Re: [obm-l] riemann

["Felipe Villela Dias" <f_d@xxxxxxxxxxxx>]

Olá a todos,
No texto do NY Times, dizem que a conjecutra de Poincaré é um dos 7
problemas mais importantes do milenio. Quais seriam os outros seis???
abraços
----- Original Message -----
From: "Nicolau C. Saldanha" <nicolau@sucuri.mat.puc-rio.br>
To: <obm-l@mat.puc-rio.br>
Sent: Wednesday, April 16, 2003 9:43 PM
Subject: Re: [obm-l] riemann


> On Wed, Apr 16, 2003 at 06:56:22PM -0300, gabriel wrote:
> > Caro Nicolau,
> > So nao entendi uma coisa; porque estes preprints nao devem ser levados
> > a serios?? Fisicos nao resolvem problemas de matematica?? : )
>
> Em muitos casos sim. Mas *estes* preprints em particular são escritos
> de maneira muito imprecisa demais.
>
> A pedido do Gugu segue abaixo o artigo do NYTimes.
>
> []s, N.
>
> ==========================================================================
>
> The New York Times
>
> April 15, 2003
>
> Celebrated Math Problem Solved, Russian Reports
>
> By SARA ROBINSON
>
>
> A Russian mathematician is reporting that he has proved the Poincaré
> Conjecture, one of the most famous unsolved problems in mathematics.
> The mathematician, Dr. Grigori Perelman of the Steklov Institute of
> Mathematics of the Russian Academy of Sciences in St. Petersburg, is
> describing his work in a series of papers, not yet completed.
> It will be months before the proof can be thoroughly checked. But if true,
> it will verify a statement about three-dimensional objects that has
haunted
> mathematicians for nearly a century, and its consequences will reverberate
> through geometry and physics.
> If his proof is accepted for publication in a refereed research journal
and
> survives two years of scrutiny, Dr. Perelman could be eligible for a $1
> million prize sponsored by the Clay Mathematics Institute in Cambridge,
> Mass., for solving what the institute identifies as one of the seven most
> important unsolved mathematics problems of the millennium.
> Rumors about Dr. Perelman's work have been circulating since November, whe
n
> he posted the first of his papers reporting the result on an Internet
> preprint server.
> Last week at the Massachusetts Institute of Technology, he gave his first
> formal lectures on his work to a packed auditorium. Dr. Perelman will give
> another lecture series at the State University of New York at Stony Brook
> starting on Monday.
> Dr. Perelman declined to be interviewed, saying publicity would be
> premature.
> For two months, Dr. Tomasz S. Mrowka, a mathematician at M.I.T., has been
> attending a seminar on Dr. Perelman's work, which relies on ideas
pioneered
> by another mathematician, Richard Hamilton. So far, Dr. Mrowka said, every
> time someone brings up an issue or objection, Dr. Perelman has a clear and
> succinct response.
> "It's not certain, but we're taking it very seriously," Dr. Mrowka said.
> "He's obviously thought about this stuff very hard for a long time, and it
> will be very hard to find any mistakes."
> Formulated by the French mathematician Henri Poincaré in 1904, the
Poincaré
> Conjecture is a central question in topology, the study of the geometrical
> properties of objects that do not change when the object is stretched,
> twisted or shrunk.
> The hollow shell of the surface of the earth is what topologists would
call
> a two-dimensional sphere. It has the property that every lasso of string
> encircling it can be pulled tight to one spot.
> On the surface of a doughnut, by contrast, a lasso passing through the
hole
> in the center cannot be shrunk to a point without cutting through the
> surface.
> Since the 19th century, mathematicians have known that the sphere is the
> only bounded two-dimensional space with this property, but what about
higher
> dimensions?
> The Poincaré Conjecture makes a corresponding statement about the
> three-dimensional sphere, a concept that is a stretch for the
> nonmathematician to visualize. It says, essentially, that the
> three-dimensional sphere is the only bounded three-dimensional space with
no
> holes.
> "The hard part is how to tell globally what a space looks like when you
can
> only see a little piece of it at a time," said Dr. Benson Farb, a
professor
> of mathematics at the University of Chicago. "It was pretty reasonable to
> think the earth was flat."
> That conjecture is notorious for the many "solutions" that later proved
> false. Indeed, Poincaré himself demonstrated that his earliest version of
> his conjecture was wrong. Since then, dozens of mathematicians have
asserted
> that they had proofs until experts found fatal flaws.
> Although many experts say they are excited and hopeful about Dr.
Perelman's
> effort, they also urge caution, noting that not all of the proof has been
> written down and that even the most reliable researchers make mistakes.
> That was the case in 1993 with Dr. Andrew J. Wiles, the Princeton
professor
> whose celebrated proof for Fermat's Last Theorem turned out to have a
> serious gap that was repaired after months of effort by Dr. Wiles and a
> former student, Dr. Richard Taylor.
> Dr. Perelman's results go well beyond a solution to the problem at hand,
as
> did those of Dr. Wiles. Dr. Perelman's results say he has proved a much
> broader conjecture about the geometry of three-dimensional spaces made in
> the 1970's. The Poincaré Conjecture is but a small part of that.
> Dr. Perelman's personal story has parallels to that of Dr. Wiles, who,
> without confiding in his colleagues, worked alone in his attic on Fermat's
> Last Theorem. Though his early work has earned him a reputation as a
> brilliant mathematician, Dr. Perelman spent the last eight years
sequestered
> in Russia, not publishing.
> In his paper posted in November, Dr. Perelman, now in his late 30's,
thanks
> the Courant Institute at New York University, SUNY Stony Brook and the
> University of California at Berkeley, because his savings from visiting
> positions at those institutions helped support him in Russia.
> His papers say that he has proved what is known as the Geometrization
> Conjecture, a complete characterization of the geometry of
three-dimensional
> spaces.
> Since the 19th century, mathematicians have known that a type of
> two-dimensional space called a manifold can be given a rigid geometric
> structure that looks the same everywhere. Mathematicians could list all
the
> possible shapes for two-dimensional manifolds and explain how a creature
> living on the surface of one can tell what kind of space he is on.
> In the 1950's, however, a Russian mathematician proved that the problem
was
> impossible to resolve in four dimensions and that even for three
dimensions,
> the question looked hopelessly complex.
> In the early 1970's, Dr. William P. Thurston, a professor at the
University
> of California at Davis, conjectured that three-dimensional manifolds are
> composed of many homogeneous pieces that can be put together only in
> prescribed ways and proved that in many cases his conjecture was correct.
> Dr. Thurston won a Fields Medal, the highest honor in mathematics, for his
> work.
> Dr. Perelman's work, if correct, would provide the final piece of a
complete
> description of the structure of three-dimensional manifolds and, almost as
> an afterthought, would resolve Poincaré's famous question. Dr. Perelman's
> approach uses a technique known as the Ricci flow, devised by Dr.
Hamilton,
> who is now at Columbia University.
> The Ricci flow is an averaging process used to smooth out the bumps of a
> manifold and make it look more uniform. Dr. Hamilton uses the Ricci flow
to
> prove the Geometrization Conjecture in some cases and outlined a general
> program of how it could be used to prove the Geometrization Conjecture in
> all cases. He ran into problems, however, coping with certain types of
large
> lumps that tended to grow uncontrollably under the averaging process.
> "What Perelman has done is to figure out some new and interesting ways to
> tame these singularities," Dr. Mrowka said. "His work relies heavily on
> Hamilton's work but makes amazing new contributions to that program."
> If Dr. Perelman succeeds in resolving Poincaré, he will probably share the
> Clay Mathematics Institute Award with Dr. Hamilton, mathematicians said.
> Even if Dr. Perelman's work does not prove the Geometrization Conjecture,
> mathematicians said, it is clear that his work will make a substantial
> contribution to mathematics.
> "This is one of those happy circumstances where it's going to be fun no
> matter what," Dr. Mrowka said. "Either he's done it or he's made some
really
> significant progress, and we're going to learn from it."
>
>
>
>
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