# Millennium Prize Problems

Seven Mathematical problems from the The Clay Mathematics Institute in 2000

The correct solution to any of these problems will be rewarded \$1 million Dollars US

The Poincaré conjecture at the moment, is the only one solved. It was solved by Grigori Yakovlevich Perelman in 2003

# Rules for the Millennium Prizes

The revised rules for the Millennium Prize Problems were adopted by the Board of Directors of the Clay Mathematics Institute on 26 September, 2018.

1. CMI does not accept direct submission of proposed solutions.
2. The document is a complete statement of the rules and procedures: CMI will not offer any futher guidance or advice.
3. Before CMI will consider a proposed solution, all three of the following conditions must be satisfied: (i) the proposed solution must be published in a Qualifying Outlet (see §6), and (ii) at least two years must have passed since publication, and (iii) the proposed solution must have received general acceptance in the global mathematics community

27 September, 2018

Taken from:

https://www.claymath.org/millennium-problems/rules-millennium-prizes

# Poincaré conjecture – Solved

Henri Poincare Conjecture
The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. … The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.

# P versus NP

P is the set of problems whose solution times are proportional to polynomials involving N’s. … NP (which stands for nondeterministic polynomial time) is the set of problems whose solutions can be verified in polynomial time. But as far as anyone can tell, many of those problems take exponential time to solve. Oct 29, 2009

# Hodge conjecture

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

https://www.claymath.org/millennium-problems/hodge-conjecture

# Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 12. … The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex.  – Wikipedia

# Yang–Mills existence and mass gap

YangMills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)).  – Wikipedia

# Navier–Stokes existence and smoothness

The NavierStokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. – Wikipedia

# Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.      –  Wikipedia