One famous list of unsolved problems is the “Millennium Prize Problems,” established by the Clay Mathematics Institute in 2000. These problems each carry a prize of $1 million for a correct solution.
The seven Millennium Prize Problems are:
- The Birch and Swinnerton-Dyer Conjecture: Elliptic curves are a type of algebraic curve defined by cubic equations with two variables. The conjecture concerns the behavior of these curves over the field of rational numbers. The rank of an elliptic curve is a measure of the number of rational points on the curve. The L-function associated with an elliptic curve is a complex analytic function that encodes information about the distribution of prime numbers related to the curve. The conjecture asserts that there is a precise relationship between the rank of an elliptic curve and the behavior of its associated L-function, but this relationship has not been fully understood.
- The Hodge Conjecture: The Hodge Conjecture concerns the cohomology groups of algebraic varieties. These groups are a way of measuring the topological structure of these objects. The conjecture asserts that certain topological properties of an algebraic variety can be computed algebraically, in terms of the structure of its cohomology groups. This would have important implications for algebraic geometry and topology, but the conjecture remains unsolved.
- The Navier-Stokes Equations: The Navier-Stokes equations describe the behavior of fluids, such as air and water, in terms of their velocity, pressure, and temperature. The equations are non-linear and complex, making them difficult to solve in general, especially in three dimensions. Solving the Navier-Stokes equations is important for understanding many real-world phenomena, such as turbulence and weather patterns.
- The P versus NP Problem: The P versus NP problem is one of the most famous open problems in computer science. It concerns the complexity of computational problems and asks whether every problem that can be solved in polynomial time by a non-deterministic algorithm can also be solved in polynomial time by a deterministic algorithm. If the answer is yes, it would have important implications for cryptography, optimization, and many other fields.
- The Poincaré Conjecture: The Poincaré Conjecture concerns the topology of three-dimensional manifolds. A manifold is a geometric object that is locally similar to Euclidean space, but which may have a more complex global structure. The conjecture asserts that any simply connected, closed, three-dimensional manifold is topologically equivalent to the three-dimensional sphere. This was famously proved by Grigori Perelman in 2002, using a novel approach based on Ricci flow.
- The Riemann Hypothesis: The Riemann Hypothesis concerns the distribution of prime numbers, which are the building blocks of the integers. The Riemann zeta function is a complex analytic function that encodes information about the distribution of prime numbers. The hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on a specific line in the complex plane, called the critical line. The Riemann Hypothesis has important implications for number theory and has been called the “holy grail” of mathematics.
- The Yang-Mills Theory: The Yang-Mills Theory is a mathematical framework for describing the behavior of elementary particles in terms of gauge fields. Gauge fields are a type of field that transform in a specific way under certain transformations, such as rotations or phase changes. The theory has important implications for particle physics and the study of the fundamental forces of nature. The problem is to find a rigorous mathematical proof of the existence and behavior of Yang-Mills fields, which is still an open problem.
Solved in past 100 years
- Hilbert’s 23 Problems: In 1900, mathematician David Hilbert presented a list of 23 unsolved problems in mathematics, which became a major driving force for research throughout the 20th century. Several of these problems were later solved, including the solution to the 10th problem by Yuri Matiyasevich in 1970.
- Four Color Theorem: In 1976, mathematicians Kenneth Appel and Wolfgang Haken proved the Four Color Theorem, which states that any map can be colored using only four colors such that no two adjacent regions have the same color.
- Fermat’s Last Theorem: In 1994, mathematician Andrew Wiles proved Fermat’s Last Theorem, a problem that had remained unsolved for over 350 years. The theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.
- Poincaré Conjecture: In 2003, mathematician Grigori Perelman proved the Poincaré Conjecture, a problem that had been open for over a century. The conjecture states that any simply connected, closed 3-dimensional manifold is homeomorphic to the 3-dimensional sphere.
- Kepler Conjecture: In 1998, mathematician Thomas Hales proved the Kepler Conjecture, which states that no packing of congruent spheres in three dimensions can have a density greater than that of the face-centered cubic packing.
During the period of 1500-1900
- Algebraic equations: In the 16th century, Italian mathematicians such as Niccolò Fontana Tartaglia and Gerolamo Cardano developed methods for solving cubic and quartic equations, laying the groundwork for modern algebra.
- Calculus: In the 17th century, mathematicians such as Isaac Newton and Gottfried Leibniz independently developed calculus, a powerful tool for solving problems in physics, engineering, and other fields.
- Probability theory: In the 17th century, mathematicians such as Blaise Pascal and Pierre de Fermat worked on probability theory, developing the concept of expected value and laying the foundations for modern probability theory.
- Number theory: In the 18th and 19th centuries, mathematicians such as Leonhard Euler and Carl Friedrich Gauss made significant contributions to number theory, including work on modular arithmetic, prime numbers, and Diophantine equations.
- Non-Euclidean geometry: In the 19th century, mathematicians such as Nikolai Lobachevsky and János Bolyai developed non-Euclidean geometry, which challenged the traditional notions of geometry and paved the way for the development of modern geometry.
- Arabic numerals: In the 10th century, Persian and Arab mathematicians developed the modern numeral system, including the use of zero and positional notation, which revolutionized arithmetic and algebra.
- Algebra: In the 9th century, the Persian mathematician Al-Khwarizmi wrote the first book on algebra, “Al-Jabr wa’l-Muqabala,” which introduced systematic methods for solving linear and quadratic equations.
- Trigonometry: In the 10th century, the Persian mathematician Al-Battani made important contributions to trigonometry, including the discovery of the sine function and the use of trigonometric tables.
- Fibonacci sequence: In the 13th century, the Italian mathematician Fibonacci introduced the famous Fibonacci sequence, a series of numbers in which each number is the sum of the two preceding numbers.
- Indian mathematics: In India, mathematicians such as Brahmagupta and Bhaskara II made significant contributions to number theory, algebra, and trigonometry, including the discovery of negative numbers and the sine function.
There have been many great mathematicians throughout history who have made significant contributions to the field.
- Euclid: A Greek mathematician who lived in the 4th century BCE, Euclid is known for his work “Elements,” which is considered one of the most influential books on geometry.
- Archimedes: A Greek mathematician and physicist who lived in the 3rd century BCE, Archimedes made important contributions to calculus, geometry, and mechanics, including the principle of buoyancy and the law of the lever.
- Isaac Newton: An English mathematician and physicist who lived in the 17th century, Newton is famous for his work on calculus, optics, and mechanics, including his laws of motion and universal gravitation.
- Leonhard Euler: A Swiss mathematician who lived in the 18th century, Euler made significant contributions to number theory, calculus, and graph theory, and is known for his prolific output and his use of notation and symbols still used in mathematics today.
- Carl Friedrich Gauss: A German mathematician who lived in the 18th and 19th centuries, Gauss made important contributions to number theory, algebra, and geometry, including the fundamental theorem of algebra and the Gauss-Bonnet theorem.
- Bernhard Riemann: A German mathematician who lived in the 19th century, Riemann made significant contributions to geometry, analysis, and number theory, including the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.
- Ada Lovelace: An English mathematician and writer who lived in the 19th century, Lovelace is known for her work on Charles Babbage’s proposed mechanical general-purpose computer, the Analytical Engine, and is considered one of the first computer programmers.
- Emmy Noether: A German mathematician who lived in the early 20th century, Noether made important contributions to abstract algebra, including Noether’s theorem, which relates symmetries in physics to conservation laws.
- Alan Turing: An English mathematician and computer scientist who lived in the 20th century, Turing is known for his work on the code-breaking efforts during World War II, as well as his development of the concept of a universal machine, which led to the development of modern computers.
- John von Neumann: A Hungarian-American mathematician who lived in the 20th century, von Neumann made important contributions to game theory, quantum mechanics, and computer science, including the development of the von Neumann architecture, a model for computer architecture still used today.
- Andrew Wiles: A British mathematician who made headlines in the 1990s for his proof of Fermat’s Last Theorem, a problem that had remained unsolved for over 350 years.