Math seems like such a well-defined subject that it is difficult to believe that there are still any big questions left unanswered. After all, for a subject that’s been studied extensively for at least 3000 years and one that builds on itself so easily, how can there be anything left unknown? However the reality is there are so many unproven conjectures and areas of research that it would be impossible for me to list them all.
At the turn of the millennium, the Clay Mathematics Institute chose 7 hugely important unanswered problems still plaguing mathematics today. Whoever solves one of the “Millenium Problems” is entitled to receive a whopping $1,000,000 for their work.
So what kinds of solutions are worth a million dollars? The following is the statement of the Poincaré Conjecture, the only Millenium Prize Problem to have been solved so far:
“Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere” [1].
So, let’s break down exactly what this statement is saying. First, what is a 3-manifold? In mathematics, a manifold is essentially a surface with the property that at any given point, the surface resembles our regular intuition of a space with some number of dimensions [2]. We actually see a manifold in action every day by living on the Earth. Although the Earth is a sphere, when we walk on it, it’s easy to believe we lie on a flat 2-dimensional surface. Thus, the earth, and any 3-dimensional sphere, is a 2-manifold since when we walk on it, we effectively walk in a 2-dimensional space.
More generally, a n-manifold is a higher dimensional (for simplicity's sake we can just assume the dimension is n+1) surface, but when an observer zooms in on a particular point on the manifold, the observer will see that an n-manifold will behave like n-dimensional Euclidean space. Thus, the 3-manifold is a 4-dimensional surface that behaves in 3-dimensions when we zero in on it.
A 3-sphere is a specific example of a 3-manifold, but what exactly is it? To understand what a 3-sphere is, it is useful to examine a 2-sphere. To begin, a 2-sphere is what we normally think of when we see the word “sphere”: a 3-dimensional surface with each point equidistant from a given center. Likewise, a 3-sphere is just the 4-dimensional analog; it is a 4-dimensional surface with each point equidistant from a given center.
The conjecture proposes that any 3-manifold that is continuous (without any holes or other weird features) can be molded in a way that transforms the 3-manifold into the 3-sphere. For a slightly more rigorous definition, each point on a 3-manifold can be mapped to a unique point on the 3-sphere and vice-versa.
Though we had to go through some complex definitions to understand the conjecture, the statement itself is simple enough and seems rather intuitive. However, it was actually proposed by Henry Poincaré in 1904, before being solved over one hundred years later by Grigori Perelman [3]. That’s not to say no one else had attempted to solve the question. By some metrics, the Poincaré Conjecture has had more false proofs than any other statement in recent history [3]. Rather, the fact that it took over a century to solve it shows how difficult proving this seemingly-simple statement is and there are six more just like it!
There are still a lot of unanswered questions in mathematics, but with each solution, we get closer to a better understanding of the universe we live in.
References
[Image] Veisdal, J. The Poincaré conjecture - cantor’s paradise - medium https://medium.com/cantors-paradise/the-poincar%C3%A9-conjecture-cb4ca7014cc5 (accessed Mar 1, 2021).
http://www.owlnet.rice.edu/~fjones/chap1.pdf (accessed Mar 1, 2021).
Prize, P. The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles by George G. Szpiro.
Poincaré Conjecture https://www.claymath.org/millennium-problems/poincar%C3%A9-conjecture (accessed Mar 1, 2021).
