“As far as the laws of mathematics refer to reality, they are not certain;-Albert Einstein
and as far as they are certain, they do not refer to reality.“
In 1900, Henri Poincare put for a conjecture that colloquially states that
“If it walks like a sphere and it quacks like a sphere, it is a sphere.” This statement, known as the Poincare conjecture, became one of the early questions in a field now called “low-dimensional topology” and proved itself to be extremely subtle and intractable. It attracted the attention of many great mathematicians and attempts to solve it lead to the development of powerful tools.
In the early 1980s, Richard Hamilton put forth an ambitious program to
attack the Poincare conjecture. He proposed a process called the Ricci flow that would deform the shape of a space and hopefully allow its curvature to dissipate throughout the space. If the space satisfied the hypotheses of the Poincare conjecture, he hoped the space would evolve into a round sphere and that he could prove the conjecture with this nicer space. He was able to make this approach work in two dimensions as a proof of concept, but he found that in three dimensions the flow would sometimes violently rip apart spaces and so was unable to finish the proof.
The rest of the story is already a part of mathematical lore.
If there is no heat source the hottest points of any surface will cool down while the coldest ones warm up. This phenomena has been studied using the heat equation, which was introduced by Joseph Fourier. This equation says that a heat distribution u will evolve over time via the equation ∂u/∂t = ∆u.
The above equation involves the Laplacian ∆u. Many of you will remember this from vector calculus as being ∇·(∇u) (or div(grad u)) but it is illustrative to think of it in terms of coordinates, where we have (in R3) ∆u = ∂2u/∂x2 + ∂2u/∂y2 + ∂2u/∂z2 .
However, for the context of the Ricci flow, we can consider the Hessian matrix,
The Heat Equation–
The heat equation is a partial differential equation given by the expression ∂u/∂t = ∆u. We think of t as being a time parameter and the right hand side as being derivatives in space. Given an initial distribution of heat a solution to this equation will give the temperature of points at a time t. We call such a solution a “heat flow.”
Background material from Ricci flow–
Hamilton introduced the Ricci flow equation, ∂g(t)/∂t = −2Ric(g(t)). This is an evolution equation for a one-parameter family of Riemannian metrics g(t) on a smooth manifold M. The Ricci flow equation is weakly parabolic and is strictly parabolic modulo the ‘gauge group’, which is the group of diffeomorphisms of the underlying smooth manifold. One should view this equation as a non-linear, tensor version of the heat equation. From it, one can derive the evolution equation for the Riemannian metric tensor, the Ricci tensor, and the scalar curvature function. These are all parabolic equations. For example, the evolution equation for scalar curvature R(x,t) is ∂R/∂t (x,t) = △R(x,t) + 2|Ric(x,t)|^2, illustrating the similarity with the heat equation.
For a better understanding I would recommend an understanding of the Riemannian metric tensor.
Grigori Perelman’s advances–
So far we have been discussing the results that were known before Perelman’s work. They concern almost exclusively Ricci flow (though Hamilton in had introduced the notion of surgery and proved that surgery can be performed preserving the condition that the curvature is pinched toward positive). Perelman extended in two essential ways the analysis of Ricci flow – one involves the introduction of a new analytic functional, the reduced length, which is the tool by which he establishes the needed non collapsing results, and the other is a delicate combination of geometric limit ideas and consequences of the maximum principle together with the non collapsing results in order to establish bounded curvature at bounded distance results. These are used to prove in an inductive way the existence of canonical neighborhoods, which is a crucial ingredient in proving that is possible to do surgery iteratively, creating a flow defined for all positive time. While it is easiest to formulate and consider these techniques in the case of Ricci flow, in the end one needs them in the more general context of Ricci flow with surgery since we inductively repeat the surgery process, and in order to know at each step that we can perform surgery we need to apply these results to the previously constructed Ricci flow with surgery. We have chosen to present these new ideas only once – in the context of generalized Ricci flows – so that we can derive the needed consequences in all the relevant contexts from this one source.
Grigori Perelman was offered The Fields Medal and one million dollars as promised by the Clay Mathematical Institute for proving the Poincare Conjecture using Ricci Flow. However he turned down the offer very humbly.