Some problems are easy, and some problems are hard.
In the world of math and computer science, there are a lot of problems that we know how to program a computer to solve "quickly" - basic arithmetic, sorting a list, searching through a data table. These problems can be solved in "polynomial time," abbreviated as "P." It means the number of steps it takes to add two numbers, or to sort a list, grows manageably with the size of the numbers or the length of the list.
But there's another group of problems for which it's easy to check whether or not a possible solution to the problem is correct, but we don't know how to efficiently find a solution. Finding the prime factors of a large number is such a problem - if I have a list of possible factors, I can multiply them together and see if I get back my original number. But there is no known way to quickly find the factors of an arbitrary large number. Indeed, the security of the Internet relies on this fact.
For historical and technical reasons, problems where we can quickly check a possible solution are said to be solvable in "nondeterministic polynomial time,"
Any problem in P is automatically in NP - if I can solve a problem quickly, I can just as quickly check a possible solution simply by actually solving the problem and seeing if the answer matches my possible solution. The essence of the P vs NP question is whether or not the reverse is true: If I have an efficient way to check solutions to a problem, is there an efficient way to actually find those solutions?
Most mathematicians and computer scientists believe the answer is no. An algorithm that could solve NP problems in polynomial time would have mind-blowing implications throughout most of math, science, and technology, and those implications are so out-of-this-world that they suggest reason to doubt that this is possible.
Of course, proving that no such algorithm exists is itself an incredibly daunting task. Being able to definitively make such a statement about these kinds of problems would likely require a much deeper understanding of the nature of information and computation than we currently have, and would almost certainly have profound and far-reaching consequences
2.THE NAVIER-STOKES EQUATION
It's surprisingly difficult to explain what happens when you stir cream into your morning coffee.
The Navier-Stokes equations are the fluid-dynamics version of Newton's three laws of motion. They describe how the flow of a liquid or a gas will evolve under various conditions. Just as Newton's second law gives a description of how an object's velocity will change under the influence of an outside force, the Navier-Stokes equations describe how the speed of a fluid's flow will change under internal forces like pressure and viscosity, as well as outside forces like gravity.
The Navier-Stokes equations are a system of differential equations. Differential equations describe how a particular quantity changes over time, given some initial starting conditions, and they are useful in describing all sorts of physical systems. In the case of the Navier-Stokes equations, we start with some initial fluid flow, and the differential equations describe how that flow evolves.
Solving a differential equation means finding some mathematical formula to determine what your quantity of interest actually will be at any particular time, based on the equations that describe how the quantity changes. Many physical systems described by differential equations, like a vibrating guitar string, or the flow of heat from a hot object to a cold object, have well-known solutions of this type.
The Navier-Stokes equations, however, are harder. Mathematically, the tools used to solve other differential equations have not proven as useful here. Physically, fluids can exhibit chaotic and turbulent behavior: Smoke coming off a candle or cigarette tends to initially flow smoothly and predictably, but quickly devolves into unpredictable vortices and whorls.
It's possible that this kind of turbulent and chaotic behavior means that the Navier-Stokes equations can't actually be solved exactly in all cases. It might be possible to construct some idealized mathematical fluid that, following the equations, eventually becomes infinitely turbulent.
Anyone who can construct a way to solve the Navier-Stokes equations in all cases, or show an example where the equations cannot be solved, would win the Millennium Prize for this problem.
3. THE RIEMANN HYPOTHESIS
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Going back to ancient times, the prime numbers - numbers divisible only by themselves and 1 - have been an object of fascination to mathematicians. On a fundamental level, the primes are the "building blocks" of the whole numbers, as any whole number can be uniquely broken down into a product of prime numbers.
Given the centrality of the prime numbers to mathematics, questions about how primes are distributed along the number line - that is, how far away prime numbers are from each other - are active areas of interest.
By the 19th century, mathematicians had discovered various formulas that give an approximate idea of the average distance between primes. What remains unknown, however, is how close to that average the true distribution of primes stays - that is, whether there are parts of the number line where there are "too many" or "too few" primes according to those average formulas.
The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the "Riemann zeta function" all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.
Like several of the Millennium Prize problems, there is significant evidence suggesting that the Riemann Hypothesis is true, but a rigorous proof remains elusive. To date, computational methods have found that around 10 trillion solutions to the zeta function equation fall along the required line, with no counter-examples found.
Of course, from a mathematical perspective, 10 trillion examples of a hypothesis being true absolutely does not substitute for a full proof of that hypothesis, leaving the Riemann Hypothesis one of the open Millennium Prize problems.
In September 2018, the esteemed mathematician Professor Michael Atiyah claimed to have found a proof of the Riemann Hypothesis in a lecture at Heidelburg Laureate Forum. However, several mathematicians have expressed skepticism about the result, and a thorough process of reviewing and evaluating Atiyah's arguments will now begin.
4 The Birch and Swinner
5 The Hodge conjecture
6 Yang-Mills theory and the quantum mass gap
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