Two mathematicians have made history- they solved a century-old problem and got awarded $3, 000, 000 for it.
They are Christopher Hacon, who currently teaches at the University of Utah, and a mathematician hailing from the University of California- James McKernan. This year, they managed to win the breakthrough prize in Mathematics. This achievement was granted to them due to their proof of an ancient conjecture about the number of solutions we can assign a polynomial equation.
A polynomial equation is one in which any of the variables are raised to a whole number exponent. For example, the equation X^3+9X-11=2.
What these two showed is that no matter how difficult or complex the polynomial may be, it must always have a finite set of solutions.
The Breakthrough Prize is the single biggest individual prize in the sciences, at least in terms of money. It is currently sponsored by Google’s co-founder and Facebook’s founder. These awards are intended to go to scientists which make a monumental achievement in one of the following three categories:
- Life Sciences
- Fundamental Physics
One day, it’s entirely plausible for this prize to go to a bright young mind that found the cure for HIV. This kind of fundamental development is what the Breakthrough Prize was made to facilitate.
Like most complex mathematical problems, this one is simple to formulate, while the proof itself requires pages upon pages of high-level math.
Now, anyone that’s ever studied quadratic equation will understand what polynomial equations are. Chances are, you’ve done more than a few in your life. The idea that they must all have a finite number of solutions feels simple, almost trivial at first.
However, the proof itself spans many, many pages, and can only be understood by few mathematicians in the world.
Ideally, this is the type of problem students should be tackling. Naturally, not one that requires dozens of pages in order to prove, but problems that are simple and involve problem-solving. If we teach our children to work on problem-solving related problems today, we might find that they’re winning a prize like this tomorrow.
Given the basic model where a polynomial equation exists, such as X^2+Y^2=r^2 then we need to figure out how many different shapes of solutions there are. In this case, X and Y are variables, while r^2 is the radius of a circle.
For every polynomial, there is a shape that it makes when graphed. The one we showed you above, for example, outlines a circle.
There are other famous polynomials that define objects in more dimensions such as spheres, cubes, hypercubes etc. The more variables we have, the more dimensions it takes us in order to represent the polynomial. This also means that the more dimensions we have, the more shapes a solution can be.
Now, while mathematicians had a feeling that every polynomial, regardless of how many dimensions its solution required to represent, still had a finite number of shapes assigned to them. Despite this, the proof of this concept, referred to as the “minimal model program in all dimensions,” was nowhere to be found.
The proof displays that this feeling was correct, though only for one kind of shape, which is those that have at least one hole. In this scenario, the duo decided to use a well-known “lemma” which is an argument that is used to solve a far lesser problem. Now, once they understood this lemma could help them find the solution for this century-old problem, they started cracking on. According to them, the solution came quickly, which in the math world means just a few years.
Now, the current form of the solution can’t tell us how many solutions to a given polynomial there are, rather than this, it can only tell us that there’s only a finite number of them.
You might be thinking “Okay that’s cool, but what does this mean in reality?” Well, like many solutions to ancient questions, this one doesn’t currently help us with anything really. At least not in the practical world we live in. On the other hand, Hacon has pointed out that this has a large importance to the world of particle physics. Specifically, theoretical physics, Hacon said the following:
“There's this string theory that suggests there should be an extra sixth dimension of the universe that we can't perceive. So one question researchers have asked is, "How may possible shapes can these extra six dimensions have and how do those shapes affect the universe we see?"
Now, the current proof doesn’t get us quite this far. This is because of the most popular versions of string theory stipulating that the world is made up of rolled-up dimensions without holes, and the current proof only applies to shapes that have holes. With that being said, Hacon has pointed out that making such a proof after their work should prove significantly easier than before.
Now, the question of visualization is an interesting one. After all, how are we supposed to visualize something in 4D, 5D or even 6-dimensional space?
To that, Hacon responded with the following:
"You cheat. You've seen abstract paintings, Picasso and whatnot. The drawing is nothing like a real person but nevertheless you can recognize the main features and it does convey something to you."
In much the same way that Picasso draws his abstract pictures, we could envision a multi-dimensional space. Now, while this method of visualization can work for some number of dimensions, once we go past 4 or 5, it gets far too complex to be modeled by the human hand. Instead, mathematicians rely on computer models and simulations.
A development like this always leaves a mark on history. It’s fascinating how finding solutions to ancient problems can present future developments. Step by step, mathematics is getting us closer to solving some of mankind’s greatest problems.
Ranging from dose calculations in medicine to the thrust of an airplane, our world is dictated and aided by math and those that learned it.