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Millennium Prize Problems
The Millennium Problems are:
The Riemann Hypothesis
A 200-year old question that is one of the most famous mathematical problems ever. Solving it will make mathematicians understand a lot more about prime numbers. It has applications in cryptography, number theory, and might even be useful in physics.
Mathematicians want to know when a certain function, called the Riemann Zeta function, written ζ(s), equals zero. There are many well known values of s where ζ(s) is zero. They are the negative even integers. The Riemann Hypothesis says that apart from the negative even integers, ζ(s) equals zero only when s is a complex number with real part 1/2 or a negative whole number.
The Yang-Mills Equations
The solution to this problem is especially important for physicists, as it has applications in quantum mechanics and particle physics, two very important branches of physics. It will, most likely, have applications in mathematics as well.
This problem will be solved if someone proves that a certain set of equations, called the Yang-Mills equations, have solutions with certain properties.
The P versus NP problem
This problem is very important to computer science. It will be solved if someone manages to discover whether or not a computer can always find a solution to a problem as fast as it can check if the solution is correct. It has applications in engineering, cryptography, economics, and a bunch of other areas. Solving it would even affect the way that online sales and purchases are done.
The Navier-Stokes equations are probably the most important equations in fluid mechanics, which studies liquids and gases. People have been using it to build better cars and airplanes, to learn how the atmosphere and sea works, and many other things. They are much used in engineering, in mathematics, and in sciences.
The prize will go to the person who discovers whether there is a solution of the equations that eventually does not make sense (goes to infinity, for example). In other words, can a smoothly flowing fluid eventually become infinitely rough?
The Hodge Conjecture
This problem has few known applications outside of mathematics. It will help mathematicians understand a lot more about algebraic geometry and algebraic topology, which are connected to many other areas of mathematics. The problem is difficult to explain using words, as it involves things that are not found in everyday life, like algebraic varieties, homology and other related things.
The Poincaré Conjecture
The only Millennium Problem that has been solved, as of 2018. A mathematician named Grigori Perelman found out that it was true.
The Poincaré Conjecture states that the sphere is the only 3D object that can be shrunk to a single point, given certain conditions.