Neither brilliant mathematicians nor artificial intelligence have yet been able to handle them
Millennium problems are seven of the greatest mathematical challenges proposed for solution by the Clay Mathematics Institute in 2000. These problems are so complex that a prize of 1 million dollars is promised for the solution of each. Despite the efforts of the world's best mathematicians, most remain unsolved and continue to pose a challenge to the scientific community.
The history of its emergence
The idea of proposing mathematical problems of this scale arose by analogy with the well-known Hilbert problems, formulated by the German mathematician David Hilbert in 1900. Many of them defined the development of mathematics in the 20th century. Inspired by this example, American mathematician Arthur Jaffe initiated the creation of a list of problems for the new millennium in 2000. These problems address key questions in areas such as topology, number theory, quantum theory, and computational mathematics.
Of all seven problems, only one—the Poincaré conjecture—has been solved (in 2003 by Russian mathematician Grigori Perelman).
Let's take a closer look at each of them and try to understand why their solutions are still challenging.
1. The Riemann Hypothesis
The essence of the problem
The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. The Riemann zeta function is a key function in number theory that is closely related to the distribution of prime numbers.
Why it remains unsolved
The problem is difficult because the Riemann zeta function contains an enormous amount of information about prime numbers, which themselves exhibit unpredictable behavior. No known tools can prove the hypothesis or disprove it on general grounds.
2. The Hodge Conjecture
The essence of the problem
The Hodge conjecture asserts that on a smooth projective variety (a high-dimensional geometric space), any class of cohomology classes is a linear combination of classes of algebraic cycles.
Why it remains unsolved
Algebraic geometry is a complex field where deep knowledge of both algebra and geometry is required. Many of the methods used to study spaces simply do not work on all classes of varieties, making the Hodge conjecture extremely difficult to prove.
3. Navier-Stokes Equations
The essence of the problem
The Navier-Stokes equations describe the behavior of fluids and gases in space and time. The problem is to prove the existence of smooth solutions to these equations or to prove that such solutions may not exist under certain conditions.
Why it remains unsolved
Current mathematical methods are not powerful enough to cope with the analysis of fluid behavior at all possible scales. This is because the Navier-Stokes equations often exhibit chaotic behavior, especially in turbulent flows, making the problem extremely complex.
4. Yang-Mills Theory and the Mass Gap Hypothesis
The essence of the problem
Yang-Mills theory describes interactions between elementary particles using so-called gauge fields. The hypothesis is that there exists a non-zero lower bound for the mass of particles described by this theory (mass gap).
Why it remains unsolved
Field theory is an incredibly complex area of physics and mathematics. Building an accurate mathematical model that could describe the real behavior of particles faces serious challenges at both algebraic and analytical levels.
5. P ≠ NP
The essence of the problem
This is one of the most famous problems in computational machine theory. It asks: are all problems whose solutions can be verified in polynomial time also solvable in that same time? It is formulated as the question of whether two complexity classes P and NP are equal.
Why it remains unsolved
The P ≠ NP problem addresses fundamental questions about what computational complexity is and how we measure the efficiency of algorithms. Mathematicians have not yet found a way to either prove that NP class problems cannot be solved in polynomial time or demonstrate that they can be.
6. Birch and Swinnerton-Dyer Conjecture
The essence of the problem
The hypothesis relates the behavior of elliptic curves (defined by algebraic equations) to their L-functions and asserts that certain information about the number of solutions of an elliptic curve is contained in this L-function.
Why it remains unsolved
Elliptic curves are complex objects studied at the intersection of algebra, analysis, and geometry. Solving the conjecture will require the development of new methods, as existing approaches do not allow for a definitive conclusion about the connection between elliptic curves and their L-functions.
7. The P vs PSPACE Problem
The essence of the problem
This is a less well-known but also important problem concerning computational complexity. It questions whether the complexity classes P and PSPACE are equal, where P consists of problems solvable in polynomial time and PSPACE consists of problems solvable using polynomial memory.
Why it remains unsolved
Although the problem is related to algorithm analysis, proving or disproving the equality of P and PSPACE requires a deep understanding of the structure of computational models and the fundamental limitations of modern algorithms.
Why can't artificial intelligence solve the millennium problems?
Artificial intelligence (AI) and machine learning are advancing at incredible rates today and are helping to solve many problems, especially in data processing, optimization, and forecasting.
