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Duke University

Millennium Prize Problems

Sam Pease

Math 89S: Mathematics of the Universe

Professor Hubert Bray

September 2016

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You’d think a million dollars would be enough incentive for someone in the world to

solve a simple math problem. The Clay Institute of Mathematics thought this would be the

case too when they proposed 7 mathematics problems at the turn of the 21st century to

emphasize the fact that there were still unsolved problems in math. The problems each carry

a 1-million-dollar prize to whomever can solve them. You’d think that the problems would

have been quickly solved by eager mathematicians looking for a quick way to make some

money, but only one has been solved to date. This is because these are no simple problems; It

has been stated that this is one of the most difficult ways to earn a million dollars that there

is. Each of these problems are important classical questions that have resisted solution for

many years. One of these, the Riemann hypothesis problem was put forth in a similar

competition back in 1900. These problems represent not just as an interesting competition for

mathematicians to show off their abilities but as a furtherment of mathematics. These

problems require new tools and ways of thinking to solve and each have profound

consequences when solved. This prize money is simply to stir up more interest in expanding

the field of mathematics. The seven problems are the Poincare Conjecture, Yang–Mills and

Mass Gap, Riemann Hypothesis, P vs NP Problem, Navier–Stokes Equation, Hodge

Conjecture, and the Birch and Swinnerton-Dyer Conjecture (Clay Math).

The only one of the seven problems that has been solved is the Poincare Conjecture.

This problem is in the fields of topology and differential geometry and was very difficult,

taking nearly a century to solve. The problem states that every simply connected closed

three-manifold is homeomorphic to the three-sphere. In two dimensions if we were to stretch

a rubber band around a ball we are able to shrink the rubber band down into a point by

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sliding it off of the ball. But if we were to put this rubber band around a doughnut we would

be unable to bring it to a point. Because of this we say the surface of the sphere is simply

connected, and any 3-dimensional shape that is also simply connected is homeomorphic to

the sphere, meaning they are of the same form. The Poincare Conjecture is that the same is

true for the three-sphere which is a four-dimensional sphere with a three-dimensional surface

(Wolfram Mathworld)

This problem was solved by Russian mathematician Dr. Grigori (Grisha) Perelman of

the Steklov Institute of Mathematics in 2002 and 2003. He, however, declined the prize

money. He is quoted saying he believed the prize was unfair. Perelman told Interfax he

considered his contribution to solving the Poincare conjecture no greater than that of

Columbia University mathematician Richard Hamilton. (math.columbia.edu) It was

Images from Wikipedia

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published in two prints on ArXiv.org and the second of which begins with “This is a

technical paper, which is the continuation of [Perelman 2002].” Signifying the papers are not

meant to be accessible. And in fact, the papers don’t contain a single reference to the

Poincare Conjecture but instead prove the broader Thurston's geometrization conjecture

which establishes the Poincare conjecture. This conjecture was proven using Ricci flow,

which is a tool to change a shape in order to better understand it (Wolfram Mathworld).

The Yang–Mills and Mass Gap problem is in the area of mathematical physics. Yang

and Mills introduced a new framework to describe elementary particles using structures that

also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of

elementary particle theory, and its predictions have been tested at many experimental

laboratories, but it does not have a rigorous mathematical background. The successful use of

Yang-Mills theory to describe the strong interactions of elementary particles depends on a

subtle quantum mechanical property called the mass gap. This means that since energy is

quantized there is a minimum amount of energy which corresponds to a smallest mass

possible of fundamental particles. This property has been discovered by physicists from

experiments and confirmed by computer simulations, but it still has not been understood

Visualization of Ricci flow, image from WikipediaVisualization of Ricci flow, image from Wikipedia

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from a theoretical point of view. Progress in establishing the existence of the Yang-Mills

theory and a mass gap will require the introduction of fundamental new ideas both in physics

and in mathematics. A rigorous proof of Yang-Mills would require much new math in order

to state the problem and would give much more credit the phenomenological physical

theories. It however would not likely to have a large influence on pure math.

The Riemann Hypothesis is the oldest of the millennium prize problems and carries

huge implications for pure mathematics. The infinite series that follows was studied by gauss

and has many interesting properties:

. ζ(2)=pi^2/6 and

ζ(4)=pi^4/90. It also can be expressed as Showing that

there is a direct relationship between a series adding the natural numbers together with a

series multiplying prime numbers together. This infinite series was then analytically

continued by G.F.B. Riemann in order so that it would equal the same value for s>1 but

modified it to converge at s<=1 and all the numbers in the complex plane beside 1. This is

where the popular misconception that the sum of the natural numbers is equal to -1/12

because ζ(-1)=-1/12. When studying this new analytically continued function we can look at

where it is equal to zero. This function has trivial zeros (easy to prove and uninteresting) at

all the negative even integers, but the Riemann Hypothesis concerns the non-trivial zeros.

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The Riemann Hypothesis states that all nontrivial zeros of the function have a real

component of ½. This has been checked for the first 10,000,000,000,000 solutions but is yet

to be proven. Their distribution is not completely understood but, more importantly, their

study yields impressive results concerning prime numbers and related objects in number

theory. An alternative hypothesis that is the equivalent to the Riemann hypothesis is that the

error in the prime counting function Li(x) is at most sqrt(x)*log(x) in its estimation for the

number of prime numbers smaller than a given number x. This along with many other

applications in number theory would be better understood with a proof of the Riemann

hypothesis (Thomas Wright).

P vs NP is a problem in Computability theory. A proof for or against it would serve to

win the prize and experts in the field are split on which they think is likely true, though most

seem to believe that P is not equal to NP (Lane A. Hemaspaandra). This problem has to do

with the ability for a computer algorithm to solve a problem vs its ability to check the

solution to a problem. Problem of class P can be solved in Polynomial time which means that

even as the problem scales up a solution can be found proportional to the size of the problem

and is generally seen as a realistic timeframe for an algorithm, as opposed to exponential

time which is impossible for larger problems. Problem of class NP or Nondeterministic

polynomial time are problems which if a solution is known it can be checked very easily.

Our current encryption is based off the fact that week can check that two partners have the

correct keys in an online transaction but an outsider could not generate such a key in

polynomial time. This example seems to imply P does not equal NP but it could just mean

that there are more efficient algorithms for finding a correct key than we know and a proof of

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P=NP would cast doubt on the safety of all our encryption. A problem is NP complete if

every NP problem can be transformed into the NP complete problem in polynomial time.

This means that if one were to prove P=NP for a single NP complete problem then the result

could be extended to all NP problems. Minesweeper has been shown to be NP complete by

imbedding logical circuits which are open NP problems into a minesweeper game. This

specific sequence of mines shows that if you are able to solve PvsNP for Minesweeper that

you could extend your result to include simple Boolean NP complete problems which means

you can extend your results to all NP problems. It is possible to win a million dollars if you

can solve Minesweeper (Ian Stewart).

The Navier-Stokes existence and smoothness problem is a problem in Mathematical

Physics, Differential Topology, and Real Analysis. The Navier-Stokes equations are well

defined but not well understood: . This formula describes the

motion of a fluid were where nu is the kinematic viscosity, u is the velocity of the fluid

parcel, P is the pressure, and rho is the fluid density. The equations are differential equations

and could, in theory, be solved for a given flow problem by using methods from calculus.

But, in practice, these equations are too difficult to solve analytically. Turbulent flow, which

is one of the last unsolved areas of classical mechanic, is believed to obeys the Navier-Stokes

equations. However, the flow is so complex that it is not possible to solve turbulent problems

from first principles with the computational tools available today, opposed to laminar flow

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which is much easier to model with Navier-Stokes.

These equations are already used and are incredibly useful to describe fluid flow but they can

only be solved in special circumstances and no general solution has been found. That is what

the millennium problem is: to find a general solution. (NASA,

https://universe-review.ca/R13-10-NSeqs.htm, Illinois Institute of Technology)

The Hodge conjecture is one of the most difficult of the problems to understand in an

intuitive way without being an expert in the field. The official statement is: Let X be a

projective manifold then a topological cycle C on X is homologous to a rational combination

of algebraic cycles if and only if C has a rotation curve of zero. This is hard to visualize

because it is a conjecture about non-Euclidean spaces that are projective manifolds. But this

problem is very important because it is bringing together Algebraic Topology and Algebraic

Geometry by relating topological cycles to algebraic cycles. (Daniel S Freed)

The Birch and Swinnerton-Dyer Conjecture is about Diophantine equations and is

very important in number theory. A Diophantine equation is one where we constrain our

solutions either rational or integer solutions. These can range from being simple and

common place such as the Pythagorean theorem: a^2+b^2=c^2 to very difficult to solve or

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proven to have no solution such as Fermat’s Last Theorem: a^n+b^n=c^n. All Diophantine

equations can be out into the classes of rational, elliptic, or general type. These correspond to

equations of degree 2, 3, and ≥ 4, respectively (except in some exceptional degenerate cases).

The problem of finding rational solutions turns out to be easy for the first class, hard but

attackable for the second, and in general impossible for the third. Rational equations usually

have infinitely many solutions and a formula can be found that gives them all. Elliptic

equations have finitely or infinitely many solutions and the Birch and Swinnerton-Dyer

Conjecture describes the structure of these equations and gives a way to predict the number

of their solutions. (Gunter Harder and Don Zagier)

As you can see these problems are not just trivial statements eluding proof. They are

all on the forefront of their respective areas and are very difficult to even understand without

a background in the specific area. One problem has been solved, and still many others don’t

even have a real method of solving them. All together the Clay Institute of Mathematics

succeeded in their job of bringing unsolved math to the vision of the public. But a one-

million-dollar reward is not going to magically make answers to these problems come out of

the woodworks. They each have their own value that makes their solutions sought after and

the reason they aren’t solved isn’t simply a problem of motivation. Whatever way you look

at it the solutions to these problems will forward the progress of mathematical and scientific

research and unlock new problems to be tackled.

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Bibliography

(n.d.). Retrieved November 01, 2016, from

https://www.grc.nasa.gov/www/k-12/airplane/bga.html

Fluid Dynamics and the Navier-Stokes Equations. (n.d.). Retrieved November 01,

2016, from https://universe-review.ca/R13-10-NSeqs.htm

Freed, D. S. (n.d.). The Hodge Conjecture. Retrieved November 01, 2016, from

http://www.ma.utexas.edu/users/dafr/HodgeConjecture/netscape_noframes.html

Harder, G., & Zagier, D. (n.d.). THE CONJECTURE OF BIRCH AND

SWINNERTON-DYER. Retrieved November 1, 2016, from http://people.mpim-

bonn.mpg.de/zagier/files/tex/BSDwHarder/fulltext.pdf

Hemaspaandra, L. A. (n.d.). Opinion poll - UMD Department of Computer Science.

Retrieved November 1, 2016, from http://www.cs.umd.edu/~gasarch/papers/poll.pdf

Ian Stewart on Minesweeper. (n.d.). Retrieved November 1, 2016, from

http://www.claymath.org/sites/default/files/minesweeper.pdf

Perelman Turns Down Millennium Prize. (n.d.). Retrieved November 01, 2016, from

http://www.math.columbia.edu/~woit/wordpress/?p=3056

Poincaré Conjecture. (n.d.). Retrieved November 01, 2016, from

http://mathworld.wolfram.com/PoincareConjecture.html

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Poincaré Conjecture Proved—This Time for Real - MathWorld. (n.d.). Retrieved

November 1, 2016, from http://mathworld.wolfram.com/news/2003-04-15/poincare/

The Millennium Prize Problems. (n.d.). Retrieved November 01, 2016, from

http://www.claymath.org/millennium-problems/millennium-prize-problems

The Navier-Stokes Equations. (n.d.). Retrieved November 1, 2016, from

https://web.iit.edu/sites/web/files/departments/academic-affairs/Academic Resource

Center/pdfs/Navier_Stokes.pdf

Wright, T. (n.d.). A Friendly Introduction to The Riemann Hypothesis. Retrieved

November 1, 2016, from http://www.math.jhu.edu/~wright/RH2.pdf