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Accretion disc jets: Why do the accretion discs surrounding certain astronomical objects, such as the nuclei of active galaxies, emit relativistic jets along their polar axes?
Accelerating universe: Why is the expansion of the universe accelerating, as we have observed? What is the nature of the dark energy driving this acceleration? If it is due to a cosmological constant, why is the constant so small, yet non-zero? Why isn't it huge, as predicted by most quantum field theories, or zero for some yet unknown symmetry reason? What is the ultimate fate of the universe?
Amorphous solids: What is the nature of the transition between a fluid or regular solid and a glassy phase? What are the physical processes giving rise to the general properties of glasses?
Twin prime conjecture
Number of Magic squares
Hilbert's sixteenth problem
Generalized star height problem
P versus NP
The Hodge conjecture
The Poincaré conjecture
The Riemann hypothesis
Yang-Mills existence and mass gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture
Originally posted by JIMC5499
If you look at many of the great inventions of the last century an astonishing number of them were invented by someone with an informal education working in a garage or home workshop. One thing that needs to be considered is what happens if someone actually solves one of these problems. I don't know what the legalities of a group like this may be. I would really hate for someone to post a solution and get screwed out of the potential personal and financial gain because he posted it here.