# What are the 7 big problems in mathematics?

Contents

The seven big problems in mathematics are: The Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, Navier-Stokes Existence and Smoothness, P versus NP Problem, Hodge Conjecture, Yang-Mills Existence and Mass gap, and the Existence and Smoothness of the solutions to the Navier-Stokes equations.

## So let us dig a little deeper

Mathematics has always been a subject of curiosity and fascination for many. The seven big problems in mathematics challenge the most talented mathematicians in the world.

The first problem on the list is the Riemann Hypothesis, named after the German mathematician Bernhard Riemann. It deals with the distribution of prime numbers, and its proof can have far-reaching applications in cryptography and data security. It remains unsolved since its formulation in 1859.

The second problem is the Birch and Swinnerton-Dyer Conjecture, which concerns the properties of elliptic curves and their relation to number theory. The conjecture was proposed by Bryan Birch and Peter Swinnerton-Dyer in 1960 and remains unsolved to this day.

The third problem is the Navier-Stokes Existence and Smoothness problem, which concerns the equations governing the motion of fluids. It deals with whether solutions exist to these equations and whether they are smooth and well-behaved. It has been one of the most challenging problems of modern mathematics and remains unsolved despite being posed in the 19th century.

The fourth problem on the list is the P versus NP Problem, which relates to the complexity of computational problems. It asks whether every problem that can be solved by a computer in polynomial time can also be verified in polynomial time. It is one of the most important open problems in computer science and mathematics.

The fifth problem is the Hodge Conjecture, which concerns the topology of complex algebraic varieties. It asks whether certain types of differential forms on such varieties can be expressed as a combination of simpler forms. It remains unsolved since its formulation in 1950.

The sixth problem is the Yang-Mills Existence and Mass gap problem, which concerns the behavior of subatomic particles and their interactions. It asks whether the fundamental particles in our universe have mass and what the nature of their interactions is. It has been an active area of research in mathematical physics and remains unsolved.

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The seventh problem on the list is the Existence and Smoothness of the solutions to the Navier-Stokes equations, which is a more specific problem related to fluid dynamics. It asks whether solutions to these equations exist and whether they are smooth and well-behaved. It is a more focused version of the third problem on the list.

As John Nash, a famous mathematician, once said: “Mathematics is the study of patterns and structures.” The seven big problems in mathematics highlight some of the most interesting and challenging patterns and structures that mathematicians have been trying to understand for decades.

Table:

| Problem | Description |

|———|————-|

| Riemann Hypothesis | Queries the numerical distribution of prime numbers |
| Birch and Swinnerton-Dyer Conjecture | Analyzes the properties of elliptic curves |
| Navier-Stokes Existence and Smoothness | Deals with equations governing the motion of fluids |
| P versus NP Problem | Concerns the complexity of computational problems |
| Hodge Conjecture | Analyzes the topology of complex algebraic varieties |
| Yang-Mills Existence and Mass gap | Deals with subatomic particles and their interactions |
| Existence and Smoothness of the solutions to the Navier-Stokes equations | Focuses on specific issues related to fluid dynamics |

Interesting facts:
– The Navier-Stokes equations were named after Claude-Louis Navier and George Gabriel Stokes in the 19th century.
– The P versus NP problem has been designated as one of the Millennium Prize Problems, with a \$1 million prize offered for its solution.
– The Birch and Swinnerton-Dyer Conjecture has been described as one of the most important unsolved problems in algebraic number theory.
– The Hodge Conjecture has been connected to the Langlands program, which is a set of conjectures linking number theory and geometry.
– The Yang-Mills Existence and Mass gap problem is connected to the behavior of quarks and gluons, the building blocks of matter.

## Some further responses to your query

The seven problems are the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equations, P versus NP, the Poincaré Conjecture, the Riemann Hypothesis, and the Yang-Mills Theory. In 2003, the Poincaré Conjecture was proven by Russian mathematician Grigori Perelman.

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute on . The problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap.

The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

Here’s a few:

P versus NP problem [ http://en.wikipedia.org/wiki/P_%3D_NP_problem ]
Hodge conjecture [ http://en.wikipedia.org/wiki/Hodge_conjecture ]
Riemann hypothesis [ http://en.wikipedia.org/wiki/Riemann_hypothesis ]
Yang–Mills existence and mass gap [ http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap ]
Navier–Stokes existence and smoothness [ http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness ]
Birch and Swinnerton-Dyer conjecture [ http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture ]

Solve any one of these and you will be awarded \$1 million.

EDIT: The asker decided to add the statement “Aside from the millennium problem” after I posted. Please refer to Amanda Urquiza’s answer to What are the biggest problems in mathematics that remain to be solved? [ https://www.quora.com/What-are-the-biggest-problems-in-mathematics-that-remain-to-be-solved/answer/Amanda-Urquiza-1 ] for a very detailed list.

## This video has the solution to your question

The video discusses the paradoxical nature of mathematics, where people have an innate understanding of basic principles but struggle with higher levels requiring abstract thinking. It explores famous paradoxes such as Fermat’s Last Theorem and the reward for solving math problems, such as the \$1 million bounties for Millennium Prize Problems offered by the Clay Mathematics Institute. It then delves into the P vs. NP problem and the Navier-Stokes equations, which are some of the problems among the Millenium Prize Problems. Andrew Wiles, the mathematician who solved Fermat’s Last Theorem, explains the years he spent working on it in secret and the impact of solving major mathematical problems on a person’s life and career.

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## You will be interested

It’s interesting that, The Millennium Prize Problems gives the official description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics. CMI/AMS publication. 165pp., Hardcover, List: \$32. AMS Members: \$25.60.
Did you know that, To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture, which was solved in 2003 by the Russian mathematician Grigori Perelman. Clay “to increase and disseminate mathematical knowledge.”

## I’m sure you will be interested

### What are the 7 most difficult math problems?

As a response to this: Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

### What is the biggest math problem?

Mathematicians worldwide hold the Riemann Hypothesis of 1859 (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

### How many of the 7 millennium problems have been solved?

Answer: Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems. To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010.
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### What math problems Cannot be solved?

The reply will be: The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve. So what is the Collatz Conjecture and what makes it so difficult?

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### What are the 7 types of mathematical problems?

Answer to this: The seven selected problems range over a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science.

### Why were the 7 problems chosen?

The seven particular problems were chosen in part because of their difficulty, but even more so because of their central importance to modern mathematics. The problems and the corresponding general areas of mathematics are as follows.

### How many math problems are still unsolved?

In 2000, the Clay Mathematics Institute, a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered \$1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

### Are there any unsolved problems in Algebra and model theory?

Response: The Erlagol Notebook ( Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory. Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

### What are the 7 problems in math?

The seven problems are the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equations, P versus NP, the Poincaré Conjecture, the Riemann Hypothesis, and the Yang-Mills Theory. In 2003, the Poincaré Conjecture was proven by Russian mathematician Grigori Perelman .

### How many math problems are still unsolved?

The answer is: In 2000, the Clay Mathematics Institute, a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered \$1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

### Are there any unsolved problems in Algebra and model theory?

The Erlagol Notebook ( Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory. Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

### What are the Millennium Problems?

Response: The Millennium Problems are a set of seven problems for which the Clay Mathematics Institute offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new millennium in May 2000. The problems all have significant impacts on their field of mathematics and beyond, and were all unsolved at the time of the offering of the prize.

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