The Seven Millennium Prize Problems

About a hundred years ago David Hilbert, a German mathematician resented twenty-three math puzzles to the International Congress of Mathematicians. Today, only three remain unsolved. Added to those were four more unsolvable problems. One is solved, but six have resisted all attempts to solve are listed here:

  1. The Poincare Conjecture [solved in 2002],
  2. The Birch and Swinnerton-Dyer Conjecture,
  3. The Navier-Stokes Equations,
  4. The Riemann Hypothesis (the oldest and most famous),
  5. The P Vs. NP Problem,
  6. The Hodge Conjecture,
  7. The Yang-Mills Existence and Mass Gap.

Many experts believe that solving these problems would lead to extraordinary advances in physics, medicine and many other unknown areas in the world of math.

On May 24, 2000 the Clay Mathematics Institute, has offered a prize of US$ 1,000,000 for the solution of one of the seven Millennium Prize Problems.

The Poincaré conjecture

The Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This problem was solved by Perelman1 in 2002.

Henri Poincaré (1854-1912)

The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation.

Sir Henry Peter Francis Swinnerton-Dyer (1927-2018) and Bryan John Birch (1931- )

The Navier–Stokes equations

The Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids.

Claude Louis Navier (1785-1836) and George Gabriel Stokes (181`9-1903)

The Riemann hypothesis

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

Bernhard Riemann (1826-1866)

The P versus NP problem

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. The informal term quickly, used here, means the existence of an algorithm solving the task that runs in polynomial time2, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time).

The Hodge conjecture

The Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
Or, in simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which can not be otherwise easily visualized.

The Yang–Mills existence and mass gap problem

The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics. The problem is phrased as follows: prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap Δ > 0. Where R4 is Euclidean 4-space and the mass gap Δ is the mass of the least massive particle predicted by the theory.

1 Grigori Yakovlevich Perelman was born 13 June 1966 and is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology.
On 22 December 2006, the scientific journal Science recognized Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year“, the first such recognition in the area of mathematics.

2 Running an algorithm can take up some computing time. It mainly depends on how complex the algorithm is. Computer scientists have made a way to classify the algorithm based on its behaviour of how many operations it needs to perform (more operations take up more time of course). One of that class shows polynomial time complexity. Operational complexity is proportional to nc while n is size of input and c is some constant. Obviously the name comes because of nc which is a polynomial.

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