The world of mathematics is based on ideas and logically sound theorems that is reflection of some larger truth of the world around us. The set of equations of formulas known starts off as an idea, a hunch inside the brains of the greatest mathematicians ,till they are written down , perfected and proved, finding its way to the pages of journals and textbook, ready to be introduced to the world.
However, not every speculation makes it way to the bottom of the page. Some of these statements remain as conjectures , ready to be debunked by some mathematician of the future. One of such conjectures in the branch of topology is the Poincare Conjecture that has tormented mathematicians for decades before Russian mathematician, Grigori Perelman solved it almost a century later . In 2004, he made a landmark contribution to the field of geometric topology by formally proving the famous conjecture making it to the pages of Science Journal as the first “Mathematical Breakthrough of the Year”. During his lifetime, Perelman turned down several accolades the most mentionable ones being Fields Medal in 2006 and Millenium Prize. Perelman refused the sum of $1 million as he felt that it was unjust for not sharing the prize with Richard S Hamilton, the mathematician who discovered the Ricci Flow.
THE POINCARE CONJECTURE
Poincare Conjecture states that ” Every simply-connected 3-manifold is homeomorphic to the 3-sphere”.
The conjecture, in very simple terms classifies finite space, based on whether all possible loops on its surface can be tightened to a point without any point on the loop extending beyond or dipping below the surface of the three dimensional space. i.e. the loop must always remain on the surface during the tightening.
Objects in which this is possible are spheres, ellipsoids and others.
Some objects that doesn’t fall in this category are torus. This is because in some points on the torus, a loop will get trapped like a lasso. A much more detailed article on the conjecture will be posted later.
In mathematics and science , classification in general, serves as an important tool that helps to sort and understand the similarities and differences of entities in a systematic manner. There isn’t a very specific example of its application, but it definitely helps understand spheres in a new way. Most of the times, some theorems in mathematics remain unsolved due to inadequacy of either fundamental or related theorems required in its proof. Perelman solved the conjecture by proving the stronger Thurston’s geometrization technique that gave a deeper understanding of 3-manifolds in general.
Another important aspect of its proof is understanding higher dimensional space. Intuitively, it is more than difficult to wrap our minds around something as bizarre as fourth dimensional space and shapes such as a 4-sphere. Yet, in solving the theorem, we are able to understand concepts of hypersphere and fourth dimensional topological analogues in a much better manner. The once known Poincare Conjecture, now a theorem, equips us with tools which will surely aid mathematicians for future research and discoveries.