$6 Million still on offer...

Jack Dikian

June 2011

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts established seven Prize Problems. As of June 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute. Grigori Perelman solved the Poincaré conjecture and was awarded the first Clay Millennium Prize in 2010 – he turned down the prize. The seven Millennium Prize Problems are:

1. Birch & Swinnerton - Dyer Conjecture

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.

2. Hodge Conjecture

Over the years mathematicians discovered powerful ways to investigate the shapes of complicated objects. The idea is to ask to what extent can we approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

3. Navier-Stokes Equation

Mathematicians have long believed that an explanation for the way turbulent air and water currents follow moving planes and boats can be found through an understanding of solutions to the Navier-Stokes equations. The challenge here is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.

4. P vs NP Problem

One of my interest areas - Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice.

This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a co-worker is satisfactory, however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe.

Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer. In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure.

5. Poincaré Conjecture (Solved)

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we 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. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin).

6. Riemann Hypothesis

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function:

ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

7. Yang-Mills and Mass Gap

The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable 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 its mathematical foundation is still unclear. 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:" the quantum particles have positive masses, even though the classical waves travel at the speed of light.

This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

Alice, Bob and Eve - Quantum Cryptography

Jack Dikian

June 2011

Quantum cryptography is the use of quantum systems to perform cryptographic tasks. A key application of quantum cryptography is quantum key distribution (QKD). QKD describes the process of using quantum communication to establish a shared key between two parties Alice and Bob without a third party Eve learning anything about that key, even if Eve can eavesdrop on all communication between Alice and Bob.

This is due to the process of measuring a quantum system in general disturbs the system. A third party trying to eavesdrop on the key must in some way measure it, thus introducing detectable anomalies.

In contrast traditional cryptography often relies on mathematical theorems that may be vulnerable to attacks by individuals armed with increasingly powerful supercomputers.

But, even though QKD is theoretically secure researchers have been aware that loopholes may arise when QKD is used in real world applications.

Now, a team of researchers at the Centre for Quantum Technologies at the National University of Singapore, the Norwegian University of Science and Technology and the University Graduate Center in Norway have created a perfect eavesdropper for QKD allowing researchers to obtain an entire shared secret key without alerting either of the legitimate parties that there had been a security breach.

The Role of Fractal Geometry In Evolution

Jack Dikian

May 2011

A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity (wikipedia).

James Gleick writes that a decade after Mandelbrot published his physiological speculations; some theoretical biologists began to find fractal organization controlling structures all through the body. The standard exponential description of a bronchial branching proved to be quite wrong; a fractal description turned out to fit the data.

In the view of the Darwinists, the endlessly exquisite designs of nature are the result of an interplay of two factors - random genetic mutation and Natural Selection. Genetic mutation proposes, Natural Selection disposes. Some maintain that mutations usually detract from the viability of an organism and that Darwin's premise that genetic mutations are random is wrong. That is, that nature selects the strongest does not hold true.

The question of "design" in nature was one that troubled Charles Darwin. In the year following the publication of the Origin, he wrote, "I am conscious that I am in an utterly hopeless muddle. I cannot think that the world, as we see it, is the result of chance; and yet I cannot look at each separate thing as the result of design."

Design according to [Darwin] is Random Mutation + Natural Selection + Time. An awful lot of time. Regardless of the species, the changes over time have to come from changes in the Deoxyribonucleic acid (DNA). The entire plan for any organism is contained in its DNA, a molecule with 4 nucleobases. A strand of DNA can have anywhere from 500 thousand bases (in the case of the smallest known parasite) to about 3 billion for man and large animals.

So Darwinism says that random mutation or copying errors in the DNA produce a modified organism, and that natural selection weeds out less adaptable ones. What [remains] are new innovations in an organism design.

Of course, one of the difficulties with Darwinism is that it takes millions of years and many billions of organisms to produce significant change over time making it difficult to empirically prove Darwinism in the short lifetime of a human being.

In a totally different scientific discipline, mathematics, Benoit Mandelbrot‘s interest in irregular and seemingly chaotic patterns developed fractal geometry in the mid 1970’s. Sure a number of geometry systems have been developed since Euclidean geometry (c.300 B.C) fractals explore the infinitely complex shapes and self-similarity across scale featuring recursion – patterns seen everywhere in nature, economics, and cell biology (and here is the rub).

An example of this is a photograph of a section of coastline from a plane will show the same ragged contours as a photograph of the whole coast taken from a space station. A photograph of a one-foot-long section of the same coast will also show the same contours. The various coastlines are "self-similar," each similar to the others in shape, but different in magnitude.

Biologists began to find fractal organization controlling structures throughout the body. The standard exponential description of bronchial branching, for example, proved to be wrong; a fractal description turned out to fit the data. Other systems include the urinary collecting system, Biliary duct in the liver, network of special fibers in the heart that carry pulses of electric current to the contracting muscles.

Ham Sandwich and Pancakes - A Theorem In Topology

Jack Dikian

May 2011

Biting into a ham sandwich today, it occurred to me that I’d almost forgotten the taste of ham. I hadn’t eaten it for years. The other thing, it reminded me of fond memories of university days, a time when I first read about the ham sandwich theorem. You can depend on Topologists to make what are otherwise abstract ideas more appealing. And so the theorem goes like this:

Regardless of the distribution of ham in a sandwich you can use one slice to divide the sandwich into two parts containing equal parts of bread and ham.

More formally, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given n measurable "objects" in n-dimensional space, it is possible to divide all of them in half with a single (n − 1) - dimensional hyperplane.

For those who enjoy pancakes - the two dimensions case is known as the pancake theorem of having to cut two infinitesimally thin pancakes on a plate each in half with a single cut.

7 Bridges

Jack Dikian

January 2011

While writing about Knot Theory last night the subject and associated problems in Topology bought back many memories that are reminiscent to a child experiencing the color and movement of a circus for the first time.

Topology and in particular Graph Theory is often not part of high school math curricula, and thus for many it sounds strange and intimidating. However, there are some readily graspable ideas at the base of topology that are interesting, fun, and highly applicable to all sorts of situations. One of these is the topology of networks, first developed by Euler in the early 1700’s inspired by the Seven Bridges of Konigsberg.

Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in Russia. The city is built around the River Pregel where it joins another river. An island named Kniephof is in the middle of where the two rivers join. There are seven bridges that join the different parts of the city on both sides of the rivers and the island.

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. No one was able to find a way to do it, and this came to the attention of a mathematician Euler.

In 1735, Euler, before the Russian Academy presented the solution to the problem explaining why crossing all seven bridges without crossing a bridge twice was impossible.

Euler simplified the bridge problem by representing each land mass as a point and each bridge as a line. He reasoned that anyone standing on land would have to have a way to get on and off. Thus each land mass would need an even number of bridges. But in Konigsberg, each land mass had an odd number of bridges. This was why all seven bridges could not be crossed without crossing one more than twice.

Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.

Untangling Knots

Jack Dikian

January 2011

Recently, over the Christmas and New Year holidays, I found myself getting increasingly irritable with the number of times I found I needed to untangle my Ipod’s headphones. The angrier I become, the harder I tried to untangle the mess, the tighter the knots seemed to get. My young niece on the other hand approached the problem very differently. For her, it was a breeze - calm, and almost mechanical – little fingers might have been a benefit.

What did interest me however was the idea of whether a heuristic approach can be taken to untangle a knot. Two things just changed in my language. One that I used the word “knot” in the singular and secondly, implicitly, the ability to identify the knot-type and use a set of steps to unwind, untangle the knot.

We all know from experience how to create knots. We do this all the time, often, like me, unwittingly. In Pure Maths, we learned about those knots whose ends are glued together and their classification form the subject of a branch of Topology known as Knot Theory.

The thread goes back almost 150 years when in the nineteenth century physicists were speculating about the underlying principles of atoms. Lord Kelvin put forward a comprehensive theory of atoms, which, seemed to explain several of the essential qualities of the chemical elements - Kelvin's theory conjectured that atoms were knotted tubes of ether.

Kelvin's theory of atoms was taken seriously for many years. Maxwell for example thought that ``it satisfies more of the conditions than any atom hitherto considers''. The topological stability and the variety of knots were thought to mirror the stability of matter and the variety of chemical elements.

This theory inspired physicist Peter Tait to undertake an extensive study and tabulation of knots in an attempt to understand when two knots were ``different''. He’s intuitive understanding of ``different'' and ``same'' is a useful notion even today. Two knots are isotopic if one can be continuously manipulated in 3-space with no self-intersections allowed until it looks like the other.

Mathematicians, then and now, continue to pose the same question. That is, how do we know when two knots are isotopically the same. This failed atomic theory also left in its wake the riches of Tait's tabulation---163 knot projections---and a rudimentary understanding of isotopic sameness in terms of how one projection could be continuously manipulated to look like another. This understanding of projection manipulation was summarized in a set of conjectures for knot projections, the famous Tait Conjectures.

Alice's Adventures in Wonderland - the Math

Jack Dikian

November 2010

Alice's Adventures in Wonderland (commonly shortened to Alice in Wonderland) is an 1865 novel written by English author, mathematician, logician, Anglican deacon and photographer Charles Lutwidge Dodgson under the pseudonym Lewis Carroll.

Much has been said of the mathematical undercurrent in Alice’s adventures in wonderland. Before reflecting upon the math it’s useful to look at the times, and backdrop of mathematics in the latter half of the nineteenth century.

How Dodgson came to write his novel has been much documented – in the way of a very brief summary, the story goes that in 1862, Dodgson, together with the Reverend Robinson Duckworth, rowed in a boat up the River Thames with three young girls, Lorina Charlotte Liddell, aged 13, Alice Pleasance Liddell, aged 10, and Edith Mary Liddell, aged 8, the daughters of Henry George Liddell, the Vice-Chancellor of Oxford University and Dean of Christ Church College, as well as headmaster of the nearby, private, Westminster School.

As they rowed, Dodgson made up and told the girls a story about a bored little girl named Alice who goes looking for an adventure. The three girls loved it, and Alice Liddell asked Dodgson to write it down for her. Two years later he did just that, and in November 1864 he gave Alice the handwritten manuscript of what he then called "Alice's Adventures Under Ground," illustrated by his own drawings.

The math

In the latter half of the nineteenth century, it was a turbulent period for mathematicians, with the subject becoming more and more abstract. Non-Euclidean geometry, Symbolic algebra and the growing use of imaginary numbers were just some of the developments that challenged classical mathematics.

Martin Gardner, whose The Annotated Alice was published in 1960, Helena Pycior of the University of Wisconsin-Milwaukee, and more recently Melanie Bayley, of the University of Oxford in England, have examined linkages between mathematics and the experiences of the characters in the novel

According to Bayley, Dodgson was dismayed by what he saw as the declining standards of mathematical rigor. She suggests Dodgson additions for publication was a wicked satire on those new developments.

What of the links with mathematics - Here are just a few examples

In the chapter "Advice from a caterpillar." Alice has fallen down the rabbit hole and eaten a cake that has shrunk her to a height of just 3 inches. The Caterpillar enters, smoking a hookah pipe, and shows Alice a mushroom that can restore her to her proper size. But one side of the mushroom stretches her neck, while another shrinks her torso, so she must eat exactly the right balance to regain her proper size and proportions. Bayley believes this expresses Dodgson's view of the absurdity of symbolic algebra.

Bayley suggests that the overall madness of Wonderland reflects Dodgson's views on the dangers of this new symbolic algebra. Alice has moved from a rational world to a land where even numbers behave erratically.

In another scene, the hallway, Alice tries to remember her multiplication tables, but they have slipped out of the base-10 number system she is used to. Yet another, perhaps, unsubtle but pointed remark.

In the caterpillar scene, Alice's height fluctuates between 9 feet and 3 inches. Alice, bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling: "Being so many different sizes in a day is very confusing," she complains.

To survive in Wonderland, Alice must act like a Euclidean geometer, keeping her ratios constant, even if her size changes.

The baby's discomfort with the whole process, and the Duchess's unconcealed violence, signpost Dodgson's virulent mistrust of "modern" projective geometry, Bayley says. Everyone in the pig and pepper scene is bad at doing their job.