The next major development was that of non-Euclidean geometry by Gauss, Riemann, Bolyai, and Lobachovsky. From the work of Lobachevsky, Gauss, Bolyai, and Riemann two new non-Euclidean geometries were discovered and each was found to be just as valid and consistent as Euclidean geometry. Also, it soon became clear that it is impossible to tell which, if any, of the three geometries is the most accurate as a mathematical representation of the real world. Thus, mathematicians were forced to abandon the cherished concept of a single correct geometry and to replace it with the concept of equally consistent and valid alternative geometries. They were also forced to realize that mathematical systems are not merely natural phenomena waiting to be discovered; instead, mathematicians create such systems by selecting consistent axioms and postulates and studying the theorems that can be derived from them.

 

Einstein used this new perspective to explain his general theory of relativity in mathematical terms. In addition, Einstein added time as a fourth dimension to the field of geometry or, more broadly speaking, to man’s concept of the universe.

 

Lastly, was the “group” concept that emerged gradually from investigations in algebra by Galois and Lagrange among others, and number theory. The extraordinary progress of algebra, analysis, geometry, mechanics, and theoretical physics is due to the idea of a group and its associated set of invariants.(5)

 

Footnotes:

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(1) Joseph Needham, Science and Civilization in China, Volume 3 - Mathematics and the Sciences of the Heavens and the Earth (Cambridge, 1959), p. 91. 

(2) Needham, p. 155.

(3) Ibid., p. 156.

(4) Ibid., p. 167-168.

(5) Isabella Bashmakova and Galina Smirnova, translated from the Russian by Abe Shenitzer, The Beginnings & Evolution of Algebra (Washington, D.C., 2000), p. 126.

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Math Problems for the 21st Century

 

As a closing note the Clay Mathematics Institute announced on May 24, 2000 seven Millennium Prize Problems. A correct solution to any of the problems results in a U.S. $1 million price awarded by the institute to the discoverer(s). Two of the problems come from people chosen for this book as the most influential, Poincaré and Riemann.

 

The seven problems are:

 

1. The Riemann Hypothesis

 

This is the only problem that remains unsolved from Hilbert’s list in 1900 (see Hilbert biography for a discussion of the whole list) excluding a few that are too imprecise to have a definite answer. Mathematicians across the world agree that this obscure-looking question about the possible solutions to a particular equation is the most significant unsolved problem in mathematics.

 

The Riemann Hypothesis: The Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ½. It asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

 

The problem was created by the German mathematician Bernhard Riemann in 1859 as part of an attempt to answer one of the oldest questions in mathematics: What, if any, is the pattern of the prime numbers among all counting numbers? Around 350 B.C.E., the famous Greek mathematician Euclid proved that the primes continue forever; that is, there are infinitely many of them. Moreover, by inspection, the primes seem to “thin out” and become less common the higher up you go through the counting numbers. But can you conclude any more than that? The answer is yes.

 

Euclid also proved that every number bigger than 1 (i.e., every positive counting number bigger than 1) is either itself a prime or else can be written as the product of prime numbers in a way that is unique apart from the order in which the primes are written. For example,

 

                                                                21 = 3 x 7,

                                                                260 = 2 x 2 x 5 x 13.

 

The expressions to the right of the equals signs are the “prime decompositions” of the numbers 21 and 260, respectively. Thus, we can express Euclid’s result by saying that every counting number bigger than 1 is either prime or else has a unique (up to changing the order) prime decomposition.

 

This fact, called the fundamental theorem of arithmetic, tells us that the primes are like the chemist’s atoms—the fundamental building blocks out of which all numbers are constructed. Just as an understanding of the unique molecular structure of a substance can tell us a lot about its properties, knowing the unique prime decomposition of a number can tell us a lot about its mathematical properties.

 

Finally one of the deepest observations about the pattern of the primes was made by Gauss, Riemann’s Ph.D. advisor. In 1791, when he was just 14 years old, Gauss noticed that the prime density DN = P (N) / N is approximately equal to 1/ln (N), where ln (N) is the natural logarithm of N. As far as Gauss could tell, the bigger N got, the better this approximation became. He conjectured that this was not just an accident, and that by making N sufficiently large, the density Dn could be made as close as you want to 1/ln (N). Gauss was never able to prove his conjecture. This was finally achieved—using some very powerful math—in 1896 by the Frenchman Jacques Hadamard and the Belgian Charles de la Vallée Poussin, working independently. Their result is known today as the Prime Number Theorem.

 

There are at least two amazing aspects of this result. First, it demonstrates that, despite the seemingly random way that the primes crop up, there is a systematic pattern to the way they thin out. The pattern is not apparent if you look at an arbitrary finite stretch of the numbers. No matter how far out along the numbers you go, you can find clusters of several primes close together as well as stretches as long as you like in which there are no primes at all. Nevertheless, when you step back and look at the entire sequence of counting numbers, you see that there is a very definite pattern:  The larger N becomes, the closer the density DN gets to 1/ln (N).

 

The second and far more important feature of the Prime Number Theorem is the nature of the pattern of the primes that is uncovers. The counting numbers are discrete objects, invented by our ancestors some 8,000 years ago as a basis for trading. The natural logarithm function was invented by sophisticated mathematicians a mere two hundred years ago. It is not discrete; rather, its definition depends upon a detailed analysis of infinite processes, and forms part of the discipline sometimes called advanced calculus and sometimes called real analysis. One of several equivalent definitions of ln(x) is as the inverse to the exponential function ex.

 

If you wanted to represent the prime numbers on a graph, the most obvious way would be to mark a point at each prime number on the x-axis, as shown in Figure 1 below.

 

____________________________________________

 

 

 

The graph of the function ln(x), on the other hand, is a smooth, continuous curve, as shown in Figure 2 below. The question is this:  Why is there a connection between the irregularly spaced points on the x-axis in in Figure 1 and the smooth curve shown in Figure 2? How is it that the function ln(x) can tell us something about the pattern of the primes?

 

 

 

 

 

 

 

 

 

 

 

 

 

To sum up, a proof of the Riemann Hypothesis would add to our understanding of the prime numbers and the way they are distributed. This would do far more than satisfy the curiosity of mathematicians. Besides having implications in mathematics well beyond the patterns of the primes, it would have ramifications in physics and modern communications technology specifically Internet security.

 

The Riemann Hypothesis and the World Wide Web

 

Every time you use an ATM machine at your bank or carry out a business transaction on the Internet, you are depending on the mathematical theory of prime numbers to keep your transaction secure. This is how it works.

 

From the moment people started to send messages to one another, the following issue arose: How can you prevent an unauthorized person who gets hold of the message from understanding what it says? The answer is you encode the message (the technical term is “encrypt”) so that only the intended receiver can access the original contents. Julius Caesar used a very simple system to encrypt the messages he sent to his generals commanding the Roman legions across Europe. He simply replaced each letter of the alphabet in each word by another, according to a fixed scheme, such as “replace each letter by the previous one in the alphabet” (with Z replacing A and A replacing B). A message encrypted this way might look completely unreadable, but to today’s cryptanalyst, Caesar’s system is easily broken.

 

Today, with the computer power available to any would-be codebreaker, it is extremely difficult to design a secure encryption system. (Much of the drive to develop computer technology during World War II came from the desire of each side to break the enemy’s codes.) If there is any kind of “recognizable” pattern to the encrypted text, a sophisticated statistical analysis using a powerful computer can usually crack the code. So, your encryption system needs to be sufficiently robust to resist computer attack.

 

These days, encryption systems invariably consist of two components: an encryption procedure and a “key.” The former is typically a computer program or, in the most widely used systems, a specially designed computer chip; the key is usually a secretly chosen number. To encrypt a message, the system required not only the message but also the chosen key. The encryption program codes the message so that the encrypted text can be decoded only with the key. Since the security depends on the key, the same encryption program may be used by many people for a long period of time, and this means that a great deal of time and effort can be put into its design. An obvious analogy is that manufacturers of safes and locks design one type of lock which may be sold to millions of users, who rely upon the uniqueness of their own key—be it a physical key or a secret number combination—to provide security. Just as an enemy may know how your lock is designed and yet be unable to break into your safe, so too the enemy may know what encryption system you are using without being able to break your coded messages.

 

In early key systems, the message sender and receiver agreed beforehand on some secret key, which they then used to send each other messages. As long as they kept this key secret, the system would (it was hoped) remain secure. An obvious drawback with such an approach was that the sender and receiver had to agree in advance on the key they would use, and since they would clearly not want to transmit that key over any interceptable communication channel, they would have to meet ahead of time and choose the key (or perhaps employ a trusted courier to communicate it). Such a system is obviously unsuitable in many situations. In particular, it won’t work in, say, international banking or commerce, where it is often necessary to send secure messages across the world to someone the sender has never met.

 

In 1975, two mathematicians, Whitfield Diffie and Martin Hellman, proposed a quite new type of encryption system: public key cryptography, in which the encryption method requires not one but two keys—one for encryption and the other for decryption. Such a system is used like this. A new user, say Nicole, obtains the standard program (or special computer chip) used by all members of the communication network concerned. She then generates two keys. One of these, her deciphering key, she keeps secret. The other key, the one used for encrypting messages sent to her by anyone else in the network, she publishes in a directory of the network users. To send a message to a network user, all that has to be done is to look up that user’s public encryption key, encrypt the message using that key, and send it. To decode the message it is of no help knowing (as anyone can) the encryption key. You need the decryption key, which only the intended receiver knows.

 

Several specific methods were developed to implement Diffie and Hellman’s general scheme. The one that gained most support, and which remains to this day the industry standard, was designed by Ronald Rivest, Adi Shamir, and Leonard Adelman, of the Massachusetts Institute of Technology. It is known by their initials as the RSA system, and is marketed by a commercial data security company, RSA Data Security, Inc., based in Redwood City, California. The secret decryption key used in the RSA method consists (essentially) of two large prime numbers (each having, say, 100 digits) chosen by the user. (The choice of the two primes is made using a computer, not chosen from any published list of primes, to which an enemy might have access. Modern computers can find large primes with ease.) The public encryption key is the product of these two primes. The system’s security depends upon the fact that there is no known quick method of factoring large numbers. This means that it is practically impossible to recover the decryption key (the two primes) from the public encryption key (their product). Message encryption corresponds to multiplication of two large primes (an easy task), decryption corresponds to the opposite process of factoring (which is hard). (This is not exactly how the system works. Some moderately sophisticated mathematics is involved.)

 

At present, the largest numbers that can be factored on a powerful computer in less than a few days have around 90 to 100 digits, so using a key obtained by multiplying together two 100-digit primes, i.e., a number with 200 digits, should make the RSA system extremely secure. But there is a danger. The methods mathematicians use to factor large numbers are not simply trial-and-error searches such as you might use if I asked you to find the prime factors of 221. Doing it that way is fine for fairly small numbers, but could take a powerful computer over a year to factor a 60-digit number. Instead, mathematicians use some highly sophisticated techniques to find prime factors. The methods they have developed are clever and powerful, and getting steadily more so. Those methods make use of much that we know about prime numbers, and every time there is an advance in our knowledge of the primes, there is always a possibility that it will lead to a new method for factoring numbers.

 

Since the Riemann Hypothesis tells us so much about the primes, a proof of that conjecture might well lead to a major breakthrough in factoring techniques. Not because we will then know the hypothesis is true. Suspecting that it is true, mathematicians have been investigating its consequences for years. Indeed, some factoring methods work on the assumption that it’s true. Rather, the fear among the encryption community is that the methods used to prove the hypothesis will involve new insights into the pattern of the primes that will lead to better factoring methods.

 

Clearly, then, with Internet security and large parts of contemporary mathematics hanging in the balance, far more is at stake in the Riemann Problem than the $1 million Millennium Prize.

 

2. Yang-Mills Theory and the Mass Gap Hypothesis

 

The Yang-Mills equations come from quantum physics. They were formulated almost fifty years ago by the physicists Chen-Ning Yang and Robert Mills to describe all of the forces of nature other than gravity. They do an excellent job. The predictions culled from these equations describe particles that have been observed at laboratories around the world. But while the Yang-Mills theory works in practical terms, it has not yet been worked out as a mathematical theory. The second Millennium Problem asks for, in part, that missing mathematical development of the theory, starting from axioms.

 

The mathematics needs to meet a number of conditions that have been observed in the laboratory. In particular, it should establish (mathematically) the “Mass Gap Hypothesis,” which concerns supposed solutions to the Yang-Mills equations. This hypothesis is accepted by most physicists, and provides an explanation of why electrons have mass. Proof of the Mass Gap Hypothesis is regarded as a good test of a mathematical development of the Yang-Mills theory. It would help the physicists as well. They cannot explain why electrons have mass either; they simply observe they do.

 

3. The P Versus NP Problem

 

This is the only Millennium Problem that is about computers. Many people will find this surprising since many people assume most math is done on computers today. However this is not true. Most numerical calculations are done on computers, but numerical calculation is only a very small part of mathematics, and not a typical part at that.

 

Although the electronic computer came out of mathematics—the final pieces of the math were worked out in the 1930s, a few years before the first computers were built—the world of computing has hitherto generated only two mathematical problems that would merit inclusion among the world’s most important. Both problems concern computing as a conceptual process rather than any specific computing devices, although this does not prevent them from having important implications for real computing. Hilbert included one of them as number 10 on his 1900 list. That problem—which asks for a proof that certain equations cannot be solved by a computer—was solved in 1970.

 

The other problem is more recent. This is a question about how efficiently computers can solve problems. Computer scientists divide computational tasks into two main categories:

(1) Tasks of type P can be tackled effectively on a computer and (2) Tasks of type E could take millions of years to complete. Unfortunately, most of the big computational tasks that arise in industry and commerce fall into a third category, NP, which seems to be intermediate between P and E. But is it? Could NP be just a disguised version of P? Most experts believe that NP and P are not the same (i.e., that computational tasks of type NP are not the same as tasks of type P? But after thirty years of effort, no one has been able to prove whether or not NP and P are the same. A positive solution would have significant implications for industry, for commerce, and for electronic communications, including the World Wide Web.

 

4. The Navier-Stokes Equations

 

The Navier-Stokes equations describe the motion of fluids and gases—such as water around a pier or air over a plane wing. They are of a kind that mathematicians call partial differential equations. College students in science and engineering routinely learn how to solve partial differential equations, and the Navier-Stokes equations look just like the kinds of equations given as exercises in a college calculus textbook. But appearances can be deceptive. To date, no one has a clue how to find a formula that solves these particular equations—or even if such a formula exists.

 

This failure has not prevented marine engineers from designing efficient boats or aeronautical engineers from building better aircraft. Although there is no general formula that solves the equations (say, in the way that the quadratic formula solves all quadratic equations), the engineers who design high-performance boats and aircraft can use computers to solve particular instances of the equations in an approximate way. Like the Yang-Mills Problem, the Navier-Stokes Problem is another case where mathematics wants to catch up with what others, in this case engineers, are already doing.

 

5. The Poincaré Conjecture

 

This problem was first raised in 1904 by the French mathematician Henri Poincaré. It starts with a seemingly simple question: How can you distinguish an apple from a doughnut? This does not seem like a question that would lead to a $1 million math problem. But what makes it hard is that Poincaré wanted a mathematical answer that could be used in more general situations. That rules out the more obvious solutions, such as simply taking a bite of each.

 

Here is how Poincaré himself answered the question. If you stretch a rubber band around the surface of an apple, you can shrink it down to a point by moving slowly, without tearing it and without allowing it to leave the surface. On the other hand, if you imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. Surprisingly, when you ask whether the same shrinking band idea distinguishes between four-dimensional analogies of apples and doughnuts—which is what Poincaré was really after—no one has been able to provide an answer. The Poincaré conjecture says that the rubber band idea does identify four-dimensional apples.

 

This problem lies at the heart of topology, one of the most fascinating branches of present-day mathematics. Besides its inherent and sometimes quirky fascination—for instance, it tells you the deep and fundamental ways in which a doughnut is the same as a coffee cup—topology has applications in many areas of mathematics, and advances in the subject have implications for the design and manufacture of silicon chips and other electronic devices, in transportation, in understanding the brain, and even in the movie industry.

 

In 2003, the Poincaré Conjecture was solved by the Russian mathematician Grigori Perelman. 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. Amazingly, he refused the prize money telling the Interfax news agency that his contribution was no greater than Richard Hamilton who had completed work on the problem earlier.

 

6. The Birch and Swinnerton-Dyer Conjecture

 

This problem is in the same general area of mathematics as the Riemann Hypothesis. Since the time of the ancient Greeks, mathematicians have wrestled with the problem of describing all solutions in whole numbers x, y, z to algebraic equations like

 

x2 + y2 = z2

 

For this particular equation, Euclid gave the complete solution—that is to say, he found a formula that produces all the solutions. In 1994, Andrew Wiles proved that for any exponent n greater than 2, the equation

 

xn + y2 = z2

 

has no nonzero whole-number solutions. (This was the result known as Fermat’s Last Theorem. See a detailed discussion of Wiles solution under the Fermat biography.) But for more complicated equations it becomes extremely difficult to discover whether there are any solutions, or what they are. The Birch and Swinnerton-Dyer Conjecture provides information about the possible solutions to some of those difficult cases.

 

As with the Riemann Hypothesis, to which it is related, a solution to this problem will add to our overall understanding of the prime numbers. Whether it would have comparable implications outside of mathematics is not clear. Proving the Birch and Swinnerton-Dyer Conjecture might turn out to be important only to mathematicians.

 

On the other hand, it would be foolish to classify this or any mathematical problem as being “of no practical use.” Admittedly, the mathematicians who work on the abstract problems of “pure mathematics” are usually motivated more by curiosity than by any practical consequences. But again and again, discoveries in pure mathematics have turned out to have important practical applications.

 

On top of that, the methods mathematicians develop to solve one problem often turn out to have applications to quite different problems. This was definitely the case with Andrew Wiles’s proof of Fermat’s Last Theorem. Similarly, a proof of the Birch and Swinnerton-Dyer Conjecture would almost certainly involve new ideas that will later be found to have other uses.

 

7. The Hodge Conjecture

 

This is another “missing piece” question about topology. The general question is about how complicated mathematical objects can be built up from simpler ones. Of all the Millennium Problems, this is perhaps the one the layperson will have the most trouble understanding. Not so much because the underlying intuitions are any more obscure than for the other problems or because it is believed to be harder than any of the other six problems. Rather, the Hodge Conjecture is a highly technical one having to do with the techniques mathematicians use to classify certain kinds of abstract objects. It arises deep within the subject, at a high level of abstraction, and the only way to reach it is by way of those layers of increasing abstraction. This is why I have put this problem last following Keith Devlin’s order in his book on the Millennium problems.

 

The path to the conjecture began in the first half of the twentieth century, when mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea was to ask to what extent you can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it was generalized in many different ways, eventually leading to powerful tools that enable mathematicians to catalogue many different kinds of objects. Unfortunately, the generalization obscured the geometric origins of the procedure, and the mathematicians had to add pieces that did not have any geometric interpretations at all. The Hodge conjecture asserts that for one important class of objects (called projective algebraic varieties), the pieces called Hodge cycles are, nevertheless, combinations of geometric pieces (called algebraic cycles).

 

 

Key References:

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1. The Millennium Problems – The Seven Greatest Unsolved Mathematical Puzzles of Our Time by Keith Devlin, 2002.

2. For precise descriptions of the seven problems, together with the official rules for the competition, consult the Clay Mathematics Institute website at www.claymath.org (The website also features a twenty minute streaming video on the Millennium Problems, presented by Keith Devlin and produced by the Clay Institute.).

3. On the Poincaré Conjecture – Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen, 2009.

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