The Millennium Prize Problems are a set of seven mathematical problems that were designated by the Clay Mathematics Institute in 2000. A prize of one million dollars is offered for the solution to each problem. These problems were chosen because of their fundamental importance and difficulty, and solving any of them would represent a significant advance in the field of mathematics. The seven Millennium Prize Problems are:
Birch and Swinnerton-Dyer conjecture: This conjecture deals with the relationship between the number of rational points on elliptic curves and the behavior of certain associated L-series.
Hodge conjecture: This problem involves the topology of algebraic cycles on non-singular projective algebraic varieties.
Navier–Stokes existence and smoothness: This problem concerns the behavior of solutions to the Navier-Stokes equations, which describe the motion of incompressible fluid flow.
P versus NP problem: This is a question about the relationship between the time it takes to verify the solution to a problem and the time it takes to solve the problem.
Poincaré conjecture: Proven by Grigori Perelman in 2003, this conjecture was the only one of the seven to be solved at the time of the Millennium Prize announcement. It dealt with the classification of three-dimensional manifolds.
Riemann hypothesis: This is a conjecture about the distribution of prime numbers and is one of the most famous open problems in mathematics.
Yang-Mills existence and mass gap: This problem concerns the existence of quantum Yang-Mills fields with a mass gap, which is essential for the consistent formulation of the quantum field theory.
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.
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.
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.
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.
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.
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.
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.