On May 24, 2000, the world was given a set of seven problems. With the dawn of the new millennium, The Grand Challenge in Mathematics was announced by Arthur Jaffe, then president of the Clay Mathematical Institute. It has been almost 20 years since the challenge was put forth, still there is 6 million USD is to be won.

  • Birch and Swinnerton-Dyer conjecture
  • Hodge conjecture
  • Navier-Stokes existence and smoothness
  • P versus NP problem
  • Poincare conjecture
  • Riemann hypothesis
  • Yang-Mills existence and mass gap.

Out of these seven problems only the Poincare conjecture has been solved.

The Poincare conjecture

The Poincare conjecture was developed by Henry Poincare. He was a French mathematician, theoretical physicist and engineer from the 19th century. By the early 20th century he gave the world this conjecture. Poincare conjecture deals with topology and geometry. This conjecture kind of states that the simplest possible closed shape in any number of dimensions is a sphere.

It might seem so simple. But it has the solutions that are secrets to understanding the shape of the universe.

Consider a ball. Loop a string around it. If we shrink the loop by pulling the ends of the twine, it would ultimately shrink to a point. This implies that there is no hole in the shape.

If we do the same thing on a Doughnut or Ulundha Vadai, we would have to cover the surface of the shape with the loop. That would imply that the string goes through the centre of the doughnut. Then, when we shrink it, the loop will get stuck at the boundary of the shape. Proving the presence of a hole in the closed surface of the shape.

You get a million dollars if you solve one problem: The seven, millennium-prize problems Torus cycles
Neither of the two coloured loops on this torus can be continuously tightened to a point. A torus is not homeomorphic to a sphere.
Image and caption: Wikipedia

Thus in three dimensions, the sphere is the simplest shape (it had no holes).  generalized that for any higher dimensions the simplest shape would be a sphere in that dimension.

About a hundred years later in 2002, a Russian mathematician named Grigori Perelman gave the first valid proof for the conjecture using Ricci flow program. But he rejected the million-dollar prize and later the Fields medal (it is like a noble prize for mathematics).

The other problems are still unsolved. Waiting for the chosen one.

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