Somewhere else on this site I already mentioned the top millenium problems of m athematics. More on this subject here:
Just one of the seven Millennium Prize Problems named 21 years ago has been solved. Answering the others would offer major new insights in fundamental mathematics and might even have real-world consequences for technologies such as cryptography.
But big questions in math have not often attracted the same level of outside interest that mysteries in other scientific areas have. When it comes to understanding what math research looks like or what the point of it is, many folks are still stumped, says Wei Ho, a mathematician at the University of Michigan. Although people often misunderstand the nature of her work, Ho says it does not have to be difficult to explain.
“My cocktail party spiel is always about elliptic curves,” she adds. Ho often asks partygoers, “You know middle school parabolas and circles? Once you start making a cubic equation, things get really hard…. There are so many open questions about them.”
One famous open problem called the Birch and Swinnerton-Dyer conjecture concerns the nature of solutions to equations of elliptic curves, and it is one of the seven Millennium Prize Problems that were selected by the founding scientific advisory board of the Clay Mathematics Institute (CMI) as what the institute describes as “some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium.” At a special event held in Paris on May 24, 2000, the institute announced a prize of $1 million for each solution or counterexample that would effectively resolve one of these problems for the first time. Rules revised in 2018 stipulate that the result must achieve “general acceptance in the global mathematics community.”
The 2000 proclamation gave $7 million worth of reasons for people to work on the seven problems: the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the P versus NP problem, the Yang-Mills existence and mass gap problem, the Poincaré conjecture, the Navier-Stokes existence and smoothness problem, and the Hodge conjecture.
Yet despite the fanfare and monetary incentive, after 21 years, only the Poincaré conjecture has been solved.
An unexpected solution
In 2002 and 2003 Grigori Perelman1, a Russian mathematician then at the St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences, shared work connected to his solution of the Poincaré conjecture online. In 2010 CMI announced that Perelman had proved the conjecture and, along the way, had also solved the late mathematician William Thurston’s related geometrization conjecture. (Perelman, who rarely engages with the public, famously turned down the prize money.)
According to CMI, the Poincaré conjecture focuses on a topological question about whether spheres with three-dimensional surfaces are “essentially characterized” by a property called “simple connectivity.” That property means that if you encase the surface of the sphere with a rubber band, you can compress that band—without tearing it or removing it from the surface—until it is just a single point. A two-dimensional sphere or doughnut hole is simply connected, but a doughnut (or another shape with a hole in it) is not.
Martin Bridson, a mathematician at the University of Oxford and president of CMI, describes Perelman’s proof as “one of the great events of, certainly, the past 20 years” and “a crowning achievement of many strands of thought and our understanding of what three-dimensional spaces are like.”
And the discovery could lead to even more insights in the future. “The proof required new tools, which are themselves giving far-reaching applications in mathematics and physics,” says Ken Ono, a mathematician at the University of Virginia.
Ono has been focused on another Millennium Problem: the Riemann hypothesis, which involves prime numbers and their distribution. In 2019 he and his colleagues published a paper in the Proceedings of the National Academy of Sciences USA that reexamined an old, formerly abandoned approach for working toward a solution. In an accompanying commentary, Enrico Bombieri, a mathematician at the Institute for Advanced Study in Princeton, N.J., and a 1974 winner of mathematics’ highest honor, the Fields Medal, described the research as a “major breakthrough.”
Yet Ono says it would be unfounded to describe his work as “anything that suggests that we’re about to prove the Riemann hypothesis.” Others have also chipped away at this problem over the years. For instance, mathematician “Terry Tao wrote a nice paper a couple years ago on [mathematician Charles] Newman’s program for the Riemann hypothesis,” Ono says.
Progress on what won’t work
The fact that just one of the listed problems has been solved is not surprising to the experts—the puzzles are, after all, long-standing and staggeringly difficult. “The number of problems that have been solved is one more than I would expect” to see by now, says Manjul Bhargava, a mathematician at Princeton University and a 2014 Fields medalist. Bhargava himself has reported multiple recent results connected to the Birch and Swinnerton-Dyer conjecture, including one in which he says he and his colleagues “prove that more than 66 percent of elliptic curves satisfy the Birch and Swinnerton-Dyer conjecture.”
None of the problems will be easy to solve, but some may prove especially intractable. The P versus NP problem appears so difficult to solve that Scott Aaronson, a theoretical computer scientist at the University of Texas at Austin, calls it “a marker of our ignorance.” This problem concerns the issue of whether questions that are easy to verify (a class of queries called NP) also have solutions that are easy to find (a class called P).
Aaronson has written extensively about the P versus NP problem. In a paper published in 2009 he and Avi Wigderson, a mathematician and computer scientist at the Institute for Advanced Study and one of the winners of the 2021 Abel Prize, showed a new barrier to proving that the P class is not the same as the NP class. The barrier that Aaronson and Wigderson found is the third one discovered so far.
“There’s a lot of progress on showing what approaches will not work,” says Virginia Vassilevska Williams, a theoretical computer scientist and mathematician at the Massachusetts Institute of Technology. “Proving that P [is] not equal to NP would be an important stepping-stone toward showing that cryptography is well founded,” she adds. “Right now cryptography is based on unproved assumptions,” one of which is the idea that P is not equal to NP. “In order to show that you cannot break the cryptographic protocols that people need in modern computers,” including ones that keep our financial and other online personal information secure, “you need to at least prove that P is not equal to NP,” Vassilevska Williams notes. “When people have tried to pin me down to a number,” Aaronson says, “I’ll give a 97 percent or 98 percent chance that P is not equal to NP.”
Climbing Mount Everest
Searching for solutions to the prize problems is similar to trying to climb Mount Everest for the first time, Ono says. “There are various steps along the way that represent progress,” he adds. “The real question is: Can you make it to base camp? And if you can, you still know you’re very far.” For problems such as the Birch and Swinnerton-Dyer conjecture and the Riemann hypothesis, Ono says, “surely we’re at Nepal”—one of the countries of departure for climbing the mountain—“but have we made it to base camp?” Mathematicians might still need additional “gear” to trek to the peak.
“We’re now trying to figure out what the mathematical analogues are for the high-tech tools, the bottles of oxygen, that will be required to help us get to the top,” Ono says. Who knows how many obstacles could be sitting between current research and possible solutions to these problems? “Maybe there are 20. Maybe we’re closer than we think,” Ono says.
Despite the difficulty of the problems, mathematicians are optimistic about the long term. “I hope very much that while I’m president of the Clay institute, one of them will be solved,” says Bridson, who notes that CMI is in the process of strategizing about how to best continue raising awareness about the problems. “But one has to accept that they’re profoundly difficult problems that may continue to shape mathematics for the rest of my life without being solved.”
The Poincaré conjecture
The Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This problem was solved by Perelman1 in 2002.
The Poincaré conjecture in simple terms
If you stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. But if you were to stretch a rubber band around the surface of a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. Therefore the surface of an apple is “simply connected,” and the one of the doughnut is not.
About a hundred years ago, Poincare knew that a two dimensional sphere is essentially characterized by this property of simple connectivity. He asked the corresponding question for the three dimensional sphere – the set of points in four-dimensional space at unit distance from the origin. As it turns out, this is an extraordinarily difficult question to be answered.
The Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation.
The Riemann hypothesis
The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
The P versus NP problem
The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. The informal term quickly, used here, means the existence of an algorithm solving the task that runs in polynomial time2, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time).
1 Grigori Yakovlevich Perelman was born 13 June 1966 and is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology.
On 22 December 2006, the scientific journal Science recognized Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year“, the first such recognition in the area of mathematics.
2 Running an algorithm can take up some computing time. It mainly depends on how complex the algorithm is. Computer scientists have made a way to classify the algorithm based on its behaviour of how many operations it needs to perform (more operations take up more time of course). One of that class shows polynomial time complexity. Operational complexity is proportional to nc while n is size of input and c is some constant. Obviously the name comes because of nc which is a polynomial.