Millennium Problems

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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.

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.

History

Announcement

The millennium problems were first announced at Millennium Meeting on May 24, 2000 at the Collège de France. Timothy Gowers first presented a lecture titled The Importance of Mathematics as an introduction. After this, the British mathematician Michael Atiyah and the American John Tate announced the prize: one million dollars to anyone who could solve one of the seven most difficult open problems at the time.

A small committee of mathematicians, selected by the scientific advisory board (SAB) of the Clay Mathematical Institute (which also had organized the meeting), had selected the problems over the previous several months. They were led by Arthur Jaffe, the first director of the CMI, the former director of the American Mathematical Society, and the incumbent of the Landon Clay Chair in Mathematics at Harvard University. This committee included such luminaries as Andrew Wiles, the aforementioned Atiyah and Tate, the American Edward Twitten, and the French Alaine Connes.

Motivation

Partly, the motive of the CMI and its founder (see "Rules and Financing") was the founder's support of mathematical research. However, specifically, the inspiration was a similar prize exactly a hundred years earlier.

Paris had seen a similar event then, at the second International Congress of Mathematicians. The famous German mathematician David Hilbert drew up a list of 23 "Hilbert Problems" on August 8, "setting the agenda for the twentieth century". (Devlin 2003, pp. 2–3) These problems he believed to be the most significant and important unsolved in mathematics.

Some of these problems were either shown to be unsolvable, indefinite, or trivial. However, many were difficult problems, and enormous prestige was given to a mathematician who solved one of them as soon as the mathematical community had pronounced his solution correct. All but one of these problems had been solved by the meeting in 2000, and thus it was natural to create a new set of such problems.

Wiles, however, notes that Hilbert's and the CMI's motivations were slightly different; "Hilbert was trying to guide mathematics by his problems; we're trying to record great unsolved problems. There are big problems in mathematics that are important but where it is very hard to isolate one problem that captures the program." (Devlin 2003, p. 3)

Solving of the Poincaré Conjecture

On April 7–11, 2003, Russian mathematician Grigori Perelman, a member of the Steklov Institute of Mathematics, a division of the Russian Academy of Sciences in St. Petersburg, presented his proof of the Poincaré Conjecture during the Simons Lecture Series at the MIT Mathematics Department. He gave three lectures, titled "Ricci Flow and Geometrization of Three-Manifolds", on April 7, 9, and 11. These were his first public presentation of the important results he had published earlier, in November 2002 and March 2003.

Perelman's paper proved not only the Poincaré Conjecture, but a generalization known as Thurston's Geometrization Conjecture. The former merely stated that every closed, simply-connected three manifold is homeomorphic to the three sphere; a sphere with a three-dimensional surface, or a four-dimensional sphere. Thurston's conjecture extended the conjecture to any positive integer n, stating that a compact n manifold is homotopy equivalent to the n sphere iff it is homeomorphic to the n sphere, or, more simply, that only a n manifold with no holes was simply connected. All cases of this conjecture had been proven up to this point except the case where $n=3$.

In 1995, Perelman had learned of Ricci flow, the key to his proof, from Hamilton in the United States. Returning to Russia, though publishing few results, he became an expert on Ricci flow and differential geometry in general. The aforementioned November 2002 and March 2003 publications were then posted on the internet, climaxing to Perelman's lectures in April of that year.

Previous proofs had been purported, most notably Dunwoody's, a year prior to Perelman's, but they had all proved false. Perelman's, however, was well received by the mathematical community, and the conjecture was declared proven in 2006 after four years of debate and three 300-page-long publications filling in details of the proof. Perelman, however, declined the Fields Medal offered to him by the IMU. He also retired from mathematics, citing in a New Yorker interview his colleagues' unethical actions, particularly the credit claimed by the authors of the third 300-page-long publication. Controversies regarding the third publication have abounded since then, resulting in the collapse of a planned January 2007 Poincaré Conjecture "All-Stars" meeting.

(References: MathWorld article, ScienceMag article)

Rules and Financing

The prize fund is financed privately by Landon Clay, the man who had established the CMI one year earlier as a nonprofit organization based in his hometown; Cambridge, MA to aid mathematical research. Clay, a well-off mutual-fund manager, though not a mathematician, was greatly interested in and supportive of the mathematical community.

The rules were laid out as follows: The SAB would consider a proof of one of the problems on several conditions. Firstly, the proof had to be complete. Secondly, it had to "be published in a refereed mathematics publication of worldwide repute and [...] general acceptance in the mathematics community two years after". (Millennium Prize Rules; as of January 19, 2005) If these conditions are met, the SAB will appoint an advisory committee to thoroughly examine the solution. This committee would consist of at least two world-renowned mathematicians and at least one member of the SAB.

After the analysis, the committee would report back to the SAB. The SAB would then report to the directors of the CMI, possibly giving recommendations of whether or not the prize should be awarded and (in the case of a group of mathematicians collaborating on one problem or multiple mathematicians solving a problem near-simultaneously) which person(s) should receive the prize. In the case of multiple prizewinners, the prize would be divided proportionally according to the judgement of the directors. Counterexamples are put through essentially the same process; again, the directors make the final decision, though the SAB can advise them.

In the special case of a problem being shown to be false as stated, but ambiguous with a small adjustment, a small prize may be awarded to the mathematician who discovered this; though the money would be taken from funds other than the Millennium Prize ones. The new problem would then replace the old in the selection of problems, with the same process and conditions.

Problems

Birch and Swinnerton-Dyer Conjecture

Main article: Birch and Swinnerton-Dyer Conjecture

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 the zero of the associated $L$-function $L(E, s)$ at $s = 1$.

As of 2005, it has been proved only in special cases, such as over certain quadratic fields (by Henri Darmon of McGill University). It has been an open problem for around 40 years, and has stimulated much research; its status as one of the most challenging mathematical questions has become widely recognized.

Hodge Conjecture

Main article: Hodge Conjecture

The Hodge conjecture asserts that structures known as Hodge classes, which can be elementarily described as geometric representations of a given manifold's topological properties, are composed of algebraic cycles. More rigorously, the common phrasing for the conjecture is "Given a projective complex manifold, every Hodge class on it is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of it."

Navier-Stokes Equations

Main article: Navier-Stokes Equations

The Navier-Stokes equations describe the motion of fluids. These equations establish that the acceleration of fluid particles is simply the product of changes in pressure and dissipative viscous forces (similar to friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

P versus NP

Main article: P versus NP

The problem of P versus NP is an important problem in computability and complexity theory relating to whether decision problems (problems admitting a yes or no answer) whose solutions can be verified in polynomial time (as a function of the input, often expressed using big-O notation) can also be solved in polynomial time. The set $P$ consists of decision problems such that there exists a deterministic computer program (or Turing machine) that decides $P$ in polynomial time. The set $NP$, informally, consists of decision problems whose "yes" instances can be verified by a deterministic program in polynomial time, given a certificate. Whether $P = NP$ unknown, though many problems can be shown to be NP-complete - that is, if a problem $L$ is NP-complete, then any NP problem can be reduced to $L$ in polynomial time. This implies that if any NP-complete problem has a polynomial time solution, then $P = NP$.

Poincaré Conjecture

Main article: Poincaré Conjecture

In elementary terms, the Poincaré conjecture states that the only three-manifold with no "holes" is the three-sphere. This would also show that the only $n$-manifold with no "holes" is the $n$-sphere; the case $n=1$ is trivial, the case $n=2$ is a classic problem, and the truth of the statement for $n\ge 4$ was verified by Stephen Smale in 1961. More rigorously, the conjecture is expressed as "Every simply connected, compact three-manifold (without boundary) is homeomorphic to the three-sphere."

Riemann Hypothesis

Main article: Riemann Hypothesis

The Riemann hypothesis is a well-known conjecture in analytic number theory that states that all nontrivial (the trivial roots are when $s=-2,-4,-6,\ldots$) zeros of the Riemann zeta function have real part $\frac{1}{2}$.

The Riemann Hypothesis is an important problem in the study of prime numbers. Let $\pi(x)$ denote the number of primes less than or equal to x, and let $\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt$. Then an equivalent statement of the Riemann hypothesis is that $\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))$.

Yang-Mills Theory

Main article: Yang-Mills Theory

The quantum Yang-Mills theory (no quarks) with a non-abelian gauge group is an exception to the general rule that nontrivial (i.e. interacting) quantum field theories that we know of in 4D are effective field theories with a cutoff scale. It has a property known as asymptotic freedom, meaning that it has a trivial UV fixed point. Because of this, this is the simplest model to pin our hopes on for a nontrivial constructive QFT model in 4D. (QCD, with its fermionic quarks is obviously more complicated).

It has already been well proven at the standards of theoretical physics, but not mathematical physics, that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement.

See Also

References

  • Devlin, Keith J (2003). Basic Books. The Millennium Problems. ISBN 978-0465017300.

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