Do any decision problems exist outside NP and NP-Hard?

What are the real-life applications to the solutions of the Millennium Prize Problems?

• The Millennium Problems are: http://www.claymath.org/millenium-problems/yang%E2%80%93mills-and-mass-gap Experiment  and computer simulations suggest the existence of a "mass gap" in the  solution to the quantum versions of the Yang-Mills equations. But no  proof of this property is known.      http://www.claymath.org/millenium-problems/riemann-hypothesis The  prime number theorem determines the average distribution of the primes.  The Riemann hypothesis tells us about the deviation from the average.  Formulated in Riemann's 1859 paper, it asserts that all the  'non-obvious' zeros of the zeta function are complex numbers with real  part 1/2.      http://www.claymath.org/millenium-problems/p-vs-np-problem If  it is easy to check that a solution to a problem is correct, is it also  easy to solve the problem? This is the essence of the P vs NP question.  Typical of the NP problems is that of the Hamiltonian Path Problem:  given N cities to visit, how can one do this without visiting a city  twice? If you give me a solution, I can easily check that it is correct.  But I cannot so easily find a solution.      http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation This  is the equation which governs the flow of fluids such as water and air.  However, there is no proof for the most basic questions one can ask: do  solutions exist, and are they unique? Why ask for a proof? Because a  proof gives not only certitude, but also understanding.      http://www.claymath.org/millenium-problems/hodge-conjecture The  answer to this conjecture determines how much of the topology of the  solution set of a system of algebraic equations can be defined in terms  of further algebraic equations. The Hodge conjecture is known in certain  special cases, e.g., when the solution set has dimension less than  four. But in dimension four it is unknown.      http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture In  1904 the French mathematician Henri Poincaré asked if the three  dimensional sphere is characterized as the unique simply connected three  manifold. This question, the Poincaré conjecture, was a special case of  Thurston's geometrization conjecture. Perelman's proof tells us that  every three manifold is built from a set of standard pieces, each with  one of eight well-understood geometries.      http://www.claymath.org/millenium-problems/birch-and-swinnerton-dyer-conjecture Supported  by much experimental evidence, this conjecture relates the number of  points on an elliptic curve mod p to the rank of the group of rational  points. Elliptic curves, defined by cubic equations in two variables,  are fundamental mathematical objects that arise in many areas: Wiles'  proof of the Fermat Conjecture, factorization of numbers into primes,  and cryptography, to name three.

• Answer:

  Not everything in mathematics must have a real-life application. In fact, most results in mathematics do not (at least not directly). These mathematical problems are interesting and important because of their intellectual value, not because of their practical value. See these questions on Quora for further information: