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## 1 Solved. 6 Millennium Prize Math Problems Remain

Grigori Perelman, a Russian mathematician, solved one of the world's most complicated math problems several years ago. The Poincare Conjecture was the first of the seven Millennium Prize Problems to be solved. Renee Montagne speaks with mathematician Keith Devlin about the problems that are worth $1 million each if solved.

RENEE MONTAGNE, host:

Grigori Perelman is one of the world's finest mathematical minds. In 2003, he solved the Poincare Conjecture, which deals with shapes that exist in four or more dimensions. A solution had evaded mathematicians for a century. The Poincare Conjecture is one the seven Millennium Prize Problems, and solving any of what you might call the Seven Wonders of the Math World brings a million-dollar award.

Subtracting Pereleman's win, that leaves six more to be solved, and to talk about those, Keith Devlin, NPR's WEEKEND EDITION Math Guy joins us.

Good morning.

Professor KEITH DEVLIN (Stanford University): Ah, good morning, Renee.

MONTAGNE: Give us a simple version of what the remaining Millennium Prize Problems are. I mean if you can put it a tweet.

Prof. DEVLIN: There aren't enough characters in a typical tweet to be able to do it. There were six of them, as you mentioned, Renee. One is about how well computers can solve certain kind of problems. One is about what pattern do the prime numbers have - how much can you know about the pattern of the primes. One is about the fundamental nature of matter, the things that we and everything around us are made of.

One is an old, 19th century problem about can you solve the equations that describe how water flows along a pipe. And then there's another one connected with prime numbers and the structure of the whole numbers.

So they're in all different areas of mathematics: physics, computational mathematics, and patterns of prime numbers.

MONTAGNE: Well, let me ask you. I mean, most of these sound beyond me - and probably, most people. But let's say the mathematicians going at it are looking for something, and one would maybe be the thrill or the prestige of solving them. But what would the larger benefits be?

Prof. DEVLIN: Oh, boy. In these different problems, one of the other problems is a thing called the P versus NP Problem. If that were solved in one direction, it would mean Internet commerce and Internet cybersecurity would collapse in an instant. That much is at stake.

We think the answer is going to go the other way. But if someone comes along and solves one of these Millennium Problems about computation, and it goes in the way we dont expect, then it will tell us that everything we assume about the security of communications over the Internet is false.

MONTAGNE: Have computers helped in any of this?

Prof. DEVLIN: Computers have affected mathematics around the edges. But these are problems that mathematicians have to sit down, paper and pencil, close their eyes, think and dream and talk to each other from time to time, and try to solve them - in exactly the way that Grigori Perelman recently solved the Poincare Conjecture. And all of the other Millennium Problems are really of that nature.

MONTAGNE: Thats a lovely thought, except now that it's sort of out in the world - unleashed, as it were - one, he doesnt want to collect his big prizes. He doesnt want to collect the money attached to it because he doesnt want the publicity. Right?

Prof. DEVLIN: He's really the mathematical equivalent of J.D. Salinger. You know, he writes "Catcher in the Rye" and then disappears from view. But the unfortunate thing is, he has now solved one of the biggest unsolved problems in mathematics. And there's going to be a ceremony in Paris this June. Most of my colleagues that know something about Perelman believe that he's not going to turn up for that.

Whether he will arrange to receive the $1 million prize, quietly out of the limelight, I think we'd all be very surprised if he publicly turned up to receive a check and have photographs of him receiving a check. You know, one of these big checks like a lottery winner.

(Soundbite of laughter)

Prof. DEVLIN: Everything I know about him suggests he's not going to go that route.

MONTAGNE: Keith Devlin is author of "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time." And you might know him also as the Math Guy on NPR's WEEKEND EDITION.

Thanks very much.

Prof. DEVLIN: OK. My pleasure, Renee.

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## Millennium Prize Problems

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The Millennium Prize Problems are seven of the most well-known and important unsolved problems in mathematics. The Clay Mathematics Institute, a private nonprofit foundation devoted to mathematical research, famously challenged the mathematical community in 2000 to solve these seven problems, and established a US $1,000,000 reward for the solvers of each. One of the seven problems has been solved, and the other six are the subject of a great deal of current research.

The timing of the announcement of the Millennium Prize Problems at the turn of the century was an homage to a famous speech of David Hilbert to the International Congress of Mathematicians in Paris in 1900. The 23 unsolved problems posed by Hilbert were studied by countless \(20^\text{th}\) century mathematicians, which led not only to solutions to some of the problems, but also to the development of new ideas and new research topics. Some of Hilbert's problems remain open--indeed, the most famous of Hilbert's problems, the Riemann hypothesis , is one of the seven Millennium Prize Problems as well.

The problems encompass a diverse group of topics, including theoretical computer science and physics, as well as pure mathematical areas such as number theory, algebraic geometry, and topology.

## Poincare Conjecture

Hodge conjecture, riemann hypothesis, yang-mills existence and mass gap, navier-stokes existence and smoothness, birch-swinnerton-dyer conjecture.

The only Millennium Problem that has been solved to date is the Poincare conjecture, a problem posed in 1904 about the topology of objects called manifolds .

A manifold of dimension \(n\) is a geometric object equipped with a topological structure such that every point has a neighborhood that looks like (is homeomorphic to) \(n\)-dimensional Euclidean space, for some \( n.\) The standard example is a sphere, the surface of a ball embedded in three-dimensional space. An ant on the surface of a sphere thinks that he is standing on flat ground, as the curvature of the sphere is not observable locally. So a sphere is a \(2\)-manifold; the flat ground looks like \(2\)-dimensional Euclidean space.

Another example of a \(2\)-manifold is a (one-holed) torus .

Two manifolds are considered to be different if one cannot be continuously deformed into the other. One way to see that the torus is different from the \(2\)-sphere is that loops on the sphere can all be contracted on the sphere to a point (imagine a rubber band on the surface of a sphere--it can be pulled to the top of the sphere without breaking the band or leaving the sphere), but loops on a torus cannot (e.g. the loop on the top of the torus, or one of the black loops in the picture).

A fundamental question in the theory of manifolds is the classification problem : is there a way to characterize when two manifolds are the same, without having to explicitly write down the map that identifies them? That is, is there a set of properties such that any two manifolds that share all these properties must be the same?

The Poincare conjecture states that any closed (boundaryless) \( n\)-manifold which is homotopy equivalent to the \(n\)-sphere must be the \(n\)-sphere. (Homotopy equivalence is a notion that is strictly weaker than being the same, in general.) This is relatively easy for \(n=1,2.\) It was proved for \( n\ge 5\) by Stephen Smale in the 1960s, and for \( n=4 \) by Michael Freedman in 1982. Both mathematicians were given Fields Medals, the highest honor a mathematician can receive.

The case \(n=3\) is equivalent to the following statement:

Any simply connected, closed 3-manifold is the same as the 3-sphere.

Here simply connected means intuitively that the manifold has no holes; a loop on its surface can always be contracted to a point. As \(n=3\) was the only case left to be proved, this was the statement of the Poincare conjecture when it was posed as a Millennium Problem.

The conjecture was proved in 2003 by the Russian mathematician Grigori Perelman, using ideas of Richard Hamilton from the early 1980s. Hamilton suggested using a vector field flow called the Ricci flow to solve the problem, and demonstrated its efficacy by proving special cases of Poincare's conjecture. Perelman announced his solution of the problem in a series of papers in 2002 and 2003. Peer review confirmed that his proof was correct, and in 2006 he was offered the Fields Medal for his work.

Perelman turned down the Fields Medal and also refused to accept the Clay Millennium Prize when it was officially offered to him in 2010, saying that his contributions were no more significant than Hamilton's. His work is by all accounts quite original and accomplished; despite his apparent modesty and shunning of the spotlight, his proof of the Poincare conjecture will be famous for a very long time.

Main article: P vs. NP

The problem of determining whether \({\mathbf P} = \mathbf{NP}\) is the most important open problem in theoretical computer science. The question asks whether computational problems whose solutions can be verified quickly can also be solved quickly. The consensus of most experts in the field is that this is not true in general, i.e. \( {\mathbf P}\ne \mathbf{NP},\) but there is very little progress toward a proof.

The class of problems in \( \mathbf P\) is the set of problems for which a solution can be found in polynomial time. That is, the problem depends on a positive integer \(n \) which represents the number of inputs (more formally, the information in the problem can be translated into a string of length \( n\)), and it is in \( \mathbf P\) if there is an algorithm that takes the problem as an input and returns a valid solution, such that the running time of the algorithm is less than \( cn^k \) for some positive numbers \( c,k\) which are independent of \( n.\)

The problem of computing the greatest common divisor of two integers \( a,b\) is in \( \mathbf P;\) in fact the Euclidean algorithm runs in time \( \le 5n,\) where \(n\) is the number of decimal digits of either of the integers.

Note that the constants \( c\) and \(k\) in the definition of polynomial time given above might be forbiddingly large in practice. For instance, the primality testing problem was shown quite recently to be in \( {\mathbf P}\); the proof exhibited an explicit algorithm, but the algorithm is not the fastest algorithm for practical purposes.

The class of problems in \( \mathbf{NP}\) is the set of problems for which a solution can be verified in polynomial time. That is, the problem depends on a positive integer \(n \) which represents the number of inputs (more formally, the information in the problem and the prospective solution can be translated into a string of length \( n\)), and it is in \( \mathbf{NP}\) if there is an algorithm that takes the prospective solution as input and returns "yes" or "no" depending on whether the prospective solution is in fact a solution to the problem, such that the running time of the algorithm is less than \( cn^k\) for some positive numbers \( c,k\) which are independent of \( n.\)

The problem of determining whether there is a Hamiltonian path on a given graph is in \( \mathbf{NP}.\) That is, it is quite easy to check whether a particular path on a graph is Hamiltonian; simply check whether it passes through each vertex exactly once. However, the problem of finding a Hamiltonian path is (conjecturally) much harder. Even the problem of determining whether a Hamiltonian path exists is in a class of problems known as \( \mathbf{NP}\)-complete problems; that is, any problem in \( \mathbf{NP}\) can be reduced in polynomial time to the Hamiltonian path problem. So if the Hamiltonian path problem is in \( \mathbf P,\) it follows that \( \mathbf{P}=\mathbf{NP}.\) An extension of the Hamiltonian path problem is the famous traveling salesperson problem .

A proof that \( {\mathbf P} = {\mathbf{NP}} \) would have far-reaching implications, as it would show that many problems thought to be hard, including problems on which many cryptosystems are based, can be solved in polynomial time. Many problems in theoretical mathematics are in \({\mathbf{NP}}\) as well, so \( {\mathbf P} = {\mathbf{NP}} \) would imply that they could be proved or disproved "mechanically" in polynomial time. It should be noted that this does not necessarily mean that these solutions would be practical, and in fact a proof that \( {\mathbf P} = {\mathbf{NP}} \) might be non-constructive; that is, it might be provable that these problems could be solved in polynomial time, via a proof that does not give any indication of the construction of an explicit algorithm that accomplishes this.

The Hodge conjecture is a statement about geometric shapes cut out by polynomial equations over the complex numbers. These are called complex algebraic varieties . An extremely useful tool in the study of these varieties was the construction of groups called cohomology groups , which contained information about the structure of the varieties. The groups are constructed quite abstractly, but have many useful relationships: for instance, a map between varieties corresponds to maps between cohomology groups. Since computations on groups are often more straightforward than computations on varieties, this gives a way to classify and study properties of complex algebraic varieties.

Some elements of these cohomology groups can be written down explicitly from geometric information about the variety, in particular subvarieties of the variety.

The unit sphere \(x^2+y^2+z^2=1\) in complex 3-space contains a curve cut out by \( z=0,\) namely the unit circle \( x^2+y^2=1\) in the \(xy\)-plane. This is the equator of the sphere, and is a subvariety.

The Hodge conjecture states that certain cohomology groups studied by Hodge over certain nice complex varieties are generated by the classes of subvarieties. The cohomology groups in question are often called the groups of Hodge classes, and classes generated by subvarieties are often called algebraic. So in these terms, the conjecture becomes

Every Hodge class on a projective complex manifold is algebraic.

The conjecture was formulated by Hodge in 1950. It is known for varieties of dimension \( \le 3,\) and certain other special cases are known. A successful proof would give a useful indication of the interplay between algebra and geometry. Correspondences between geometric structures (varieties) and algebraic structures (groups) often yield very powerful results: for another example of this phenomenon, see Wiles' proof of Fermat's last theorem , which used the Taniyama-Shimura conjecture relating elliptic curves to modular forms.

The Riemann hypothesis is perhaps the most famous unsolved problem in mathematics. It concerns the nontrivial zeroes of the Riemann zeta function , which is defined for \( \text{Re } s > 1 \) by the infinite sum \[ \zeta(s) = \sum_{n=1}^\infty \frac1{n^s}. \] It can be shown that \( \zeta \) can be analytically continued to a function which is defined and differentiable everywhere in the complex plane, except for a simple pole at \( s=1.\) This function has trivial zeroes on the negative real line, at \( s=-2,-4,-6,\ldots.\) The location of its other zeroes is more mysterious; the conjecture is that

The nontrivial zeroes of the zeta function lie on the line \( \text{Re }s=\frac12.\)

The beauty of the Riemann hypothesis is that it has strong implications about the distribution of primes . In particular, it implies strong bounds on the error term in the prime number theorem , as well as many other results from number theory . For instance, the Riemann hypothesis is equivalent to any of the following three statements:

(1) \( \sigma(n) < e^{\gamma} n \log \log n\) for all \( n > 5040,\) where \( \sigma(n) \) is the sum of divisors of \(n\) and \( \gamma\) is the Euler-Mascheroni constant .

(2) \( \sigma(n) < H_n + e^{H_n} \log(H_n)\) for all \( n \ge 2,\) where \( H_n\) is the \(n^\text{th}\) harmonic number .

(3) \( \sum\limits_{n\le x} \mu(n) = O\big(x^{\frac12 + \epsilon}\big)\) for any \( \epsilon > 0,\) where \(\mu\) is the Möbius function . (See the wiki on big O notation for an explanation of the right side of the equation.)

The generalized Riemann hypothesis is a statement about the zeroes of certain functions known as \( L\)-functions, defined by Dirichlet series , which are generalizations of the Riemann zeta function. The generalized Riemann hypothesis can be used to prove many open questions in number theory, including Artin's conjecture on primitive roots and the so-called weak Goldbach conjecture that every odd prime greater than 5 is the sum of three odd primes.

There are some known results about nontrivial zeroes; they all lie in the critical strip \( 0 < \text{Re } s < 1;\) infinitely many of them lie on the critical line \( \text{Re } s = \frac12;\) the first \( 10^{13} \) nontrivial zeroes, ordered by size of imaginary part, are all on the critical line. The Riemann hypothesis itself still appears to be quite difficult to attack in any meaningful way.

A Yang-Mills theory in quantum physics is a generalization of Maxwell's work on electromagnetic forces to the strong and weak nuclear forces. It is a key ingredient in the so-called Standard Model of particle physics. The Standard Model provides a framework for explaining electromagnetic and nuclear forces and classifying subatomic particles. It has so far proved to be consistent with experimental evidence, but questions remain about its internal consistency.

In particular, successful applications of the theory to experiments and simplified models have involved a "mass gap," which is formally defined as the difference between the default energy in a vacuum and the next lowest energy state. So this quantity is the mass of the lightest particle in the theory. A solution of the Millennium Problem will include both a set of formal axioms that characterize the theory and show that it is internally logically consistent, as well as a proof that there is some strictly positive lower bound on the masses of particles predicted by the theory.

Generally speaking, the current state of the problem is that researchers are successfully obtaining results consistent with experimental evidence by using ideas and models that come from Yang-Mills theory, but there is no rigorous, axiomatized theory that coherently explains the experimental data and successfully predicts results about nuclear forces. There does not appear to be a compelling reason to believe that the problem will be solved soon, but it is of great interest to the physics and mathematics community at large, and will be the subject of extensive research in the coming decades.

The Navier-Stokes equations are partial differential equations modeling the motion of liquids or gases. The fluid is acted on by forces including pressure \( p({\mathbb x},t),\) viscous stress \( \nu,\) and a specified external force. The Navier-Stokes equations are the result of writing down Newton's second law for the fluid with respect to these forces, in terms of partial derivatives of the velocity \(v({\mathbf x},t)\) of the fluid as a function of position and time. The Millennium Problem has two parts; there are existence and smoothness questions for solutions in \( {\mathbb R}^3,\) and existence and smoothness questions for solutions in \( \frac{{\mathbb R}^3}{{\mathbb Z}^3},\) a three-dimensional torus--this is the so-called periodic case. One common explanation for why the problem is so difficult is that solutions to these equations include turbulence, which is a little-understood area of fluid dynamics.

The problem is

Given any initial condition \( v_0 = v({\mathbf x},0)\), a vector field on \( {\mathbb R}^3, \) is there a velocity vector function \( v({\mathbf x},t) \) and a pressure function \( p({\mathbf x},t) \) that satisfy the equations? Is there a smooth solution if \( v_0\) is smooth?

In the non-periodic case, some extra requirements are also imposed on the initial condition and the solution so that they do not grow too large as the length of the position vector in \( {\mathbb R}^3 \) tends to infinity.

Not very much seems to be known about the answer to this question. Smooth solutions exist for the analogous problem in two dimensions (known since the 1960s), but this does not give much of a clue about how to proceed in three dimensions. In three dimensions, smooth solutions are known to exist if \( v_0 \) is "small" in a certain sense, and it is known that smooth solutions exist in general defined for values of the time parameter \( t \) in \( [0,T),\) where \( T\) is a finite time depending on \( v_0\) called the "blowup time." The problem asks for solutions defined for all \( t \in [0,\infty),\) which is more stringent.

There has been some progress on weak solutions to the equation, which are velocity functions \( v({\mathbf x},t) \) that satisfy the equations "on average," rather than for all points \( {\mathbf x}.\) But this has yet to lead to a convincing program to find solutions to the general equations.

The Birch-Swinnerton-Dyer conjecture concerns the rational points (points with all coordinates rational numbers ) on elliptic curves . Elliptic curves are, from a Diophantine perspective, the most interesting curves by far. Associated to every plane curve is a nonnegative integer called the genus. Genus-0 curves are well-understood, and their points are easily parameterized. Curves of genus \( \ge 2 \) have only finitely many rational points, by an extremely deep theorem from the 1980s due to Faltings. Curves of genus 1 with a rational point are precisely the elliptic curves, which have a myriad of applications and a very interesting structure on their sets of rational points. See the elliptic curves wiki for details.

In particular, it is a fact that, given an elliptic curve \( E,\) there is a nonnegative integer \( n\) and a set of rational points \( P_1,\ldots,P_n\) on \( E\) such that every rational point on \(E\) can be written uniquely as an integer linear combination of the \( P_i \) plus a torsion point \( T.\) The torsion points are the points of finite order , and there are finitely many of them. Here the linear combination involves the group law on the elliptic curve, which is nontrivial to write down explicitly (but note that it is not the same thing as coordinate-wise addition). The integer \( n\) is called the rank of \( E,\) and half of the Birch-Swinnerton-Dyer conjecture concerns the computation of that rank.

There is a function \( L(E,s)\) defined by a certain Dirichlet series , which is similar to the Riemann zeta function . The order of vanishing of \( L(E,s) \) at \( s=1\) is called the analytic rank of \( E,\) and the first half of the Birch-Swinnerton-Dyer conjecture is that

The rank of \( E\) equals its analytic rank.

The second half of the conjecture is more technical; it involves the coefficient of \( (s-1)^r \) in the Taylor series for \( L(E,s)\) around \( s=1.\) This coefficient is conjecturally equal to an expression involving products and quotients of several fundamental constants relating to the elliptic curve (for instance, one of them is the number of torsion points).

The first half of the conjecture has been proved in the case when the analytic rank is \( 0 \) or \( 1.\) The second half has been proved for certain special classes of elliptic curves with analytic rank \( 0.\) There is quite a lot of computational evidence for the conjecture (some of which dates back to computer computations done by Birch and Swinnerton-Dyer in the 1960s), but there is not very much progress toward a general proof. Establishing the conjecture would help with theoretical results about the structure of points on elliptic curves, as well as practical applications including finding generators \( P_1,\ldots,P_n\) of the set of rational points.

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## The Millennium Prize Problems

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems . The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France. Three lectures were presented: Timothy Gowers spoke on The Importance of Mathematics ; Michael Atiyah and John Tate spoke on the problems themselves.

The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years.

Following the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem.

It is of note that one of the seven Millennium Prize Problems, the Riemann hypothesis, formulated in 1859, also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.

The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.

## Unsolved problems

Birch and swinnerton-dyer conjecture.

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles’ proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

## Hodge Conjecture

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

## Navier-Stokes Equation

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

## Riemann Hypothesis

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.

## Yang-Mills & The Mass Gap

Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

## Solved problems

Poincaré conjecture.

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston’s geometrization conjecture. Perelman’s proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

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## Seven Millennium Prize Problems

The Seven Millennium Prize Problems are the most well-known and important unsolved problems in mathematics. A private nonprofit foundation Clay Mathematics Institute that is devoted to mathematical research, famously challenged the mathematical community in the year 2000 to solve these unique seven problems, and a sum of US $1,000,000 reward was established for the solvers of each of the seven problems. Out of the seven Millennium prize problems, one of the problems has been solved, and the other six are a great deal of current research.

With the spin of the century , the timing of the announcement of the Millennium Prize Problems was a homage to a famous speech of the famous David Hilbert to the International Congress of Mathematicians in the year 1900 in the city of Paris. The 23 unsolved problems that were posed by Hilbert were studied by countless 20 th century mathematicians, which led not only to solutions to some of these difficult problems but it also led to the development of new ideas as well as new research topics. There are some of Hilbert's problems that still remain open-- namely the famous Riemann hypothesis.

These seven problems encompass a diverse group of topics, which include theoretical computer science as well as physics, as well as topics of pure mathematical areas such as number theory, algebraic geometry, as well as topics of topology .

## 7 Millennium Prize Problems

1. yang-mills and mass gap.

Computer simulations as well as various experiments suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known. A Yang-Mills theory is known to be a theory in quantum physics that is a generalization of Maxwell's work on electromagnetic forces to the strong as well as weak nuclear forces. It is a key ingredient in the Standard Model of particle physics. This Standard Model is said to provide a framework for explaining electromagnetic as well as providing nuclear forces and also classifying subatomic particles.

In particular, successful applications of the theory to experiments as well as simplified models have involved a "mass gap," which can be formally defined as the difference between the default energy in a vaccum as well as also the energy in the next lowest energy state. So this quantity is also known as the mass of the lightest particle in the theory. A solution to the Millennium Problem will include both a set of formal axioms that characterize the theory as well as will show that it is internally logically consistent.

## 2. Riemann Hypothesis

The prime number theorem determines the average distribution of the prime numbers . Whereas the Riemann hypothesis basically describes the deviation from the average. It was formulated in Riemann's 1859 paper, which asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

## 3. P vs NP Problem

If it is easy to check that a solution to a problem is right, can you say that it is also easy to solve the problem? This is said to be the exact essence of the NP question vs P question. Typical of the NP problems is that of the Hamiltonian Path Problem: let’s suppose given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that the problem is correct, but it is difficult to find a solution.

## 4. Navier–Stokes Equation

The Navier-Stokes equation is said to be the equation that governs the flow of fluids such as water as well as air. However, there is no proof for the most basic questions one can ask: do solutions exist as well as are they unique? Why ask for proof? Because proof gives not only certitude but proof also gives understanding.

## 5. Hodge Conjecture

Hodge conjecture, the answer to this determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture comes in picture in certain special cases, for example, when the solution set has a dimension less than four. But in dimension four it is unknown.

This conjecture is also known to be a statement about geometric shapes cut out by polynomial equations over complex numbers. These are also known as complex algebraic varieties. An extremely useful tool in the study of these varieties was the construction of groups which is also known as cohomology groups, which contained information about the structure of the varieties.

## 6. Poincaré Conjecture

The French mathematician Henri Poincaré in the year 1904. He was the one who asked if the three-dimensional sphere is characterized as the unique simply connected three-manifold. The Poincaré conjecture is known as a special case of Thurston's geometrization conjecture. This Poincaré conjecture proof tells us that every three-manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

## 7. Birch and Swinnerton-Dyer Conjecture

This z conjecture is basically supported by much experimental evidence that relates the number of points on an elliptic curve mod p to the rank of the group of rational points.

## FAQs on Millennium Problems

1. How Many Millennium Problems are Left?

Answer. The Seven Millennium Prize Problems are the most well-known and important unsolved problems in mathematics. A private nonprofit foundation Clay Mathematics Institute that is devoted to mathematical research, famously challenged the mathematical community in the year 2000 to solve these unique seven problems, and a sum of US $1,000,000 reward was established for the solvers of each of the seven problems. Out of the seven Millennium prize problems, one of the problems has been solved, and the other six are a great deal of current research.

2. Who Solved the Millennium Problem?

Answer. Grigori Perelman

To date, Poincaré conjecture is the only Millennium Prize problem that has been solved. This problem was solved in the year 2003 by a Russian mathematician named Grigori Perelman.

- Main content

## An eminent mathematician claims to have solved one of math's greatest mysteries — and it's one of 6 problems with a $1 million prize

- At the turn of the 21st century, the Clay Mathematics Institute announced a list of seven of the most important unsolved math problems.
- The problems all have a $1 million prize awarded to whoever solves them.
- One of the seven problems has been solved, while the other six remain open questions.
- In September 2018, the mathematician Michael Atiyah claimed to have a solution to another one of the problems, but the validity of his solution remains to be seen.

In 2000, the Clay Mathematics Institute announced the Millennium Prize problems . These were a collection of seven of the most important math problems that remain unsolved.

Reflecting the importance of the problems, the Institute offered a $1 million prize to anyone who could provide a rigorous, peer-reviewed solution to any of the problems.

While one of the problems, the Poincare Conjecture, was famously solved in 2006 (with the mathematician who solved it, Grigori Perelman, equally famously turning down both the million dollar prize and the coveted Fields Medal), the other six problems remain unsolved.

In September 2018, the mathematician Michael Atiyah claimed to have a proof of the Riemann Hypothesis , one of the six remaining unsolved problems. However, it will take time for the mathematical community to evaluate Atiyah's claims.

Here are the six math problems so important that solving any one of them is worth $1 million.

Some problems are easy, and some problems are hard.

In the world of math and computer science, there are a lot of problems that we know how to program a computer to solve "quickly" — basic arithmetic, sorting a list, searching through a data table. These problems can be solved in "polynomial time," abbreviated as "P." It means the number of steps it takes to add two numbers, or to sort a list, grows manageably with the size of the numbers or the length of the list.

But there’s another group of problems for which it’s easy to check whether or not a possible solution to the problem is correct, but we don’t know how to efficiently find a solution. Finding the prime factors of a large number is such a problem — if I have a list of possible factors, I can multiply them together and see if I get back my original number. But there is no known way to quickly find the factors of an arbitrary large number. Indeed, the security of the Internet relies on this fact.

For historical and technical reasons, problems where we can quickly check a possible solution are said to be solvable in “nondeterministic polynomial time,” or “NP.”

Any problem in P is automatically in NP — if I can solve a problem quickly, I can just as quickly check a possible solution simply by actually solving the problem and seeing if the answer matches my possible solution. The essence of the P vs NP question is whether or not the reverse is true: If I have an efficient way to check solutions to a problem, is there an efficient way to actually find those solutions?

Most mathematicians and computer scientists believe the answer is no. An algorithm that could solve NP problems in polynomial time would have mind-blowing implications throughout most of math, science, and technology, and those implications are so out-of-this-world that they suggest reason to doubt that this is possible.

Of course, proving that no such algorithm exists is itself an incredibly daunting task. Being able to definitively make such a statement about these kinds of problems would likely require a much deeper understanding of the nature of information and computation than we currently have, and would almost certainly have profound and far-reaching consequences.

Read the Clay Mathematics Institute's official description of P vs NP here.

## The Navier-Stokes equations

It’s surprisingly difficult to explain what happens when you stir cream into your morning coffee.

The Navier-Stokes equations are the fluid-dynamics version of Newton’s three laws of motion. They describe how the flow of a liquid or a gas will evolve under various conditions. Just as Newton's second law gives a description of how an object's velocity will change under the influence of an outside force, the Navier-Stokes equations describe how the speed of a fluid's flow will change under internal forces like pressure and viscosity, as well as outside forces like gravity.

The Navier-Stokes equations are a system of differential equations . Differential equations describe how a particular quantity changes over time, given some initial starting conditions, and they are useful in describing all sorts of physical systems. In the case of the Navier-Stokes equations, we start with some initial fluid flow, and the differential equations describe how that flow evolves.

Solving a differential equation means finding some mathematical formula to determine what your quantity of interest actually will be at any particular time, based on the equations that describe how the quantity changes. Many physical systems described by differential equations, like a vibrating guitar string , or the flow of heat from a hot object to a cold object, have well-known solutions of this type.

The Navier-Stokes equations, however, are harder. Mathematically, the tools used to solve other differential equations have not proven as useful here. Physically, fluids can exhibit chaotic and turbulent behavior: Smoke coming off a candle or cigarette tends to initially flow smoothly and predictably, but quickly devolves into unpredictable vortices and whorls.

It's possible that this kind of turbulent and chaotic behavior means that the Navier-Stokes equations can't actually be solved exactly in all cases. It might be possible to construct some idealized mathematical fluid that, following the equations, eventually becomes infinitely turbulent.

Anyone who can construct a way to solve the Navier-Stokes equations in all cases, or show an example where the equations cannot be solved, would win the Millennium Prize for this problem.

Read the Clay Mathematics Institute's official description of the Navier-Stokes equations here .

## Yang-Mills theory and the quantum mass gap

Math and physics have always had a mutually beneficial relationship. Developments in mathematics have often opened new approaches to physical theory, and new discoveries in physics spur deeper investigations into their underlying mathematical explanations.

Quantum mechanics has been, arguably, the most successful physical theory in history. Matter and energy behave very differently at the scale of atoms and subatomic particles, and one of the great achievements of the 20th century was developing a theoretical and experimental understanding of that behavior.

One of the major underpinnings of modern quantum mechanics is Yang-Mills theory , which describes the quantum behavior of electromagnetism and the weak and strong nuclear forces in terms of mathematical structures that arise in studying geometric symmetries. The predictions of Yang-Mills theory have been verified by countless experiments, and the theory is an important part of our understanding of how atoms are put together.

Despite that physical success, the theoretical mathematical underpinnings of the theory remain unclear. One particular problem of interest is the " mass gap ," which requires that certain subatomic particles that are in some ways analogous to massless photons instead actually have a positive mass. The mass gap is an important part of why nuclear forces are extremely strong relative to electromagnetism and gravity, but have extremely short ranges.

The Millennium Prize problem, then, is to show a general mathematical theory behind the physical Yang-Mills theory, and to have a good mathematical explanation for the mass gap.

Read the Clay Mathematics Institute's official description of the Yang-Mills theory and mass gap problem here .

## The Riemann Hypothesis

Going back to ancient times, the prime numbers — numbers divisible only by themselves and 1 — have been an object of fascination to mathematicians. On a fundamental level, the primes are the "building blocks" of the whole numbers, as any whole number can be uniquely broken down into a product of prime numbers .

Given the centrality of the prime numbers to mathematics, questions about how primes are distributed along the number line — that is, how far away prime numbers are from each other — are active areas of interest.

By the 19th century, mathematicians had discovered various formulas that give an approximate idea of the average distance between primes . What remains unknown, however, is how close to that average the true distribution of primes stays — that is, whether there are parts of the number line where there are "too many" or "too few" primes according to those average formulas.

The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the "Riemann zeta function" all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.

Like several of the Millennium Prize problems, there is significant evidence suggesting that the Riemann Hypothesis is true, but a rigorous proof remains elusive. To date, computational methods have found that around 10 trillion solutions to the zeta function equation fall along the required line, with no counter-examples found.

Of course, from a mathematical perspective, 10 trillion examples of a hypothesis being true absolutely does not substitute for a full proof of that hypothesis, leaving the Riemann Hypothesis one of the open Millennium Prize problems.

In September 2018, the esteemed mathematician Professor Michael Atiyah claimed to have found a proof of the Riemann Hypothesis in a lecture at Heidelburg Laureate Forum . However, several mathematicians have expressed skepticism about the result, and a thorough process of reviewing and evaluating Atiyah's arguments will now begin.

Read the Clay Mathematics Institute's official description of the Riemann Hypothesis here .

## The Birch and Swinnerton-Dyer conjecture

One of the oldest and broadest objects of mathematical study are the diophantine equations , or polynomial equations for which we want to find whole-number solutions. A classic example many might remember from high school geometry are the Pythagorean triples , or sets of three integers that satisfy the Pythagorean theorem x 2 + y 2 = z 2 .

In recent years, algebraists have particularly studied elliptic curves , which are defined by a particular type of diophantine equation. These curves have important applications in number theory and cryptography , and finding whole-number or rational solutions to them is a major area of study.

One of the most stunning mathematical developments of the last few decades was Andrew Wiles' proof of the classic Fermat's Last Theorem , stating that higher-power versions of Pythagorean triples don't exist. Wiles' proof of that theorem was a consequence of a broader development of the theory of elliptic curves.

The Birch and Swinnerton-Dyer conjecture provides an extra set of analytical tools in understanding the solutions to equations defined by elliptic curves.

Read the Clay Mathematics Institute's official description of the Birch and Swinnerton-Dyer conjecture here .

## The Hodge conjecture

The mathematical discipline of algebraic geometry is, broadly speaking, the study of the higher-dimensional shapes that can be defined algebraically as the solution sets to algebraic equations.

As an extremely simple example, you may recall from high school algebra that the equation y = x 2 results in a parabolic curve when the solutions to that equation are drawn out on a piece of graph paper. Algebraic geometry deals with the higher-dimensional analogues of that kind of curve when one considers systems of multiple equations, equations with more variables, and equations over the complex number plane, rather than the real numbers.

The 20th century saw a flourishing of sophisticated techniques to understand the curves, surfaces, and hyper-surfaces that are the subjects of algebraic geometry. The difficult-to-imagine shapes can be made more tractable through complicated computational tools.

The Hodge conjecture suggests that certain types of geometric structures have a particularly useful algebraic counterpart that can be used to better study and classify these shapes.

Read the Clay Mathematics Institute's official description of the Hodge conjecture here .

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