What are they?
These are 7 mathematical problems, some of the most important in mathematics, that if solved, will earn the solver a million-dollar prize and a reward equivalent to the Nobel Prize, as there is no similar award in this field of knowledge.
The Clay Mathematics Institute (CMI) proposed these prizes in 2000 for anyone who solves these problems, in honor of the 23 Hilbert problems proposed in 1900. One of these 23 problems is also included in the CMI list.
What are these problems?
As mentioned earlier, these problems are seven:
- The P versus NP problem.
- The Hodge conjecture.
- The Riemann hypothesis.
- The Poincaré conjecture (SOLVED).
- The Navier-Stokes equations.
- The Birch and Swinnerton-Dyer conjecture.
- Yang-Mills and the mass gap.
The P versus NP Problem
This problem refers to the field of computational complexity. Alan Turing proposed it in the 1970s. This problem classifies problems into two categories: those solvable in a determined amount of resources (time and memory required) and those that are not.
P refers to the first type of problems (easily solvable by computers), and NP refers to the second type (hard to find or solve, potentially taking thousands of years, but easy to verify once a solution is found).
The P versus NP problem asks if all NP problems are also P problems. If P equals NP, all NP problems would have a hidden shortcut that would allow computers to quickly find perfect solutions and have unlimited problem-solving power. However, if P does not equal NP, there are no such shortcuts, which would prove that the problem-solving power of computers is limited.
The Hodge Conjecture
This conjecture involves two mathematical fields: differential geometry and algebraic geometry. It was proposed by Scottish mathematician William Hodge in 1950. It is one of the most abstract and difficult-to-explain theories in mathematics.
The basic idea of the conjecture asks to what extent we can approximate the shape of a given object by constructing it from increasingly larger simple blocks. To this day, it remains an open problem. In fact, even among mathematicians, there is considerable division about whether the theory can be proven or disproven.
The Riemann Hypothesis
Bernhard Riemann first formulated this hypothesis in 1859. Due to its relationship with the distribution of prime numbers within the natural numbers, it is one of the most important open problems in contemporary mathematics. Riemann suggested that the distribution of these numbers is related to the behavior of the “Riemann zeta function,” which has two types of zeros: the “trivial” zeros, which are all even and negative integers, and the “non-trivial” zeros, whose real part is always between 0 and 1.
The Riemann hypothesis asserts that all non-trivial zeros of the zeta function lie on the line x = 1/2. To date, more than ten trillion zeros have been calculated for the zeta function, all aligned along the critical line, supporting Riemann’s suspicion. However, no one has yet been able to prove that the zeta function has no non-trivial zeros outside this line.
The Poincaré Conjecture
The Poincaré conjecture was established in 1904 by French mathematician Henri Poincaré. It was one of the most difficult problems to solve among the 7 Millennium Problems. We say “was” because it was solved in 2006, becoming the Poincaré Theorem, thanks to the work of Russian mathematician Grigori Perelman, who declined the prize.
The theorem asserts that the 4-dimensional sphere, also called a 3-sphere or hypersphere, is the only compact 4-dimensional manifold in which every loop or closed circle (1-sphere) can be deformed (transformed) into a point. This statement is equivalent to saying that there is only one closed and simply connected manifold: the 4-dimensional sphere.
The Navier-Stokes Equations
These are a set of nonlinear partial differential equations that describe the motion of a viscous fluid. These equations govern Earth’s atmosphere, ocean currents, the flow around vehicles or projectiles, and in general, any phenomenon involving Newtonian fluids.
Since their formulation, and in the correct form, these equations describe the movement of fluids, whether chaotic (turbulent flow) or smooth (laminar flow). However, there are still unknowns to resolve, such as the transition between laminar and turbulent flow and vice versa. According to Newtonian mechanics, these equations should predict the movement of a fluid based on its initial state, but it has been impossible to confirm or deny this up to now.
The Birch and Swinnerton-Dyer Conjecture
This is a mathematical conjecture, proposed in 1965 by English mathematicians Bryan Birch and Peter Swinnerton-Dyer, although it was first posed in the 10th century in an Arabic manuscript. The conjecture describes the set of rational solutions to the equations that define an elliptic curve. The solution to this conjecture would involve finding a criterion to distinguish elliptic curves.
Yang-Mills and the Mass Gap
The Yang-Mills hypothesis laid the foundation for the theory of elementary particles of matter, in which the quantum version describes massless particles (gluons). However, several experiments have concluded that there exists what scientists call a “mass gap,” a phenomenon not observed in nature but demonstrated in quantum theory. The resolution of the problem involves determining the existence of the Yang-Mills theory. In other words, determining whether all the particles in this theory (gluons) have mass or not.
Well, these are the 7 Millennium Problems. Would you dare to try solving one?
