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World's toughest math problem has been solved.
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| Demoted |
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Journey to the 248th dimension
Map of weird mathematical entity may point way for string theory.
John Whitfield
A map of one of the strangest and most complex entities in mathematics should be a powerful new tool for both mathematicians and physicists pursuing a unified theory of space, time and matter.
The strange 'thing' that has been mapped is a 'Lie group' called E8 — a set of maths that describes the symmetry of an (unimaginable to most) 57-dimensional object.
The creation of this map, which took 77 hours on a supercomputer, resulted in a matrix of 453,060 ? 453,060 cells, containing more than 205 billion entries — "all related in intricate and complex ways", says Jeffrey Adams, the project leader and a mathematician at the University of Maryland.
This represents 60 gigabytes of data, enough data to store 45 days of MP3 music files, or fill a piece of paper the size of Manhattan (about 60 square kilometres). The human genome takes up 1 gigabyte.
The finished product is essentially a database of information, which should come in very handy to theoretical physicists tackling grand unified theories of everything. "Now that it's done, mathematicians and physicists can use the results very easily," says Ian Stewart of the University of Warwick, UK. Adams agrees: "It's going to be a fabulous tool."
Weird exception
A Lie group is a collection of mathematical descriptors that help to illustrate the symmetry of a smooth object. The Lie group for a sphere, for example, describes all the mathematical operations that can be performed on the sphere without changing its appearance. There are an infinite number of straightforward Lie groups. But there are also five 'exceptional groups': weird one-offs of which E8, discovered in 1887, is one.
It gets stranger: E8, which represents the symmetries of a particular 57-dimensional object, has 248 dimensions itself.
"It's perhaps the most beautiful structure in all of mathematics, but it's very complex," says physicist Hermann Nicolai of the Max Planck Institute for Gravitational Physics in Potsdam, Germany.
Adams's team spent two years working out how the problem could be rendered in a form that wouldn't overwhelm the memory of a computer. The rest of the time was taken writing the code and testing the map, probing the mathematical properties of different regions to see if they provided the expected answer.
"The calculation was known to be possible in principle, but it was thought to be hopeless in practice," says Adams. "But four years ago a group of us said let's really try to do it. We're pretty sure we've got it right, but it's hard to be 100% sure."
"It's probably one of the most complicated pure mathematical calculations anyone's ever done," says Stewart. "Each entry is difficult to calculate — it's amazing they managed to do this."
Balls and string
Besides pure mathematicians, the people most familiar with E8 are physicists, and they might get the most out of the new map.
The mathematics of symmetry lies at the heart of both relativity and quantum physics. String theorists trying to unify these two areas are casting around for a type of symmetry that will let them deal with the troublesome extra dimensions thrown up by their models.
"A unified theory needs unique mathematics," says Nicolai. "What we'd like is a structure with very special properties. E8 has a flavour of this, although we don't know how the symmetry is realized in physical theory — we have to study it in more detail."
"Nobody knows what pieces of mathematics string theorists are going to need, but this will be an important piece of the toolkit," agrees Stewart. "It gives a better chance of making new and unexpected predictions."
The map will be included in the Atlas of Lie Groups and Representations, and will be available online at www.liegroups.org. The researchers also plan to publish their methodologies in a scientific journal.
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My sister doesn't know how to do long division. |
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| Sushipunk |
| quote: | Originally posted by Demoted
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My sister doesn't know how to do long division. |
Wow, that's pretty cool.
Except for your sister, I guess. |
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| Demoted |
| quote: | Originally posted by Sushipunk
Except for your sister, I guess. |
Yeah man, that bitch. And her festering, brooding, vaginatrap. |
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| Sushipunk |
| quote: | Originally posted by Demoted
Yeah man, that bitch. And her festering, brooding, vaginatrap. |
In complete honesty, that is the very last thing I expected you to say :wtf: :stongue: |
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| tubularbills |
1+1=2
I WON! |
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| Demoted |
| quote: | Originally posted by Sushipunk
In complete honesty, that is the very last thing I expected you to say :wtf: :stongue: |
Well, she is a bitchpunt.
jkjkjkrowling familllly löve <333 |
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| Demoted |
| quote: | Originally posted by tubularbills
1+1=2
I WON! |
Nay, you didn't get 42 as the answer. |
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| LazFX |
| quote: | Originally posted by Demoted
Nay, you didn't get 42 as the answer. |
42 is the answer ;) |
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| Subey |
All those numbers confuse me!
Demoted should be demoted for not providing a Cor Version! |
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| Lira |
| Yes, yes, finally this century is getting started :) |
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| SuspicionVandit |
| so, what do you guys think about string theory? |
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| Omega_M |
That's not the toughest problem. :o
The Clay Institute offers million $$$ per problem to anyone who can these.
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Birch and Swinnerton-Dyer Conjecture
Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like
x^2 + y^2 = z^2
Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function ζ(s) near the point s=1. In particular this amazing conjecture asserts that if ζ(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points. |
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Hodge Conjecture
In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles. |
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Navier-Stokes Equation
Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. |
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P vs NP Problem
Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory (i.e., no pair taken from your coworker's list also appears on the list from the Dean's office), however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer. In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer. Stephen Cook and Leonid Levin formulated the P (i.e., easy to find) versus NP (i.e., easy to check) problem independently in 1971. |
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Poincaré Conjecture
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since. |
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Yang-Mills and Mass Gap
The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics. |
And the most famous of them all...
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Riemann Hypothesis
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function
ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...
called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation
ζ(s) = 0
lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. |
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