings
have a value which differs in degree only, and not in kind, from
that of the creations of the great mathematicians, or of any other
artists, great or small, who have left some kind of memorial behind
them."
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