The Poincare conjecture is another great math problem that had yet to be proven for over 100 years. Conjectured by the French mathematician Henri Poincare in 1904, he asked “If a three-dimensional shape is simply connected, is it homeomorphic to the three-dimensional sphere?”
We’ll need to begin by clarifying what we mean by dimensions when we are discussing topology. The premise of this book is learning mathematical concepts by transcending dimensions, but those are Euclidean dimensions. For instance a sphere has three Euclidean dimensions because those are the number of dimensions needed to contain the object (within its space). Topological dimensions are different -- if we want to convey the number of dimensions on the surface of a sphere, we would be describing the number of independent directions needed to describe a given point on that object (the sphere). If we were to have the sphere covered in a graph with Cartesian coordinates, we would only need two points to locate it -- so it’s really a two-dimensional manifold.
A sphere as we know it, is really called a 2-sphere in 3-dimensional space (because it has two dimensions or orientation on the surface of the sphere. A disk would be a one-sphere and what Poincare is conjecturing, is a three-sphere in four dimensions. While a two-dimensional sphere models a world, a three-dimensional sphere models a universe. It is finite (the boundary does not extend forever like an abstract Cartesian plane, so it can loop back on itself) and has no boundary (there’s no edge). In fact because of these properties, one can make the argument that every loop within this sphere can be shrunk to a single point -- Poincare describes this to us in a thought experiment using a slipknot.
Before we dive into his thought experiment, we’ll want to explain a few things with topology. Two geometrical objects are called homeomorphic if the first can be deformed into the second by stretching and bending -- but not by cutting. The idea can be simple to conceive: you take a square and you inflate it to become a sphere. You can take a cylinder and blow it up to make a sphere. Can you blow up a donut and make it into a sphere? No, you cannot. In this case the hole is the culprit as you cannot make it into a sphere without cutting an end of it to morph the pieces together. This is easy to understand in 3 dimensions, but extrapolating this to higher dimensions gets difficult (remember, in higher dimensions we’re left with only coordinates & our mind as our devices).
Topology is all about morphing shapes into much simpler shapes and holes are usually what makes some transformations impossible. In Poincare’s thought experiment, he had us imagine wrapping a slipknot around a sphere and pulling on the string only to find that it always closed into a single point. This makes the sphere simply connected. Molding the shapes to produce as simple a shape as possible is called a manifold and in this case, the sphere is a simply-connected 3-dimensional manifold. This can be done for all the shapes that are able to morph into the sphere with the exception of the donut. If you wrapped a string around the donut and tried to pull on it, you cannot distill it to a single point and it’s because of the hole it contains.
This idea is critical to answering the question of what shape our universe is if you imagine wrapping that string around the entire end of the universe. This is known as homology, which is a mathematical way of counting different types of loops and holes in topological spaces. Using homology and the idea of simple connectedness, Poincare conjectured that all closed (no holes), simply connected, three-dimensional manifolds could be reshaped and smoothed into spheres. Intuitively this makes sense with the examples we provided, but mathematical hypotheses/conjectures that are easiest to state are often the hardest to solve. Remember, trying to answer this question about our universe while we are in it makes solving this notoriously difficult.
$1 Million was offered by the Clay Institute for several years if anyone can prove this conjecture and who/how it was proven caused an even bigger splash than the actual solving of the prize. Russian mathematician Grigori Perelman subtly published his proof on the internet, as it made its way around math circles. Once it was confirmed as a proof, the $1 Million prize was offered to him, only to be rejected by him. In addition, Perelman was offered the Fields Medal (the mathematician’s Nobel Prize offered once every four years since there are no Nobel Prizes for math), only to be rejected by him, saying the award was “completely irrelevant” and only solving the proof was prize enough.
Perelman solved this conjecture by using Ricci Flow, a technique that was originally used by Richard Hamilton in his attempt to prove the Poincare Conjecture. At a high level, Ricci Flow is an equation which evolves and morphs a manifold into a more understandable shape. Describing a curvature requires several numbers and the sophistication of Ricci Flow, which is a series of six equations elegantly stitched together similar to Einstein’s equation of general relativity, proved to be a useful tool in this endeavor. Ensuring those numbers describing the curve are positive implies a positive curvature and so as one morphs these objects, one can eventually do so until it becomes a sphere. This was shown by Hamilton, but proving that those variations into positive curvature stay constant was an issue. When morphing the blob, some areas would deviate from the general behavior of the flow, causing a singularity.
Let’s take an example, imagine you are pouring water from a pitcher into a glass and you spill some on the table. As you set the pitcher down, you see the water self-arranging, a process known as cohesion (it’s when you see two closely located drops suddenly stick together). That gradual movement is similar to Ricci Flow in which an object gradually smoothens itself out, it’s a concept quite similar to how heat/energy is evenly dissipated within a medium. If you were to heat a metal pipe, the entire pipe would eventually be evenly warmed up. The heat tends towards a certain state, similarly to homeostasis -- Ricci Flow is no different.
In order to overcome the issue of singularity when using Ricci Flow, Perelman applied what was known as “Ricci Flow by surgery”, where he would halt the flow, take the problematic area, and apply a specially-tailored function to correct it, therefore resuming the flow. Hamilton was the first to devise it but Perelman was able to build on it and perfect it. This was a rote process done by hand, but it was effective.
Perelman submitted his proof in November 2002, which first verified that the curvature was uniformly bounded and second had demonstrated that those singularities always occur in the same way from growing too rapidly -- in which he then created his special function to overcome those.
What’s amazing about Perelman’s proof is that, solved over 99 years after Poincare had conjectured, the solution wasn’t even possible until mathematician Richard Hamilton devised the particulars of Ricci Flow in 1982. This is why it’s important to continue doing math in its pure, abstract form -- you never know what solution or proof it may inspire, or what new fields in mathematics it can open. For now, there is no practical application in Perelman’s work or in Poincare’s Conjecture having been proved. But the methods in which Perelman cleverly created to prove this conjecture may have some application years or decades down the line when solving an analogous problem -- either in math or science.