Millennium Problems

The Millennium Problems are the unsolved mathematics problems listed in 2000 by the Clay Mathematics Institute (CMI) in Cambridge ( Massachusetts ) . The institute has awarded prize money of one million US dollars each for solving one of the seven problems .

List of problems

The list includes the following seven problems:

1. the proof of the Birch and Swinnerton-Dyer conjecture ,
2. the proof of Hodge's conjecture ,
3. Analysis of the existence and regularity of solutions to the initial value problem of the three-dimensional incompressible Navier-Stokes equations . Several variants are formulated, including the question of the existence of smooth ( ) solutions (speed), (pressure) of the incompressible force-free Navier-Stokes equation in three dimensions for all positive times, whereby the energy remains limited . The initial value is assumed to be smooth ( ), with a growth restriction on the spatial derivatives. In another variant, periodic boundary conditions are specified (solutions on the three-dimensional torus instead of in ). It is also asked whether there are smooth force fields (with restrictions on the growth of the derivatives of and ) and initial conditions for which there are no such smooth solutions for pressure and velocity for all positive times (as above with finite energy). ${\displaystyle C^{\infty }}$ ${\displaystyle u(x,t)}$ ${\displaystyle p(x,t)}$ ${\displaystyle \textstyle \left(\int u^{2}{\text{d}}V<C\right)}$ ${\displaystyle u_{0}=u(x,0)}$ ${\displaystyle C^{\infty }}$ ${\displaystyle \mathbb {R} ^{3}/\mathbb {Z} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle u_{0}}$ ${\displaystyle p}$ ${\displaystyle u}$
4. the solution of the P-NP problem ,
5. the proof of the Poincaré conjecture (solved in 2002 by Grigori Jakowlewitsch Perelman ),
6. the proof of the Riemann Hypothesis ,
7. exploring the equations of Yang-Mills . More precisely, a strict justification (in the sense of the axiomatic quantum field theory ) of the quantized Yang-Mills theory is asked for any compact simple gauge groups in four dimensions (Euclidean space-time) and the existence of a mass gap (i.e. the predicted energetically lowest particles have finite positive mass). This corresponds to the expectation in the case of quantum chromodynamics (QCD), where Glueballs have finite, non-zero mass, even if the gauge bosons ( gluons ) are massless. The problem was also chosen to approximate the most important unsolved problem of Yang Mills theories like QCD, the confinement problem. ${\displaystyle G}$

This Millennium List is in the tradition of the list of 23 previously unsolved problems of mathematics that led to the development of mathematics in the 20th century, drawn up on August 8, 1900 by the German mathematician David Hilbert at the International Congress of Mathematicians in Paris has significantly enriched and advanced. The Riemann Hypothesis is the only problem on both lists.

solutions

Poincaré conjecture

The Poincaré conjecture was proven in 2002 by Grigori Jakowlewitsch Perelman . For his groundbreaking work, he was awarded the Fields Medal in 2006 , which he rejected (as the first mathematician in history). The Clay Institute awarded him the $ 1 million prize in 2010, but he declined.

literature

• Pierre Basieux : The Top Seven Mathematical Guesswork. rororo, Reinbek near Hamburg 2004. ISBN 3-499-61932-6 .
• James A. Carlson: The Millennium Prize Problems. American Math. Soc., Providence 2006, ISBN 0-8218-3679-X .
• Keith J. Devlin: The millennium problems - the seven greatest unsolved mathematical puzzles of our time. Basic Books, New York 2002, ISBN 0-465-01729-0 .

Web links

• CMI page on the Millennium Problems
• John Baez on the Millennium Problems, English

Videos

• Millennium Meeting claymath.org

Individual evidence

1. ↑ Genie refuses prize money. In: Focus Online. July 2, 2010, accessed December 27, 2013 .