By: Amanda Zheng
Do you want to win one cool million dollars? All you need is to solve one math problem...
ONE MILLION DOLLARS?
Recently I walked past my calculus teacher’s office and I saw seven math problems neatly posted on his door. With a closer look, I realized that these problems, filled with complex symbols and confusing titles, were way past the level of general calculus. I was curious as to what the problems were for. My calculus teacher noticed me pondering over the problems on his door, “They’re interesting problems right? If you solve one of them you’ll get one million dollars.” These seven problems are called the Millennium Prize Problems, designed by the Clay Mathematics Institute in 2000 to commemorate the turn of the century. A group of top mathematicians in the world contributed some of the most puzzling math problems of the time, and invited the entire world to join in solving them.
THE POINCARÉ CONJECTURE
Before you run for the pen and paper to start working on your one million dollar dream, you have been warned about how difficult the problems are. During the 18 years since they have been published, only one problem has been successfully solved, the Poincaré Conjecture. The problem was a topography theory posed by a French mathematician in 1904 but has not been proved until 2006. This extremely complex question deals with a sphere-like object that exists in 4 or more dimensions. I know you must be thinking, what could a 4 dimensional sphere possibly look like? This is a fundamental question in the field of topography, and the answer is not as straightforward as one would hope. Humans see life in 3 dimensions while the 4th dimension is purely theoretical as of now. If you would like to see more about 4 dimensional objects, and what they would hypothetically look like in our 3 dimensional world, check out this video by The Action Lab on Youtube. The normal 3 dimensional sphere has the property that any loop on it can be contracted to a point (if a rubber band is wrapped around the sphere, it is possible to slide it down to a point). Other spaces do not have this property, for example the donut: a rubber band that goes around the whole donut once cannot be slid down to a point without it leaving the surface. The Poincaré Conjecture asks if this property would hold for spheres in 4 dimensions or more. To learn more, watch this video by Numberphile in which topologists explain this prominent problem.
The Poincaré Conjecture was solved by Grigori Perelman, a Russian mathematician, in 2006, but what is more surprising is that he didn’t take the monetary prize . Perelman modeled his work off the Ricci flow process developed by Richard S. Hamilton, a Columbia math professor, to create a modified process called Ricci flow with surgery. He said he was not interested in the prize or fame, “I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me.'"
REMAINING CHANCES AT ONE MILLION DOLLARS
Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.
If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct or not. But it’s not so easy find a solution.
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most fundamental questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But at dimension four it is unknown.
Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, just to name three.
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References
[1] http://www.claymath.org/millennium-problem. 10/20/18