The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe, who conceived and implemented its structure and mission. The Institute is run according to a standard structure comprising a board of directors that decides on grant-awarding and research proposals, and a scientific advisory committee that oversees and approves the board's decisions. As of May, 2006, the board is integrated by members of the Clay family (including Landon Clay), whereas the advisory committee is composed of leading authorities in mathematics, namely Sir Andrew Wiles, Yum-Tong Siu, Richard Melrose, Gregory Margulis,and Simon Donaldson A non-profit organization (often called non-profit org or simply non-profit or not-for-profit) can be seen as an organization that doesnt have a goal to make a profit. ... A charitable foundation is a legal categorization of nonprofit organizations that either donate funds and support to other organizations, or provide the sole source of funding for their own activities. ... Location in Massachusetts Coordinates: , Country United States State Massachusetts County Middlesex County Settled 1630 Incorporated 1636 Government  - Type Mayor-council city  - Mayor Kenneth Reeves (D) Area  - City  7. ... Official language(s) English Capital Boston Largest city Boston Area  Ranked 44th  - Total 10,555 sq mi (27,360 km²)  - Width 183 miles (295 km)  - Length 113 miles (182 km)  - % water 13. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Harvard University (incorporated as The President and Fellows of Harvard College) is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League. ... Arthur Jaffe is an American mathematical physicist and a professor at Harvard University. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Yum-Tong Siu (Chinese: ; pinyin: Xiāo Yìntáng; born May 6, 1943 in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University. ... Gregori Aleksandrovich Margulis (first name often given as Gregory, Grigori or Grigory) (born February 24, 1946) is a mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. ...

Millennium Prize Problems

The institute is best known for its establishment on May 24, 2000 of Millennium Prize Problems. These seven problems are considered by CMI to be "important classic questions that have resisted solution over the years". The first person to solve each problem will be awarded $1,000,000 by the CMI. In announcing the prize, CMI drew a parallel to Hilbert's problems, which were proposed in 1900, and had a substantial impact on 20th century mathematics. Of the initial twenty-three Hilbert problems, most of which have been solved, only one (the Riemann hypothesis, formulated in 1859) is one of the seven Millennium Prize Problems.^[1] Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

P versus NP

Main article: Complexity classes P and NP

The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time), it can also find that solution quickly. This is generally considered the most important open question in theoretical computer science. Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... Look up computation in Wiktionary, the free dictionary. ...

The Hodge conjecture

Main article: Hodge conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles. The Hodge conjecture is a major unsolved problem of algebraic geometry. ... This article does not cite its references or sources. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In mathematics, a Hodge cycle is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic...

The Poincaré conjecture

Main article: Poincaré conjecture

In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every 2-dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds. A solution to this conjecture was proposed by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was awarded the Fields Medal for his solution. Perelman declined the award.^[2] In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... A sphere is a perfectly symmetrical geometrical object. ... An open surface with X-, Y-, and Z-contours shown. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ...

The Riemann hypothesis

Main article: Riemann hypothesis

The Riemann hypothesis is that all nontrivial zeros of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert's eighth problem, and is still considered an important open problem a century later. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ...

Yang-Mills existence and mass gap

Main article: Yang-Mills existence and mass gap

In physics, classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the deictic phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap. It has been suggested that this article or section be merged with Yang-Mills existence and mass Gap. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... Colour confinement (often just confinement) is the physics phenomenon that color charged particles (such as quarks) cannot be isolated. ... A quantum field theory model is said to have a mass gap if the energy spectrum not including zero has a positive greatest lower bound. ... In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i. ... See gauge theory for the classical prelimanaries. ...

Navier-Stokes existence and smoothness

Main article: Navier-Stokes existence and smoothness

The Navier-Stokes equations describe the movement of liquids and gases. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations. This article or section is in need of attention from an expert on the subject. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... A liquid will usually assume the shape of its container A liquid is one of the main states of matter. ... This article does not cite any references or sources. ... This article or section is in need of attention from an expert on the subject. ... This article or section is in need of attention from an expert on the subject. ...

The Birch and Swinnerton-Dyer conjecture

Main article: Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions. In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... Hilberts tenth problem is the tenth on the list of Hilberts problems of 1900. ...

Other activities

Besides the Millennium Prize Problems, the Clay Mathematics Institute also supports mathematics via the awarding of research fellowships (which range from two to five years, and are aimed at younger mathematicians), as well as shorter-term scholarships for programs, individual research, and book writing. The Institute also has a yearly Clay Research Award, recognizing major breakthroughs in mathematical research. Finally, the Institute also organizes a number of summer schools, conferences, workshops, public lectures, and outreach activities aimed primarily at junior mathematicians (from the high school to postdoctoral level). The Clay Research Award is given annually by the Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research. ...

References

1. ^ Arthur Jaffe's first-hand account of how this Millennium Prize came about can be read in The Millennium Grand Challenge in Mathematics
2. ^ http://news.bbc.co.uk/2/hi/science/nature/5274040.stm

• Keith J. Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books (October, 2002), ISBN 0-465-01729-0.

External links

• The Clay Mathematics Institute
• The Millennium Grand Challenge in Mathematics
• The Millennium Prize Problems
• QEDen: Millennium Problems Wiki