If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere. 

Clay Mathematics Institute

PROBLEM STATEMENT

If a compact three-dimensional manifold M³ has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M³ is homeomorphic to the sphere S³?

Henri Poincare’ commented, with considerable foresight, “Mais cette question nous entraˆınerait trop loin”. Since then, the hypothesis that every simply connected closed 3-manifold is homeomorphic to the 3-sphere has been known as the Poincare’ Conjecture. It has inspired topologists ever since, and attempts to prove it have led to many advances in our understanding of the topology of manifolds. From the first, the apparently simple nature of this statement has led mathematicians to overreach. Four years earlier, in 1900, Poincare’ himself had been the first to err, stating a false theorem that can be phrased as follows.

HIGHER DIMENSIONS

The fundamental group plays an important role in all dimensions even when it is trivial, and relations between generators of the fundamental group correspond to two-dimensional disks, mapped into the manifold. In dimension 5 or greater, such disks can be put into general position so that they are disjoint from each other, with no self-intersections, but in dimension 3 or 4 it may not be possible to avoid intersections, leading to serious difficulties. Stephen Smale announced a proof of the Poincare’ Conjecture in high dimensions in 1960. He was quickly followed by John Stallings, who used a completely different method, and by Andrew Wallace, who had been working along lines quite similar to those of Smale.

The Ricci Flow

Let M be an n-dimensional complete Riemannian manifold with the Riemannian metric g_ij . The Levi-Civita connection is given by the Christoffel symbols

where g^ij is the inverse of g_ij . The summation convention of summing over repeated indices is used here and throughout the book. The Riemannian curvature tensor is given by

We lower the index to the third position, so that

The curvature tensor R_ijkl is anti-symmetric in the pairs i, j and k, l and symmetric in their interchange:

Also the first Bianchi identity holds

The Ricci tensor is the contraction

and the scalar curvature is

We denote the covariant derivative of a vector field v = v^j (∂/∂x^j)by

and of a 1-form by

These definitions extend uniquely to tensors so as to preserve the product rule and contractions. For the exchange of two covariant derivatives, we have

and similar formulas for more complicated tensors. The second Bianchi identity is given by

For any tensor T = Tⁱ_jk we define its length by

and we define its Laplacian by

the trace of the second iterated covariant derivatives. Similar definitions hold for more general tensors.

The Ricci flow of Hamilton is the evolution equation

for a family of Riemannian metrics g_ij (t) on M. It is a nonlinear system of second order partial differential equations on metrics.

A COMPLETE PROOF OF THE POINCARE’ AND GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW

As a pure mathematician Newton reached his climax in the invention of the calculus, an invention also made independently by Leibniz. But do you know that, Fermat conceived and applied the leading idea of the differential calculus thirteen years before Newton was born and seventeen before Leibniz was born.

Today we are going delve into Fermat’s last work, whose proof took three and a half centuries to hail again after Fermat. Its nothing but Fermat’s Last Theorem.

We will start to take our steps towards the brilliance of this theorem by learning a bit about the Pythagorean Triples.Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². The equation a² + b² = c² is a Diophantine equation.

The Theorem

The proposition was first conjectured by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin.

Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2.

The Big Picture

Wiles’ three groundbreaking lectures, in June 1993, at the Isaac Newton Institute in Cambridge, UK were inspirational and shaped mathematics in its own way.

Theorem. Every semistable elliptic curve over Q is modular.

The strategy that ultimately led to a successful proof of Fermat’s Last Theorem was the connection with elliptic curves which arose from the “astounding” Taniyama–Shimura–Weil conjecture, proposed around 1955 which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat’s equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat’s Last Theorem, as well as leading the way to a full proof by others of what is now the modularity theorem.

The Taniyama-Shimura conjecture

The Taniyama-Shimura conjecture, since its proof now sometimes known as the modularity theorem, is very general and important conjecture (and now theorem) connecting topology and number theory which arose from several problems proposed by Taniyama in a 1955 international mathematics symposium.

The Shimura-Taniyama-Weil conjecture relates elliptic curves (cubic equations in two variables of the form y2 = x3 + ax + b, where a and b are rational numbers) and modular forms, objects (to be defined below) arising as part of an ostensibly different circle of ideas. An elliptic curve E can be made into an abelian group in a natural way after adjoining to it an extra “solution at infinity” that plays the role of the identity element. This is what makes elliptic curves worthy of special study, for they alone, among all projective curves (equations in two variables, compactified by the adjunction of suitable points at infinity) are endowed with such a natural group law. If one views solutions geometrically as points in the (x, y)-plane, the group operation consists in connecting two points on the curve by a straight line, finding the third point of intersection of the line with the curve and reflecting the resulting point about the x-axis.

The entire proof is not possible to provide through this post but a link will suffice it.

Courtesies- https://www.wikipedia.org/, THE PROOF OF FERMAT’S LAST
THEOREM by Nigel Boston University of Wisconsin – Madison.