Pi in the sky problems that no one can solve leads one to believe that perhaps there is nothing new under the sun. Perhaps just stirring the pot of the imagination will lead to discovering the solutions to seven of the toughest mathematical problems that no mathematician, no computer, has yet to solve. The Clay Mathematics Institute has offered a million dollars for each solution.
Birch and Swinnerton-Dyer Conjecture (Number Theory)
Shorthand version of problem: Prove that an ellipse, which is a curve, possesses infinite solutions in relation to the L-series (functions, part of the analytic number theory) with a value=0 (fixed point).
Hodge Conjecture (Algebraic Geometry-Topology)
Shorthand version of problem: If X is a projective complex manifold, which cohomology (algebraic invariant) classes emanate from complex subvarieties Z?
Navier-Stokes Equations (Fluid Mechanics)
Shorthand version of problem: Prove equations describing the flow (motion/velocity) of incompressible fluids in three dimensions always are in existence and do not contain any smoothness.
P vs. NP (Computer Science)
Shorthand version of problem: Complexity Theory; are all NP (non-deterministic polynomial time) problems P (polynomial time) problems?
Poincaré Conjecture (Topology: solved in 2006)
Grigori Perelman proved the three-dimensional conjecture, the only one of the seven "Millenium" listed that has been solved. The conjecture, now a theorem, involved three dimensions, three manifolds, and a sphere.
Riemann Hypothesis (Prime Numbers)
Shorthand version of problem: Non-trivial zeros, real parts equal a half. Proven to 10 million zeros on the line by computer, but not fully solved.
Yang-Mills Theory (Quantum Physics)
Shorthand version of problem: Prove that the smallest known particle is positive. This is known as the mass gap or the difference in energy from energy in a vacuum to the next level of energy.
Equations such as the above Millennium problems are important because they describe the world and have the potential to change it. Some contend that most equations are set up and not solved, noting it is the path to solving the equations which is the most important element, not the solution, which brings the creativity and change from a math problem.
Will solving the "unsolvable" math problems create new vision for the future of humankind? Is there anything new under the sun or is it all just recycling old ideas? New ideas build off of the old foundational ideas of math that govern the world. Standing on the shoulders of mathematicians past, today's mathematicians will determine how the tiniest particle behaviors, the infinite and the finite, equations depicting the flow of energy at levels unseen, for instance, will be solved. At least the road getting there will be paved with discovery and rediscovery that will affect the world's future science and economies, perhaps changing the way humans live forever.