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Category:
Science and Technology
Domain:
Keywords:
Computer Science - mathematics, computer security, physics
Outlook:
Solution of the seven maths problems named by the Clay Mathematics Institute as its Millennium Prize Problems may blur the line between 'pure' and 'applied' mathematics and could also have implications for computer and network architectures and security.
Summary Analysis:
In 2000 the Clay Mathematics Institute issued a million-dollar challenge for solutions to each of the seven most important unsolved problems in mathematics, as selected by a committee of distinguished mathematicians. By 2050, at least some of the problems may have been solved. The methods devised for the proofs and solutions are expected to have an impact on the computational, physical, and natural sciences. The purpose of the challenge is not solely to find solutions to the actual problems, but to encourage research in the most difficult areas of mathematics: they provide targets to aim for; their solution is in many ways secondary.

The problems are as follows:

  1. The Riemann hypothesis: This hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of the Riemann zeta function. The zeroes of the Riemann zeta function have been linked to another theorem regarding the distribution of prime numbers among integers. It is considered the most important unsolved problem in mathematics. Because primes are widely used in cryptography, the methods that are likely to be used in the proof will impact computer security algorithms. Though many fear that proving the Riemann hypothesis and finding the prime numbers pattern could cause a failure in computer security, this is a myth. Proving the Riemann hypothesis has no immediate industrial applications. There may also be links to quantum physics. In 2004 De Branges of Purdue claimed a proof, but it is currently unverified. Solution expected.
  2. The Poincare conjecture: This conjecture concerns the characterisation of the three-dimensional sphere. Similar conjectures have been proven in higher dimensions, yet all proposed proofs have been disproved in the three-dimensional case. In 2002, Grigori Perelman of Russia published a paper proving not only this conjecture but also the more general Thurston Geometrization conjecture. Perelman's proof is still being checked, but a growing consensus believes Perelman to be correct.
  3. P versus NP problem: This problem concerns the complexity of calculations in theoretical computer science. There are computational tasks that require searching for an object in a large space of possibilities, such that it would require a geologic time to find the object. The P versus NP problem asks whether there is a method that avoids searching by brute force -- that is, whether there are any problems for which a computer can check an answer quickly (in polynomial time) but cannot find the answer quickly (in polynomial time). Polynomial time means that the time required to solve a problem or verify a solution is on the order of a polynomial function of the size of the input, as opposed to an exponential function for instance. Most simply operations, such as addition, subtraction, multiplication, division, square roots, powers, and logarithms, can be computed in polynomial time. Most mathematicians believe the answer is not decidable. If it is found that all problems that can be checked in polynomial time can also be solved in polynomial time, it would increase the efficiency with which certain solutions can be searched and found, opening up a whole new array of solutions to problems previously unsolvable.
  4. Navier-Stokes equations: Do analytic solutions to these equations exist? The Navier-Stokes equations are the fundamental partial differential equations that describe the flow of incompressible fluid. Solutions to reduced forms of the equations have been in use for a while, but currently the best approach is the use of numerical methods, computational fluid dynamics. Some experts believe that by 2050 the equations will be shown to be unsolvable in 'exotic situations'. General solutions to the Navier-Stokes equations would lead to better designs of vehicles and have an impact on models of weather and ocean currents, allowing for more accurate predictions.
  5. Yang-Mills Mass Gap Hypothesis: This hypothesis describes particles with positive mass having classical waves traveling at the speed of light. This is called the mass gap. The goal is to establish the existence of the Yang-Mills theory and a mass gap. Experts believe that it will be settled by 2050, but by that point, it will not have a large impact on physics.
  6. Hodge conjecture and Birch/Swinnerton-Dyer conjecture: These deal with issues in algebraic geometry. As of 2005, the Birch/Swinnerton-Dyer conjectures have been proved only in special cases.

Implications:

  • Blurring of the line between 'pure' and 'applied' mathematics as computational techniques are applied to proof of exotic problems and gain use in more practical domains
  • With solution of the P versus NP problem, increase in the efficiency with which certain solutions can be searched and found by a computer
  • With solution of the Navier-Stokes equations, better designs of vehicles and more accurate prediction of weather and ocean currents, along with an increase in demand for technologies and instruments that use these improvements

Early Indicators:

  • Use by Kenneth Appel and Wolfgang Haken of a computer in proving the "four color theorem" in 1976; development of a second computer-based proof more completely reliant on automated techniques by Georges Gonthier in 2005
  • Acceptance for publication in a 'pure' maths journal in 2004 of Thomas Hale's 1998 use of computers to prove Kepler's conjecture

What to Watch:

  • Proofs to these problems that are currently under review, such as the De Branges proof of the Riemann hypothesis and the Perelman solution to the Poincare conjecture, are proven or disproven.

Parallels/Precedents:

  • The proof of Fermat's Last Theorem by Princeton Professor Andrew J. Wiles published in 1993 (which required modification by Wiles and Dr. Richard Taylor to fill gaps that were discovered on first publication)
  • Hilbert's problems: a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them turned out to be very influential for 20th century mathematics. [link]

Enablers/drivers:

  • Offer by the Clay Mathematics Institute of $1 million for each problem to the first person with a solution
  • Advances in grid computing and pattern recognition techniques
  • Continuing research into complex systems

Leaders:

  • Dr. Keith Devlin, Stanford University [link]
  • Dr. Marcus du Sautoy, Oxford University [link]
  • Dr. Ian Stewart, Warwick University [link]

Figures:
Sources:

  • "Clay Mathematics Institute Millennium Prize Problems" Clay Mathematics Institute Millennium Prize Problems [link]
  • Stewart, Ian. "The Mathematics of 2050" in Brockman, John, ed., The Next Fifty Years. New York: Vintage, 2002. [link]
  • Sipser, Michael. "The History and Status of the P versus NP Question." Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, (1992) 603-619.
  • Chang, Kenneth. "In Math, Computers Don't Lie. Or Do They?" New York Times 6 April 2004. [link]
  • "Proof and Beauty: Just what does it mean to prove something?" The Economist 31 March 2005. [link]
  • "Does the proof of the Riemann hypothesis really bring the whole e-commerce to its knees?" Recent Developments in Cryptography web site. Universite Catholique de Louvain (Belgium) Crypto Group. [link]
  • "The P-versus-NP Web Page," Dr. Gerhard J. Woeginger, Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands. [link]
  • Text of Hilbert's 1900 lecture [link]
  • The Millennium Problems: The seven greatest unsolved mathematical puzzles of our time, Keith Devlin, Granta Books 2005, ISBN 1 86207 735 5 [link]
  • Navier-Stokes equations [link]
  • 50 Years of Yang-Mills Theory [link]
  • Riemann Hypothesis [link]
  • Brief description of the seven problems by John Baez [link]
  • Wiki on the Birch and Swinnerton-Dyer Conjecture [link]
  • London Mathematical Society [link]


At A Glance:
When:
3-50 yrs
Where:
Global
How Fast:
Years
Likelihood:
Medium-High
Impact:
Medium-Low
Controversy:
Medium


Related Outlooks:

About this outlook: An outlook is an internally consistent, plausible view of the future based on the best expertise available. It is not a prediction of the future. The AT-A-GLANCE ratings suggest the scope, scale, and uncertainty associated with this outlook. Each outlook is also a working document, with contributors adding comments and edits to improve the forecast over time. Please see the revision history for earlier versions.

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