As the millennium approached, the Clay Mathematics Institute (CMI), a research institution based in the United States, gathered leading global mathematicians for an ambitious undertaking. Under the guidance of the institute's scientific council, a set of seven novel mathematical challenges was selected, giving rise to the so-called 'Millennium Prize Problems.'
The CMI's proposal is clear: to offer a one-million-dollar prize to the researcher who solves any of these problems for the first time. It is important to note that none of the seven problems are considered simple, as they were chosen to represent some of the most complex issues faced by mathematicians during the transition into the second millennium.
On May 24, 2000, the Institute released the definitive list of the seven problems: the Birch and Swinnerton-Dyer Conjecture; the Hodge Conjecture; the problem of existence and uniqueness for Navier-Stokes equations; the Poincaré Conjecture; the P versus NP problem; the Riemann Hypothesis; and the mass gap problem for Yang-Mills quantum theory. According to the CMI, the purpose of this initiative is to raise public awareness about the fact that mathematics has open frontiers filled with crucial unanswered questions.
Currently, the Millennium Prize Problems are among the most studied topics in the field, and the CMI has not imposed a final deadline for their resolution. Over the past 26 years, only one has been officially solved: the Poincaré Conjecture, in 2003, by the Russian mathematician Grigori Perelman. However, recently, OpenAI announced that one of its artificial intelligences managed to solve the Navier-Stokes problem.
To illustrate mathematical concepts, a two-dimensional sphere is considered a single closed and connected surface. A practical example is an apple: if an elastic band is stretched around its surface, it can be reduced to a single point without being broken or leaving the surface. In contrast, when imagining the same elastic band stretched around a screw shape, it becomes impossible to reduce it to a point without breaking the band or the screw itself.
In 1904, Henri Poincaré identified that this dynamic can be represented by the two-dimensional surfaces of objects: the surface of an apple is classified as 'simply connected,' while that of a screw is not. The Poincaré Conjecture was a continuation of this investigation, seeking to know whether this discovery about two-dimensional surfaces would also apply to three-dimensional shapes.
Between 2002 and 2003, Perelman demonstrated that any closed, hole-free three-dimensional space is equivalent to a sphere, thus confirming the validity of Poincaré's discovery in three dimensions.
The P versus NP problem is currently considered the biggest unsolved challenge in computer science. It is based on the following question: if it is relatively easy to verify whether a solution to a given problem is correct, does that imply that finding that solution is also easy?
The Clay Mathematics Institute (CMI) itself offers an example: imagine organizing accommodation for 400 university students, where only 100 will be selected for the dormitory, and there are restrictions that certain students cannot live together. In this scenario, it is simple to check if a specific selection meets the criteria, but the process of generating that list from scratch is extremely complicated.
Mathematicians believe that the Navier-Stokes equations could allow for the prediction of both wind and turbulence, since these equations describe the movement of fluids, such as water. Although formulated in the 19th century, the understanding of these equations remains quite limited. The Navier-Stokes problem questions whether these equations always generate a valid solution for all times or if they can produce a singularity under certain conditions.
The claim made by OpenAI suggests having defined exactly one situation of this latter type.
The distribution of prime numbers among natural numbers does not follow a predictable pattern. However, the German mathematician G.F.B. Riemann (1826-1866) noted that the frequency of these prime numbers is linked to the behavior of a complex function. The Riemann Hypothesis postulates that all non-trivial zeros of the equation $\zeta(s) = 0$ are located on a specific vertical line. Although the statement has been proven for the first 10 trillion solutions, its truth for all solutions remains unknown.
In 1954, Chen Ning Yang and Robert Mills introduced a new structure to describe elementary particles using structures found in geometry. Although Yang-Mills theory is well established, its mathematical foundation still lacks clarity. Simulations and experiments point to the existence of a 'mass gap,' where the lowest energy state above the vacuum would have a minimum positive energy. The challenge lies in mathematically proving the rigorous existence of this theory and the occurrence of this gap.
The Birch and Swinnerton-Dyer problem proposes a new premise within Number Theory applied to elliptic curves. Elliptic curves are defined by specific equations, such as $y^2 = x^3 + ax + b$, and can possess a finite or infinite number of points with rational coordinates. This conjecture establishes a connection between the quantity of these points and the behavior of a mathematical function associated with the curve.
As explained by the CMI, 'when the solutions are points of an abelian variety, the Birch and Swinnerton-Dyer conjecture states that the size of the group of rational points is related to the behavior of an associated zeta function $\zeta(s)$ near the point $s=1$.'
The Hodge Conjecture investigates to what extent the topological characteristics of certain geometric spaces can be explained by elements defined by algebraic equations. It maintains that certain topological properties of these spaces correspond to combinations of subspaces that can also be described by algebraic equations.
