The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of June 2012, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. The Poincaré conjecture, the only Millennium Prize Problem to be solved so far, was solved by Grigori Perelman, but he declined the award in 2010.

The seven problems are:

1. P versus NP problem
2. Hodge conjecture
3. Poincaré conjecture (Solved)
4. Riemann hypothesis
5. Yang–Mills existence and mass gap
6. Navier–Stokes existence and smoothness
7. Birch and Swinnerton-Dyer conjecture

P versus NP[link]

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. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not all problems in NP are also in P. This is generally considered the most important open question in theoretical computer science as it has far-reaching consequences in mathematics, biology, philosophy^[1] and cryptography (see P versus NP problem proof consequences).

If the question of whether P=NP were to be answered affirmatively it would trivialise the rest of the Millennium Prize Problems (and indeed all but the unprovable propositions in mathematics)^{[dubious – discuss]} because they would all have direct solutions easily solvable by a formal system.

"If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."

— Scott Aaronson, MIT

Most mathematicians and computer scientists expect that P≠NP.

The official statement of the problem was given by Stephen Cook.

The Hodge conjecture[link]

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

The official statement of the problem was given by Pierre Deligne.

The Poincaré conjecture (proven)[link]

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 two-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 had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.

The official statement of the problem was given by John Milnor.

A proof of this conjecture was given by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was selected to receive the Fields Medal for his solution. Perelman declined that award.^[2] Perelman was officially awarded the Millennium Prize on March 18, 2010.^[3] On July 1, 2010, it was reported that Perelman declined the award and associated prize money from the Clay Mathematics Institute.^[4] In rejecting the Millennium Prize, Perelman stated that he believed the decisions by the organized mathematics community to be unjust and that his contribution to solving the Poincaré conjecture was no greater than that of Columbia University mathematician Richard Hamilton (who first suggested a program for the solution).^[5]

The Riemann hypothesis[link]

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation 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.

The official statement of the problem was given by Enrico Bombieri.

Yang–Mills existence and mass gap[link]

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 postulated 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.

The official statement of the problem was given by Arthur Jaffe and Edward Witten.

Navier–Stokes existence and smoothness[link]

The Navier–Stokes equations describe the motion of fluids. 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 insight into these equations.

The official statement of the problem was given by Charles Fefferman.

The Birch and Swinnerton-Dyer conjecture[link]

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.

The official statement of the problem was given by Andrew Wiles.

Works[link]

• Devlin, Keith J. (2003) [2002]. The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. New York: Basic Books. ISBN 0-465-01729-0. 
• Carlson, James; Jaffe, Arthur; Wiles, Andrew, eds. (2006). The Millennium Prize Problems. Providence, RI: American Mathematical Society and Clay Mathematics Institute. ISBN 978-0-8218-3679-8. http://www.claymath.org/library/. 

See also[link]

• Hilbert's problems
• Smale's problems
• List of unsolved problems in mathematics
• Paul Wolfskehl (offered a cash prize for the solution to Fermat's Last Theorem)

References[link]

• This article incorporates material from Millennium Problems on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links[link]

• The Millennium Grand Challenge in Mathematics
• The Millennium Prize Problems

Millennium Prize may refer to:

• Millennium Prize Problems of Clay Mathematics Institute
• Millennium Technology Prize of Finland

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Michael Fredric Sipser is a professor of Applied Mathematics in the Theory of Computation Group at the Massachusetts Institute of Technology. He received his Ph.D. in 1980 from the University of California, Berkeley under the direction of Manuel Blum.

His research area is complexity theory, but he is also active in other related areas. In joint work with Farhi, Goldstone, and Gutmann, he introduced Adiabatic quantum computing.

He is the author of Introduction to the Theory of Computation (ISBN 0-534-95097-3), a standard textbook in the study of theoretical computer science.

He is currently (2011) serving as head of the MIT Mathematics department.^[1]

Notes[link]

1. ^ http://www-math.mit.edu/people/listing.php

External links[link]

• Sipser's personal homepage at MIT
• Michael Sipser at the Mathematics Genealogy Project
• Description of Sipser's research by Richard J. Lipton

This article about an American mathematician is a stub. You can help Wikipedia by expanding it. v t e

Nicholas Jerome Bennett (born 7 May 1949 in Hampstead, London) is a British politician. A member of the Conservative Party, he served as a Member of Parliament (MP) from 1987 to 1992 representing the constituency of Pembrokeshire, and was a Parliamentary Under-Secretary of State at the Welsh Office from 1990 to 1992. He is a staunch Roman Catholic.

Career[link]

Bennett was a school teacher from 1976–1985, and also worked for a period in educational publishing.

Bennett was a Conservative Councillor in the London Borough of Lewisham from 1974 to 1982, serving as the Leader of the Opposition in the Borough from 1979 to 1981. He was also a member of the Inner London Education Authority (ILEA) from 1978 to 1981, where he served on the Development and Schools Committees.

Bennett was elected to Parliament as the MP for Pembrokeshire at the 1987 general election. He served as member of the Select Committee on Welsh Affairs from 1987 to 1990, and the Select Committee on Procedure from 1988 to 1990, and was Vice-Chairman (Wales) of the Conservative Backbench Party Organisation Committee in 1990. He was appointed the Parliamentary Private Secretary to the Minister of State at the Department of Transport, Roger Freeman, in 1990, before becoming a Parliamentary Under-Secretary of State at the Welsh Office from 1990 to 1992.

At the 1992 general election, Bennett was defeated by the Labour Party candidate Nick Ainger. Bennett stood as a candidate in the constituency of Reading West at the 1997 general election, but was defeated by Labour's Martin Salter.

Bennett served as a member of the Further Education Funding Council for England (FEFCE) from 1992 to 1997. He was an advisor on public affairs to Price Waterhouse from 1993 to 1998, before becoming Chief Executive of the Association of Consulting Engineers from 1998 to 2002. In 2003, he formed his own company, Kent Refurbishment Ltd, which undertakes property development and education and public affairs consultancy.

In 2006 and 2010, Bennett was elected as a Councillor for the West Wickham ward in the London Borough of Bromley. He is a local magistrate in Battersea.

Education[link]

Bennett was educated at Sedgehill School, before taking his O-Levels and A-Levels at further education colleges. He completed a BA in Philosophy at the Polytechnic of North London, a Postgraduate Certificate in Education (PGCE) at the Institute of Education, University of London, and an MA in Educational Management at the University of Sussex.^[1]

Personal life[link]

Bennett's wife Ruth is a Councillor for the Bromley Common and Keston ward.

References[link]

• The Times Guide to the House of Commons, Times Newspapers Ltd, 1992
• Leigh Rayment's Peerage Pages ^{[self-published source?][better source needed]}

Alec Jeffreys
Born	(1950-01-09) 9 January 1950 (age 62) Oxford, United Kingdom
Nationality	British
Fields	Genetics
Institutions	University of Amsterdam, University of Leicester
Alma mater	Merton College, Oxford
Known for	Inventor of genetic fingerprinting
Notable awards	Albert Lasker Award for Clinical Medical Research (2005), Royal Medal awarded by the Royal Society (2004), Great Briton Award, Greatest Briton of 2006., EMBO member[1]
Spouse	Susan (née Miles) m. 1971

Sir Alec John Jeffreys, FRS (born 9 January 1950 in Oxford, Oxfordshire, England) is a British geneticist, who developed techniques for DNA fingerprinting and DNA profiling which are now used all over the world in forensic science to assist police detective work, and also to resolve paternity and immigration disputes.^[2][3] He is a professor of genetics at the University of Leicester,^[4][5] and he became an honorary freeman of the City of Leicester on 26 November 1992.^[6] In 1994, he was knighted by her Majesty Queen Elizabeth II of the United Kingdom, for Services to Science and Technology .

Early life[link]

Jeffreys was born on 9 January 1950 in Oxford.^[7] He came from a middle-class family and has one brother and sister. He spent the first six years of his life in Oxford until 1956 when the family moved to Luton. His curiosity and inventiveness were probably gained from his father, as well as his paternal grandfather who had a number of patents to his name.^[3] When he was eight years old his father gave him a large chemistry set which was enhanced over the next few years with extra chemicals including a bottle of concentrated sulphuric acid, bought from a pharmacy, at a time when pharmacists were less regulated than now.^[3] He liked making small explosions, but an accidental splash of sulphuric acid caused a burn and a permanent scar on his chin (now under his beard).^[3] When he was about eight or nine years old, his father bought him a beautiful Victorian brass microscope,^[8] which he used to examine biological specimens, furthering his interest in biology.^[3] At about 12 years old he made a small dissecting kit (including a scalpel crafted from a flattened pin) which he used to dissect a bumblebee, but he got into trouble with his parents when he progressed to dissecting a larger specimen. One Sunday morning he found a dead cat on the road while doing his paper round and took it home in his bag. He started to dissect it before Sunday lunch on the dining room table causing a foul smell throughout the house, which was particularly bad after he ruptured its intestines.^[3]

Jeffreys was a pupil at Luton Grammar School and then Luton Sixth Form College.^[9] He followed the youth culture of the time and initially became a Mod while owning a Vespa 150 cc motor-scooter and wearing a parka jacket.^[3] He was then a Hippie for a while before buying a Matchless 350 cc motorcycle and becoming a Rocker.^[3] From 1968 he was a student at Merton College, Oxford and in 1972 he graduated with first-class honours degree in biochemistry.^[7]

Career[link]

Alec Jeffreys

Jeffreys enjoys being at the laboratory bench, and prepared his PhD thesis entitled "Studies on the mitochondria of cultured mammalian cells" as a postgraduate student at the Genetics Laboratory, University of Oxford. After finishing his PhD, he moved to the University of Amsterdam, where he worked on mammalian genes as a research fellow.^[9] He moved on to the University of Leicester in 1977, where he found an academically stimulating and helpful environment. Working in Leicester he discovered a method of showing variations between individual's DNA in 1984, and invented and developed genetic fingerprinting.^[3][10]

Jefferys was special guest at – and officially opened – the new Soar Valley College building Soar Valley College in April 2010.

Genetic fingerprinting[link]

Jeffreys had a "eureka moment" in his lab in Leicester after looking at the X-ray film image of a DNA experiment at 9:05 am on Monday 10 September 1984, which unexpectedly showed both similarities and differences between the DNA of different members of his technician's family.^[2][3][10] Within about half an hour, he realised the possible scope of DNA fingerprinting, which uses variations in the genetic code to identify individuals. The method has become important in forensic science to assist police detective work, and it has also proved useful in resolving paternity and immigration disputes.^[3] The method can also be applied to non-human species, for example in wildlife population genetics studies. Before his methods were commercialised in 1987 his laboratory was the only centre carrying out DNA fingerprinting in the world, and during this period of about two or three years it was very busy, receiving inquiries from all over the globe.^[3][10]

Jeffreys' DNA method, which is often called DNA fingerprinting, was first put to use when he was asked to help in a disputed immigration case to confirm the identity of a British boy whose family was originally from Ghana.^[3] The case was resolved when the DNA results proved that the boy was closely related to the other members of the family, and Jeffreys saw the relief in the mother's face when she heard the results.^[3] DNA fingerprinting was first used as a police forensic test to identify the rapist and killer of two teenagers, Lynda Mann and Dawn Ashworth, who were both murdered in Narborough, Leicestershire, in 1983 and 1986 respectively. Colin Pitchfork was identified and convicted of murder after samples taken from him matched semen samples taken from the two dead girls.^[3] This turned out to be a specifically important identification for without it, British Authorities believe that Richard Buckland, the main suspect, would have inevitably been convicted. Therefore, not only did Jeffrey's work in this case prove who the real killer was, but exonerate someone who likely would have spent his life in prison otherwise. In 1992, Jeffreys' methods were used to confirm the identity for German prosecutors of the Nazi Dr. Josef Mengele, who had died in 1979, by comparing DNA obtained from a femur bone of his exhumed skeleton,^[8] and DNA from his widow and son, in a similar way to paternity testing.^[3]

DNA profiling[link]

DNA profiling based on typing individual highly variable minisatellites in the human genome was also developed by Alec Jeffreys and his team in 1985,^[11][12] with the term DNA fingerprinting being retained for the initial test that types many minisatellites simultaneously. By focussing on just a few of these highly variable minisatellites, DNA profiling made the system more sensitive, more reproducible, and amenable to computer databasing, and soon became the standard forensic DNA system used in criminal case work and paternity testing worldwide.

The development of DNA amplification by the Polymerase Chain Reaction (PCR) opened up new approaches to forensic DNA testing, allowing automation, greatly increased sensitivity and a move to alternative marker systems. The most commonly used markers are now variable microsatellites, also known as Short Tandem Repeats (STRs), that Jeffreys first exploited in 1990 in the Joseph Mengele case.^[13] STR profiling was further refined by a team of scientists led by Peter Gill at the Forensic Science Service in the 1990s, allowing the launch of the UK National DNA Database (NDNAD) in 1995. With highly automated and sophisticated equipment, modern-day DNA profiling can process hundreds of samples each day. Ten microsatellites plus a marker for sex determination are used with the current system developed for the NDNAD, giving a discrimination power of one in over a billion. Under British law, anyone arrested in England, Wales or Northern Ireland has their DNA profile taken and stored on the database whether or not they are convicted (different rules apply in Scotland).^[14] The national database now contains the DNA information of nearly five million people. Jeffreys has opposed the current use of DNA profiling, where the government has access to that database,^[15] and has instead proposed a database of all people's DNA, whose access would be controlled by an independent third party.^[16]

Personal life[link]

Jeffreys married Sue (née Miles) on 28 August 1971,^[9][17] and they have two daughters, born in 1979 and 1983.^[17] Their two daughters were still very young and growing up when Jeffreys' work life became hectic for the two or three years following his genetic fingerprinting breakthrough.^[3]

Awards and recognition[link]

• 20 March 1986 – Fellow of the Royal Society.^[18]
• 1989 – Midlander of the Year.
• 1991 – Appointed as a Royal Society Research Professor.^[19]
• 26 November 1992 – Honorary freeman of the City of Leicester.^[6]
• 1994 – Knighted.
• 1996 – Albert Einstein World Award of Science.^[20]
• 1998 – Australia Prize.
• 1999 – Stokes Medal
• 2004 – Honorary doctorate awarded by the University of Leicester, where Jeffreys is a member of staff.^[21]
• 2004 – Royal Medal of the Royal Society.^[22]
• 2004 – Pride of Britain Award for Lifetime Achievement.^[23]
• 2004 – Louis-Jeantet Prize for Medicine.^[24]
• 2005 – Albert Lasker Award for Clinical Medical Research,^[25] jointly with Edwin Southern of the University of Oxford.^[26]
• 2005 – United States National Academy of Science,^[27] elected member.
• December 2006 – Degree of Doctor of Science (Honoris Causa) by the University of Liverpool.
• 2006 – Morgan Stanley Great Briton Award for the Greatest Briton of the year, winner in the category of Science and Innovation, as well as the overall winner.
• 2006 – Dr A.H. Heineken Prize for Biochemistry and Biophysics.^[28]
• 8 March 2007 – Honorary degree from King's College London.^[29]
• 23 January 2008 – Graham Medal of the Glasgow Philosophical Society, awarded after he gave his lecture "DNA Profiling; Past, present and future", which was nominated as the Graham Lecture.^[30]
• 16 November 2009 – Awarded Honorary Doctor of Science by the University of Huddersfield^[31]
• 14 April 2010 – Awarded Edinburgh Medal^[32]
• 21 February 2011 – Awarded ABRF Annual Award^[33]

References[link]

External links[link]

Wikimedia Commons has media related to: Alec Jeffreys

• BBC Leicester – Jeffreys reflects on DNA finding