Back in May 2000, the Clay Mathematics Institute published a now-famous list of seven major unsolved mathematics problems, collectively titled the “Millennium Prize Problems.” As an encouragement for scholars to devote their attention and efforts towards those specific problems in the coming years, the CMI offered a one million dollar prize for each successful solution–a substantial amount of money for almost anyone, and certainly for most scholars. According to the CMI:
“The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.” (1)
To earn one of those cool million dollar prizes, a successful solution must 1) have been published in an appropriate scholarly journal; 2) have been in print for at least two years; and 3) have garnered a general consensus among mathematicians in that time. In other words, the work has to be thoroughly vetted by the mathematical community before the CMI will hand over the prize. (2)
Even the formal statements of six of the seven problems (the Yang-Mills and Mass Gap, the Riemann Hypothesis, the Navier-Stokes Equation, the Hodge Conjecture, the Poincaré Conjecture, and the Birch and Swinnerton-Dyer Conjecture) are likely beyond the comprehension of anyone lacking advanced degrees in mathematics. Only the P vs NP Problem is somewhat likely to be solved by a non-mathematician, given its transformational implications for computer science and cybersecurity. In case you are interested in learning more about the history of the problems themselves, the CMI, along with the American Mathematical Society, has published an edited collection appropriately titled The Millennium Prize Problems. Edited by James Carlson, Arthur Jaffe, and Andrew Wiles, the volume is now available freely online as a PDF.
In the twenty years since the problems were originally stated, only one–the Poincaré conjecture–has been solved. In 2010, Russian mathematician Grigori Perelman was awarded the first of the Millennium Prizes for work published in 2002-2003, though surprisingly, he declined the prize on the grounds that his work built on that of another scholar who was equally deserving.
Since that time, a number of scholars (and amateurs) have come forward with proposed solutions, but to date, none has been accepted by the CMI and the remaining six problems remain officially unsolved. A Kazakh scholar named Mukhtarbay Otelbaev drew a lot of attention in 2014 for a proposed solution to the Navier-Stokes equations that he published in Mathematical Journal, but a team of scholars vetting his work found an error that had likely tanked the solution beyond repair. (3) More recently, Michael Atiyah put forward a proposal regarding the Riemann hypothesis, but that too met with significant skepticism and criticism, and Atiyah has since passed away. (4)
Whatever happens with those specific attempts, it seems likely that interest in the Millennium Prize Problems will continue to grow as the prizes remain unclaimed and the problems remain unsolved.
(1) “The Millennium Prize Problems,” Clay Mathematics Institute, accessed October 27, 2020, https://www.claymath.org/millennium-problems/millennium-prize-problems.
(2) “Rules for the Millennium Prizes,” Clay Mathematics Institute, accessed October 27, 2020, https://www.claymath.org/millennium-problems/rules-millennium-prizes.
(3) Katia Moskvitch, “Fiendish million-dollar proof eludes mathematicians: Proposed solution to Navier-Stokes equations, used to model fluids, is wrong,” Nature, August 5, 2014, accessed October 27, 2020, https://www.nature.com/news/fiendish-million-dollar-proof-eludes-mathematicians-1.15659.
(4) Frankie Schembri, “Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis,” Science, September 24, 2018, accessed October 27, 2020, https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis.