Who wants to be a millionaire?
Sometimes winning $1,000,000 is simple. You can scratch a ticket, or beat the statistical odds of a lottery drawing, or earn a spot on Who Wants to be a Millionaire? Easy, right?! A not so easy way to win $1,000,000 is to solve one of the six remaining Millennium Problems. It used to be seven problems until Mathematician Grigori Perelman solved the Poincare conjecture, and then, in Perelman style, denounced his award and refused the money. The backstory on the rejection of his award is that he believed that his contribution to solving the conjecture was really no greater than Columbia University’s Mathematician Richard Hamilton, whose theories Perelman utilized to solve Poincare’s conjecture. Okay, so he’s a man with convictions. Let’s give him that.
So now, who is ready to give up on Scratchers and solve the remaining problems?
First, we have P versus NP, introduced by American-Canadian Computer Scientist and Mathematician, Professor Stephen Cook of the University of Toronto. This particular problem is a computer science problem that asks whether every problem that can be solved and verified quickly can also be quickly solved and verified again.
In short, P versus NP is like comparing a set of problems that are easy to solve and verify ℗ against problems that are easy to verify but cannot be quickly solved (NP). We know that an algorithm exists for problems in the P set, but we don’t quite know if an algorithm exists for the problems in the NP set.
Professor Pierre René Viscount Deligne of the Institute for Advanced Study introduced Hodge conjecture. Briefly, the Hodge conjecture is a topological problem and it states that complicated algebraic varieties can be reduced to algebraic cycles that are combinations of simpler forms. It incorporates many concepts in mathematics that include topology, geometry, calculus, and algebra. David Metzler of Albuquerque Academy has a really great series of six videos that helps explain Hodge conjecture starting here:
Riemann Hypothesis, based in number theory and introduced by Professor Emeritus Enrico Bombieri of the Institute for Advanced Study, asks for the proof or disproof that all non-trivial zeros of the Riemann zeta function lie on the critical line of ½ with the exception of the trivial zeros along the negative axis. I love this one! One of my favorite videos on Riemann zeta function is an explanation by Professor Edward Frenkel of the University of California Berkeley. It’s a fantastic video produced by Numberphile and Brady Haran.
The Yang-Mills existence and mass gap was presented by Mathematical Physicists Arthur Jaffe of Harvard University and Edward Witten of the Institute for Advanced Study. Jaffe and Witten created a theory that addresses strong and weak nuclear forces. The theory is based on differential geometry and quantum mechanics. In the Yang-Mills theory, there is a mass gap, also known as a non-zero Mass. It is literally a gap between strong and weak nuclear forces! This gap represents the difference between the lowest energy state and the highest energy state. Even though physical experiments show that there is a mass gap, and even though computer-based mathematical models show that there is a mass gap, we are still trying to explain, mathematically, the Mass Gap in the quantum applications of their formulas. In my opinion, this is one of the most exciting Millennium problems presented. Some theorists say it is much like the early days of calculus or even quantum physics when tangible results were being discovered even though the process of fully reducing logical aberrations through theoretically rigorous mathematical explanations hadn’t been established yet. The Yang-Mills existence and the mass gap is like forwarding to the end of the movie, seeing the most amazing ending, yet missing the climactic adventure at the end of Act 2! The excitement is yet to come!
The Navier-Stokes existence and smoothness problem is a lovely and wonderful problem for mathematicians like myself who enjoy Partial Differential Equations. Mathematics Professor Charles Fefferman of Princeton University introduced this equation.
I personally became emotionally attached to the Navier-Stokes equation as an undergrad, when a peer jokingly proposed that half of the class solve it, just to prove to the world of mathematics that sometimes undergrads know what they are doing. By the time we reached finals, we rather realized that we had no idea what we were doing.
This gif depicts Leonhard Euler’s (18th century) equation, which describes the flow of frictionless and incompressible fluids. It gets trickier with Claude-Louis Navier’s application of viscosity. Finally, Sir George Gabriel Stokes created solutions, but they were only applicable in two-dimensional flow. Now, throw in pressure, gradient turbulence, viscous flow, non-steady flows that depend on time, vortices, chaos, and three-dimensions, and you have a smoothness problem that either never occurs or presents pockets of energy per unit mass.
In 2014, Mathematics Professor Mukhtarbay Otelbayev of the Eurasian National University presented a possible solution utilizing an abstract approach. However, his results are still being analyzed. Math Professor Terence Tao of the University of California Los Angeles has presented some wonderful analysis on Navier-Stokes and Otelbayev’s solution that you can read here: https://terrytao.wordpress.com/tag/navier-stokes-equations/. Additionally, in 2016 Otelbayev presented more work on Navier-Stokes (https://aip.scitation.org/doi/abs/10.1063/1.4959619?journalCode=apc) that “clarified the correct formulation of the boundary conditions for the pressure in the environment.”
Finally, we have the Birch and Swinnerton-Dyer conjecture. Who doesn’t love donuts?!!!
Oxford University’s Regius Professor Sir Andrew John Wiles, who proved Fermat’s Last Theorem, presented the Birch and Swinnerton-Dyer conjecture. The conjecture was born out of numerical investigations of elliptic curves conducted by Professors Bryan Birch and Peter Swinnerton-Dyer at the University of Cambridge, using the EDSAC computer. This beautiful conjecture says that there must be a mathematically simple way to determine whether an elliptic curve, which is a cubic curve (to the order of 3) that is restricted to a torus (a donut) has either finite or infinite rational solutions, based on whether the function is not equal to zero or equal to zero. The details of this conjecture are beautiful (or elegant as some would say), and for budding mathematicians, it is delightful to dig into as you learn about Mordell’s Theorem, the rank of a curve and multiple consequences.
So, here are six Millennium Problems to solve and win $1,000,000, thanks to the Clay Mathematics Institute! Remember: they are called the Millennium Problems, not Millennial Problems. Millennial Problems are much harder! Those include figuring out how to pay down the exponentially rising interest on your student loans, or how to work five jobs and still get sleep. No, Millennium Problems and its glorious mathematics are much easier!
For budding mathematicians, you can read about the Millennium Problems at the Clay Mathematics Institute website at http://www.claymath.org/millennium-problems