Author: IMC

Our best efforts to gauge threats may be counter-productive.

Assessing risk is something everyone must do every day. Yet few are very good at it, and there are significant consequences of the public’s collective inability to accurately assess risk.

As a first and very important example, most people presume, as an indisputable fact, that the past century has been the most violent in all history — two devastating world wars, the Holocaust, the Rawanda massacre, the September 11 attacks and more — and that we live in a highly dangerous time today.

And yet, as Canadian psychologist (now at Harvard) Steven Pinker has exhaustively documented in his new book The Better Angels of Our Nature: Why Violence Has Declined, the opposite is closer to the truth, particularly when normalised by population.

As Pinker himself puts it:

“Believe it or not — and I know most people do not — violence has been in decline over long stretches of time, and we may be living in the most peaceful time in our species’ existence. The decline of violence, to be sure, has not been steady; it has not brought violence down to zero (to put it mildly); and it is not guaranteed to continue.

“But I hope to convince you that it’s a persistent historical development, visible on scales from millennia to years, from the waging of wars and perpetration of genocides to the spanking of children and the treatment of animals.”

How could the public perception be so wrong? The news media is partly to blame — good news doesn’t sell much advertising space. But the problem might go even deeper: we may be psychologically disposed to miscalculate risk, perhaps as an evolutionary response to danger.

One well-known problem is the “conjunction fallacy” — the common predilection to assign greater probability to a more specialised risk.

One indication of our inability to objectively assess risk is the fanatical and often counter-productive measures taken by parents nowadays to protect children. Some 42 years years ago, 67% of American children walked or biked to school, but today only 10% do, in part stemming from a handful of highly publicised abduction incidents.

Yet the number of cases of real child abduction by strangers (as opposed to, say, a divorced parent) has dwindled from 200-300 per year in the 1990s to only about 100 per year in the US today.

Even if one assumes all of these children are harmed (which is not true), this is still only about 1/20 the risk of drowning and 1/40 of the risk of a fatal car accident.

Such considerations many not diminish the tragedy of an individual loss, but they do raise questions of priority in prevention. Governments worldwide often agonise over marginal levels of additives in certain products (agar in apples in the 1980s and asbestos insulation in well-protected ceilings), while refusing to spend money or legislate for clear social good (smoking in the developing world, gun control, infectious disease control, needle exchange programs and working conditions in coal mines).

One completely absurd example is the recent surge of opposition in the U.S. (supposedly on health concerns) to “smart meters,” which once an hour send usage statistics to the local electric or natural gas utility.

The microwave exposure for these meters, even if you are standing just two feet from a smart meter when it broadcasts its data, is 550 times less than standing in front of an active microwave oven, up to 4,600 times less than holding a walkie-talkie at your ear, and up to 1,100 times less than holding an active cell phone at your ear.

It is even less than sitting in a WiFi cyber cafe using a laptop computer.

A much more serious example is the ongoing hysteria, especially in the UK and the US, over childhood vaccinations. Back in 1998, a study was published in the British medical journal Lancet claiming that vaccination shots with a certain mercury compound may be linked to autism, but other studies showed no such link.

In the meantime, many jumped on the anti-vaccination bandwagon, and several childhood diseases began to reappear, including measles in England and Wales, and whooping cough in California. We should note the rate of autism is probably increasing.

Finally, in January 2011, Lancet formally acknowledged that the original study was not only bad science (which had been recognised for years), but further an “elaborate fraud”.

Yet nearly one year later, opposition to vaccination remains strong, and irresponsible politicians such as would-be-US-President Michele Bachmann cynically (or ignorantly?) milk it.

A related example is the worldwide reaction to the Fukushima reactor accident. This was truly a horrible incident, and we do not wish to detract from death and environmental devastation that occurred. But we question decisions such as that quickly made by Germany to discontinue and dismantle its nuclear program.

Was this decision made after a sober calculation of relative risk, or simply from populist political pressure? We note this decision inevitably will mean more consumption of fossil fuels, as well as the importation of electricity from France, which is 80% nuclear.

Is this a step forward, or a step backward? We also note that concern about global warming is, if anything, more acute than ever in light of accelerating carbon consumption.

This kind of over-reaction — to which many of us are prey — is exacerbated by cynical and exploitive individuals, such as Bill and Michelle Deagle and Jeff Rense, who profit from such fears by peddling bogus medical products, speaking at conspiracy conventions for hefty fees, and charging for elite information.

This is just one instance of a large, growing and dangerous co-evolution of creationist, climate-denial and other anti-science movements.

How do we protect against such misinformation and misperceptions? The complete answers are complex but several things are clear.

First of all, science education must be augmented to address the assessment of risk — this should be a standard part of high school mathematics, as should be more attention to the information needed to make informed assessment.

Second, the press needs to be significantly more vigilant in critically commenting on dubious claims of public risk by citing literature, consulting real experts, and so on. Ideally, we should anticipate scientifically trained and certified scientific journalists.

Third, mathematicians and scientists themselves need to recognise their responsibility to help the public understand risk. Failure to do so, well, poses a serious risk to society.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jonathan Borwein (Jon)*

Basketball fans everywhere recognize the following scenario: Their favourite player scores a three-point shot. A short time later he regains control of the ball. But does the fact that he scored the last time make him more likely to try another three-pointer? Does it change the probability that he will score again?

New research by Dr. Yonatan Loewenstein and graduate student Tal Neiman at the Hebrew University in Jerusalem shatters the myth that a player who scores one or more three-pointers improves his odds of scoring another.

Dr. Loewenstein is at the Edmond and Lily Safra Center for Brain Sciences and the Department of Neurobiology at the Hebrew University.

Appearing in the latest issue of the journal Nature Communications, the report raises doubts about the ability of athletes in particular, and people in general, to predict future success based on past performance.

Loewenstein and Neiman examined more than 200,000 attempted shots from 291 leading players in the National Basketball Association (NBA) in the 2007-2008 and 2008-2009 regular seasons, and more than 15,000 attempted shots by 41 leading players in the Women’s National Basketball Association (WNBA) during the 2008 and 2009 regular seasons.

The researchers studied how scores or misses affected a player’s behaviour later in the game, and found that after a successful three-pointer, players were significantly more likely to attempt another three-pointer.

In other words, a successful three point shot provided players with positive reinforcement to attempt additional three point shots later in the game.

Surprisingly, the researchers discovered the exact opposite of what players and fans tend to believe: players who scored a three-pointer and then attempted another three-pointer were more likely to miss the follow-up shot.

On the other hand, players who missed a previous three-pointer were more likely to score with their next attempt.

According to Dr. Loewenstein, “The study shows that despite many years of intense training, even the best basketball players over-generalize from their most recent actions and their outcomes. They assume that even one shot is indicative of future performance, while not taking into account that the situation in which they previously scored is likely to be different than the current one.”

The behaviour of basketball players shows the limitations of learning from reinforcement, especially in a complex environment such as a basketball game.

“Learning from reinforcement may not improve performance, and may even damage it, if it is not based on an accurate model of the world,” explains Dr. Loewenstein. “This affects everyone’s behaviour: brokers make investments according to past market performance and commanders make military moves based on the results of past battles. Awareness of the limitations of this kind of learning can help them improve their decision-making processes — as well as those of basketball players.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Hebrew University of Jerusalem

This showas a few of the 2.3 million possible 2-D designs — planar nets — for a truncated octahedron (right column). The question is: Which net is best to make a self-assembling shape at the nanoscale?

Material chemists and engineers would love to figure out how to create self-assembling shells, containers or structures that could be used as tiny drug-carrying containers or to build 3-D sensors and electronic devices.

There have been some successes with simple 3-D shapes such as cubes, but the list of possible starting points that could yield the ideal self-assembly for more complex geometric configurations gets long fast. For example, while there are 11 2-D arrangements for a cube, there are 43,380 for a dodecahedron (12 equal pentagonal faces). Creating a truncated octahedron (14 total faces – six squares and eight hexagons) has 2.3 million possibilities.

“The issue is that one runs into a combinatorial explosion,” said Govind Menon, associate professor of applied mathematics at Brown University. “How do we search efficiently for the best solution within such a large dataset? This is where math can contribute to the problem.”

In a paper published in the Proceedings of National Academy of Sciences, researchers from Brown and Johns Hopkins University determined the best 2-D arrangements, called planar nets, to create self-folding polyhedra with dimensions of a few hundred microns, the size of a small dust particle. The strength of the analysis lies in the combination of theory and experiment. The team at Brown devised algorithms to cut through the myriad possibilities and identify the best planar nets to yield the self-folding 3-D structures. Researchers at Johns Hopkins then confirmed the nets’ design principles with experiments.

“Using a combination of theory and experiments, we uncovered design principles for optimum nets which self-assemble with high yields,” said David Gracias, associate professor in of chemical and biomolecular engineering at Johns Hopkins and a co-corresponding author on the paper. “In doing so, we uncovered striking geometric analogies between natural assembly of proteins and viruses and these polyhedra, which could provide insight into naturally occurring self-assembling processes and is a step toward the development of self-assembly as a viable manufacturing paradigm.”

“This is about creating basic tools in nanotechnology,” said Menon, co-corresponding author on the paper. “It’s important to explore what shapes you can build. The bigger your toolbox, the better off you are.”

While the approach has been used elsewhere to create smaller particles at the nanoscale, the researchers at Brown and Johns Hopkins used larger sizes to better understand the principles that govern self-folding polyhedra.

The researchers sought to figure out how to self-assemble structures that resemble the protein shells viruses use to protect their genetic material. As it turns out, the shells used by many viruses are shaped like dodecahedra (a simplified version of a geodesic dome like the Epcot Center at Disney World). But even a dodecahedron can be cut into 43,380 planar nets. The trick is to find the nets that yield the best self-assembly. Menon, with the help of Brown undergraduate students Margaret Ewing and Andrew “Drew” Kunas, sought to winnow the possibilities. The group built models and developed a computer code to seek out the optimal nets, finding just six that seemed to fit the algorithmic bill.

The students got acquainted with their assignment by playing with a set of children’s toys in various geometric shapes. They progressed quickly into more serious analysis. “We started randomly generating nets, trying to get all of them. It was like going fishing in a lake and trying to count all the species of fish,” said Kunas, whose concentration is in applied mathematics. After tabulating the nets and establishing metrics for the most successful folding maneuvers, “we got lists of nets with the best radius of gyration and vertex connections, discovering which nets would be the best for production for the icosahedron, dodecahedron, and truncated octahedron for the first time.”

Gracias and colleagues at Johns Hopkins, who have been working with self-assembling structures for years, tested the configurations from the Brown researchers. The nets are nickel plates with hinges that have been soldered together in various 2-D arrangements. Using the options presented by the Brown researchers, the Johns Hopkins’s group heated the nets to around 360 degrees Fahrenheit, the point at which surface tension between the solder and the nickel plate causes the hinges to fold upward, rotate and eventually form a polyhedron. “Quite remarkably, just on heating, these planar nets fold up and seal themselves into these complex 3-D geometries with specific fold angles,” Gracias said.

“What’s amazing is we have no control over the sequence of folds, but it still works,” Menon added.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Karolina Grabowska/Pexels,

Elliptic curves have a long and distinguished history that can be traced back to antiquity. They are prevalent in many branches of modern mathematics, foremost of which is number theory.

In simplest terms, one can describe these curves by using a cubic equation of the form

where A and B are fixed rational numbers (to ensure the curve E is nice and smooth everywhere, one also needs to assume that its discriminant 4A³ + 27B² is non-zero).

To illustrate, let’s consider an example: choosing A=-1 and B=0, we obtain the following picture:

At this point it becomes clear that, despite their name, elliptic curves have nothing whatsoever to do with ellipses! The reason for this historical confusion is that these curves have a strong connection to elliptic integrals, which arise when describing the motion of planetary bodies in space.

The ancient Greek mathematician Diophantus is considered by many to be the father of algebra. His major mathematical work was written up in the tome Arithmetica which was essentially a school textbook for geniuses. Within it, he outlined many tools for studying solutions to polynomial equations with several variables, termed Diophantine Equations in his honour.

One of the main problems Diophantus considered was to find all solutions to a particular polynomial equation that lie in the field of rational numbers Q. For equations of “degree two” (circles, ellipses, parabolas, hyperbolas) we now have a complete answer to this problem. This answer is thanks to the late German mathematician Helmut Hasse, and allows one to find all such points, should they exist at all.

Returning to our elliptic curve E, the analogous problem is to find all the rational solutions (x,y) which satisfy the equation defining E. If we call this set of points E(Q), then we are asking if there exists an algorithm that allows us to obtain all points (x,y) belonging to E(Q).

At this juncture we need to introduce a group law on E, which gives an eccentric way of fusing together two points (p₁ and p₂) on the curve, to obtain a brand new point (p₄). This mimics the addition law for numbers we learn from childhood (i.e. the sum or difference of any two numbers is still a number). There’s an illustration of this rule below:

Under this geometric model, the point p₄ is defined to be the sum of p₁ and p₂ (it’s easy to see that the addition law does not depend on the order of the points p₁, p₂). Moreover the set of rational points is preserved by this notion of addition; in other words, the sum of two rational points is again a rational point.

Louis Mordell, who was Sadleirian Professor of Pure Mathematics at Cambridge University from 1945 to 1953, was the first to determine the structure of this group of rational points. In 1922 he proved

where the number of copies of the integers Z above is called the “rank r(E) of the elliptic curve E”. The finite group Τ_E(Q) on the end is uninteresting, as it never has more than 16 elements.

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*Credit for article given to Daniel Delbourgo*

The problem’s been solved … but the sweet treats were declined. Back to the Cutting Board

In 1904, French mathematician Henri Poincaré asked a key question about three-dimensional spaces (“manifolds”).

Imagine a piece of rope, so that firstly a knot is tied in the rope and then the ends are glued together. This is what mathematicians call a knot. A link is a collection of knots that are tangled together.

It has been observed that DNA, which is coiled up within cells, occurs in closed knotted form.

Complex molecules such as polymers are tangled in knotted forms. There are deep connections between knot theory and ideas in mathematical physics. The outsides of a knot or link in space give important examples of three-dimensional spaces.

Torus. Fropuff

Back to Poincaré and his conjecture. He asked if the 3-sphere (which can be formed by either adding a point at infinity to ordinary three-dimensional Euclidean space or by gluing two solid three-dimensional balls together along their boundary 2-spheres) was the only three-dimensional space in which every loop can be continuously shrunk to a point.

Poincaré had introduced important ideas in the structure and classification of surfaces and their higher dimensional analogues (“manifolds”), arising from his work on dynamical systems.

Donuts to go, please

A good way to visualise Poincaré’s conjecture is to examine the boundary of a ball (a two-dimensional sphere) and the boundary of a donut (called a torus). Any loop of string on a 2-sphere can be shrunk to a point while keeping it on the sphere, whereas if a loop goes around the hole in the donut, it cannot be shrunk without leaving the surface of the donut.

Many attempts were made on the Poincaré conjecture, until in 2003 a wonderful solution was announced by a young Russian mathematician, Grigori “Grisha” Perelman.

This is a brief account of the ideas used by Perelman, which built on work of two other outstanding mathematicians, Bill Thurston and Richard Hamilton.

3D spaces

Thurston made enormous strides in our understanding of three-dimensional spaces in the late 1970s. In particular, he realised that essentially all the work that had been done since Poincaré fitted into a single theme.

He observed that known three-dimensional spaces could be divided into pieces in a natural way, so that each piece had a uniform geometry, similar to the flat plane and the round sphere. (To see this geometry on a torus, one must embed it into four-dimensional space!).

Thurston made a bold “geometrisation conjecture” that this should be true for all three-dimensional spaces. He had many brilliant students who further developed his theories, not least by producing powerful computer programs that could test any given space to try to find its geometric structure.

Thurston made spectacular progress on the geometrisation conjecture, which includes the Poincaré conjecture as a special case. The geometrisation conjecture predicts that any three-dimensional space in which every loop shrinks to a point should have a round metric – it would be a 3-sphere and Poincaré’s conjecture would follow.

In 1982, Richard Hamilton published a beautiful paper introducing a new technique in geometric analysis which he called Ricci flow. Hamilton had been looking for analogues of a flow of functions, so that the energy of the function decreases until it reaches a minimum. This type of flow is closely related to the way heat spreads in a material.

Hamilton reasoned that there should be a similar flow for the geometric shape of a space, rather than a function between spaces. He used the Ricci tensor, a key feature of Einstein’s field equations for general relativity, as the driving force for his flow.

He showed that, for three-dimensional spaces where the Ricci curvature is positive, the flow gradually changes the shape until the metric satisfies Thurston’s geometrisation conjecture.

Hamilton attracted many outstanding young mathematicians to work in this area. Ricci flow and other similar flows have become a huge area of research with applications in areas such as moving interfaces, fluid mechanics and computer graphics.

Ricci flow. CBN

He outlined a marvellous program to use Ricci flow to attack Thurston’s geometrisation conjecture. The idea was to keep evolving the shape of a space under Ricci flow.

Hamilton and his collaborators found the space might form a singularity, where a narrow neck became thinner and thinner until the space splits into two smaller spaces.

Hamilton worked hard to try to fully understand this phenomenon and to allow the pieces to keep evolving under Ricci flow until the geometric structure predicted by Thurston could be found.

Perelman

This is when Perelman burst on to the scene. He had produced some brilliant results at a very young age and was a researcher at the famous Steklov Institute in St Petersburg. Perelman got a Miller fellowship to visit UC Berkeley for three years in the early 1990s.

I met him there around 1992. He then “disappeared” from the mathematical scene for nearly ten years and re-emerged to announce that he had completed Hamilton’s Ricci flow program, in a series of papers he posted on the electronic repository called ArXiv.

His papers created enormous excitement and within several months a number of groups had started to work through Perelman’s strategy.

Eventually everyone was convinced that Perelman had indeed succeeded and both the geometrisation and Poincaré conjecture had been solved.

Perelman was awarded both a Fields medal (the mathematical equivalent of a Nobel prize) and also offered a million dollars for solving one of the Millenium prizes from the Clay Institute.

He turned down both these awards, preferring to live a quiet life in St Petersburg. Mathematicians are still finding new ways to use the solution to the geometrisation conjecture, which is one of the outstanding mathematical results of this era.

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*Credit for article given to Hyam Rubinstein*

When the statistician for UC Irvine’s innovative Down syndrome program retired last year, its researchers were left in a bind. The group is studying ways to prevent or delay the onset of Alzheimer’s-type dementia in people with Down syndrome, including examining possible links between seizures and cognitive decline.

“We were mid-study when we found ourselves with no statistician and little budget with which to pay one,” explains program manager Eric Doran.

Statistical analysis for the project was critical and especially difficult. Some of the subjects’ dementia had progressed to the point that they could no longer be tested on performance-based cognitive measures. They couldn’t respond to questions, making it hard for clinicians to evaluate them. But that resulted in missing data. How, then, could the team accurately quantify change over time and see whether seizures might play a role?

Enter Vinh Nguyen, then a doctoral student in statistics at the Donald Bren School of Information & Computer Sciences and now the new head of the UCI Center for Statistical Consulting, which aims to help researchers across campus and Orange County with such challenges. He proposed a model to gauge how quickly people were becoming untestable, instead of how fast they declined. Rather than including test scores – which would have been zero for those who couldn’t be quizzed – Nguyen designed a variable to show when they became unable to respond.

“My part of it was to help them find a way to look at patients with and without seizures, to see if those with seizures might have a shorter time before they became untestable,” he says. “That’s what we found.”Although the findings are preliminary, without his involvement they wouldn’t have been possible. The work resulted in a paper that has been accepted for publication in the Journal of Alzheimer’s Disease. Nguyen, as of October an assistant professor-in-residence of statistics, is a co-author.

“We’re very fortunate to have Vinh’s assistance,” Doran says. “Quite frankly, some of the statistical analysis he’s doing goes well beyond the skill level of even the most seasoned investigators. Vinh was able to pick up where our previous statistician left off, and he was pretty ingenious. His creative look at the data enabled us to complete our analysis.”

Nguyen was glad to help: “I’m excited to be involved in studies that not only advance science but also make a meaningful impact in people’s lives.”

He looks forward to doing more such work through the center, providing state-of-the-art statistical expertise in grant preparation, the design of studies and experiments, and data analysis. The center this spring will offer free statistical consulting for campus researchers via a course taught by Dr. Nguyen. Graduate students in the class will be assigned to projects based on their interests and skills.

“It’s a huge benefit to the university because it’s free, and it’s a huge benefit to the statistics graduate program because it gives our master’s and Ph.D. students a chance to exercise their knowledge and training in real-world applications,” Nguyen says. “Learning how to communicate, how to collaborate with folks outside your field – you can’t just lecture about that. It’s got to be a hands-on experience.”

Colleagues say Nguyen, 26 – whose research interests include survival analysis, robust statistical methods, sequential clinical trials and prediction – was the right choice to run the center.

“It’s a big set of responsibilities for someone so young, but he’s got the ability and maturity level to succeed,” says associate professor of statistics Dan Gillen, who directs statistics research at the Institute for Memory Impairments & Neurological Disorders. It was Gillen who introduced Nguyen, whom he was advising on his doctoral thesis, to the Down syndrome team. “Vinh understands the role of statistics across multiple branches of science, and he’s extremely good at translating a seemingly vague hypothesis into a precise statistical framework.”

A native of Vietnam, Nguyen immigrated to the United States at age 5 and grew up in Garden Grove. A true-blue Anteater, he earned all his degrees at UCI, graduating magna cum laude with a B.S. in mathematics and a B.A. in economics, then obtaining an M.S. and a Ph.D. in statistics. In 2010, he received an Achievement Rewards for College Scientists scholar award, which recognizes UCI’s academically superior doctoral students who exhibit outstanding promise as scientists, researchers and public leaders.

“I feel very fortunate to be here,” Nguyen says. “I’m honoured to be given this opportunity to lead the center and help it grow, and to work in a field and a setting that allow me to apply my knowledge.”

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Credit of the article given to Rizza Barnes, University of California, Irvine

Deciding whether a statement is true is a computational head-scratcher.

In the 1930s, Alan Turing showed there are basic tasks that are impossible to achieve by algorithmic means. In modern lingo, what he showed was that there can be no general computer program that answers yes or no to the question of whether another computer program will eventually stop when it is run.

The amazing unsolvability of this Halting Problem contains a further perplexing subtlety. While we have no way of finding in advance if a program will halt, there is an obvious way, in principle, to demonstrate that it halts if it is a halting program: run it, wait, and witness it halting!

In other words, Turing showed that, at the broadest level, deciding whether a statement is true is computationally harder than demonstrating that it’s true when it is.

A question of efficiency

Turing’s work was a pivotal moment in the history of computing. Some 80 years later, computing devices have pervaded almost every facet of society. Turing’s original “what is computable?” question has been mostly replaced by the more pertinent, “what is efficiently computable?”

But while Turing’s Halting Problem can be proved impossible in a few magical lines, the boundary between “efficient” and “inefficient” seems far more elusive. P versus NP is the most famous of a huge swathe of unresolved questions to have emerged from this modern take on Turing’s question.

So what is this NP thing?

Roughly speaking, P (standing for “polynomial time”), corresponds to the collection of computational problems that have an efficient solution. It’s only an abstract formulation of “efficient”, but it works fairly well in practice.

The class NP corresponds to the problems for which, when the answer is “yes”, there is an efficient demonstration that the answer is yes (the “N” stands for “nondeterministic”, but the description taken here is more intuitive). P versus NP simply asks if these two classes of computational problems are the same.

It’s just the “deciding versus demonstrating” issue in Turing’s original Halting Problem, but with the added condition of efficiency.

A puzzler

P certainly doesn’t look to be the same as NP. Puzzles are good examples of the general intuition here. Crossword puzzles are popular because it’s a challenge to find the solution, and humans like challenge. But no-one spends their lunchtime checking already completed crosswords: checking someone else’s solution offers nowhere near the same challenge.

Even clearer is Sudoku: again it is a genuine challenge to solve, but checking an existing solution for correctness is so routine it is devoid of entertainment value.

The P=NP possibility is like discovering that the “finding” part of these puzzles is only of the same difficulty to the “checking” part. That seems hard to believe, but the truth is we do not know for sure.

This same intuition pervades an enormous array of important computational tasks for which we don’t currently have efficient algorithms. One particularly tantalising feature is that, more often than not, these problems can be shown to be maximally hard among NP problems.

These so-called “NP-complete” problems are test cases for P versus NP: if any one of them has an efficient algorithmic solution then they all do (and efficient checking is no harder than efficient finding).

But if even just one single one can be shown to have no efficient solution, then P does not equal NP (and efficient finding really is, in general, harder than efficient checking).

Here are some classic examples of NP-complete problems.

• Partition (the dilemma of the alien pick-pockets). On an alien planet, two pick-pockets steal a wallet. To share the proceeds, they must evenly divide the money: can they do it? Standard Earth currencies evolved to have coin values designed to make this task easy, but in general this task is NP-complete. It’s in NP because, if there is an equal division of the coins, this can be easily demonstrated by simply showing the division. (Finding it is the hard part!)
• Timetabling. Finding if a clash-free timetable exists is NP-complete. The problem is in NP because we can efficiently check a correct, clash-free timetable to be clash-free.
• Travelling Salesman. A travelling salesman must visit each of some number of cities. To save costs, the salesman wants to find the shortest route that passes through all of the cities. For some given target distance “n”, is there a route of length at most “n”?
• Short proofs. Is there a short proof for your favourite mathematical statement (a Millennium Prize problem perhaps)? With a suitable formulation of “short”, this is NP-complete. It is in NP because checking formal proofs can be done efficiently: the hard part is finding them (at least, we think that’s the hard part!).

In every case, we know of no efficient exact algorithm, and the nonexistence of such an algorithm is equivalent to proving P not equal to NP.

So are we close to a solution? It seems the best we know is that we don’t know much! Arguably, the most substantial advances in the P versus NP saga are curiously negative: they mostly show we cannot possibly hope to resolve P as different to NP by familiar techniques.

We know Turing’s approach cannot work. In 2007, Alexander Razborov and Steven Rudich were awarded the Gödel Prize (often touted as the Nobel Prize of Computer Science) for their work showing that no “natural proof” can prove P unequal to NP.

Of course, we’ll keep looking!

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*Credit for article given to Marcel Jackson*

If one grossly divides mathematics into two parts they would be: tools for measuring and tools for recognition.

To use an analogy, tools for measuring are the technologies for collecting data about an object, the process of “taking a blurry photograph”. Tools for recognition deal with the following: if you are given a pile of data or a blurry photograph, how can the object that it came from be recognised from the data?

The Hodge Conjecture – a major unsolved problem in algebraic geometry – deals with recognition.

William Vallance Douglas Hodge was a professor at Cambridge who, in the 1940s, worked on developing a refined version of cohomology – tools for measuring flow and flux across boundaries of surfaces (for example, fluid flow across membranes).

The classical versions of cohomology are used for the understanding of the flow and dispersion of electricity and magnetism (for example, Maxwell’s equations, which describe how electric charges and currents act as origins for electric and magnetic fields). These were refined by Hodge in what is now called the “Hodge decomposition of cohomology”.

Hodge recognised that the actual measurements of flow across regions always contribute to a particular part of the Hodge decomposition, known as the (p,p) part. He conjectured that any time the data displays a contribution to the (p,p) part of the Hodge decomposition, the measurements could have come from a realistic scenario of a system of flux and change across a region.

Or, to put this as an analogy, one could say Hodge found a criterion to test for fraudulent data.

If Hodge’s test comes back positive, you can be sure the data is fraudulent. The question in the Hodge conjecture is whether there is any fraudulent data which Hodge’s test will not detect. So far, Hodge’s test seems to work.

But we haven’t understood well enough why it works, and so the possibility is open that there could be a way to circumvent Hodge’s security scheme.

Hodge made his conjecture in 1950, and many of the leaders in the development of geometry have worked on this basic recognition problem. The problem itself has stimulated many other refined techniques for measuring flow, flux and dispersion.

Tate’s 1963 conjecture is another similar recognition question coming out of another measurement technique, the l-adic cohomology developed by Alexander Grothendieck.

The strongest evidence in favour of the Hodge conjecture is a 1995 result of Cattani, Deligne & Kaplan which studies how the Hodge decomposition behaves as a region mutates.

Classical cohomology measurements are not affected by small mutations, but the Hodge decomposition does register mutations. The study of the Hodge decomposition across mutations provides great insight into the patterns in data that must occur in true measurements.

In the 1960s, Grothendieck initiated a powerful theory generalising the usual concept of “region” to include “virtual regions” (the theory of motives on which one could measure “virtual temperatures” and “virtual magnetic fields”.

In a vague sense, the theory of motives is trying to attack the problem by trying to think like a hacker. The “Standard Conjectures” of Grothendieck are far-reaching generalisations of the Hodge conjecture, which try to explain which virtual regions are indistinguishable from realistic scenarios.

The question in the Hodge conjecture has stimulated the development of revolutionary tools and techniques for measurement and analysis of data across regions. These tools have been, and continue to be, fundamental for modern development.

Imagine trying to building a mobile phone without an understanding of how to measure, analyse and control electricity and magnetism. Alternatively, imagine trying to sustain an environment without a way to measure, analyse and detect the spread of toxins across regions and in waterways.

Of course, the tantalising intrigue around recognition and detection problems makes them thrilling. Great minds are drawn in and produce great advances in an effort to understand what makes it all work.

One might, very reasonably, claim that the longer the Hodge conjecture remains an unsolved problem the more good it will do for humanity, driving more and more refined techniques for measurement and analysis and stimulating the development of better and better methods for recognition of objects from the data.

The Clay Mathematics Institute was wise in pinpointing the Hodge conjecture as a problem that has the capacity to stimulate extensive development of new methods and technologies and including it as one of the Millennium problems.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Arun Ram*

Among the seven problems in mathematics put forward by the Clay Mathematics Institute in 2000 is one that relates in a fundamental way to our understanding of the physical world we live in.

It’s the Navier-Stokes existence and uniqueness problem, based on equations written down in the 19th century.

The solution of this prize problem would have a profound impact on our understanding of the behaviour of fluids which, of course, are ubiquitous in nature. Air and water are the most recognisable fluids; how they move and behave has fascinated scientists and mathematicians since the birth of science.

But what are the so-called Navier-Stokes equations? What do they describe?

The equations

In order to understand the Navier-Stokes equations and their derivation we need considerable mathematical training and also a sound understanding of basic physics.

Without that, we must draw upon some very simple basics and talk in terms of broad generalities – but that should be sufficient to give the reader a sense of how we arrive at these fundamental equations, and the importance of the questions.

From this point, I’ll refer to the Navier-Stokes equations as “the equations”.

The equations governing the motion of a fluid are most simply described as a statement of Newton’s Second Law of Motion as it applies to the movement of a mass of fluid (whether that be air, water or a more exotic fluid). Newton’s second law states that:

Mass x Acceleration = Force acting on a body

For a fluid the “mass” is the mass of the fluid body; the “acceleration” is the acceleration of a particular fluid particle; the “forces acting on the body” are the total forces acting on our fluid.

Without going into full details, it’s possible to state here that Newton’s Second Law produces a system of differential equations relating rates of change of fluid velocity to the forces acting on the fluid. We require one other physical constraint to be applied on our fluid, which can be most simply stated as:

Mass is conserved! – i.e. fluid neither appears nor disappears from our system.

The solution

Having a sense of what the Navier-Stokes equations are allows us to discuss why the Millennium Prize solution is so important. The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equations. The second focuses on whether these solutions are bounded (remain finite).

It’s not possible to give a precise mathematical description of these two components so I’ll try to place the two parts of the problem in a physical context.

1) For a mathematical model, however complicated, to represent the physical world we are trying to understand, the model must first have solutions.

At first glance, this seems a slightly strange statement – why study equations if we are not sure they have solutions? In practice we know many solutions that provide excellent agreement with many physically relevant and important fluid flows.

But these solutions are approximations to the solutions of the full Navier-Stokes equations (the approximation comes about because there is, usually, no simple mathematical formulae available – we must resort to solving the equations on a computer using numerical approximations).

Although we are very confident that our (approximate) solutions are correct, a formal mathematical proof of the existence of solutions is lacking. That provides the first part of the Millennium Prize challenge.

2) The second part asks whether the solutions of the Navier-Stokes equations can become singular (or grow without limit).

Again, a lot of mathematics is required to explain this. But we can examine why this is an important question.

There is an old saying that “nature abhors a vacuum”. This has a modern parallel in the assertion by physicist Stephen Hawking, while referring to black holes, that “nature abhors a naked singularity”. Singularity, in this case, refers to the point at which the gravitational forces – pulling objects towards a black hole – appear (according to our current theories) to become infinite.

In the context of the Navier-Stokes equations, and our belief that they describe the movement of fluids under a wide range of conditions, a singularity would indicate we might have missed some important, as yet unknown, physics. Why? Because mathematics doesn’t deal in infinites.

The history of fluid mechanics is peppered with solutions of simplified versions of the Navier-Stokes equations that yield singular solutions. In such cases, the singular solutions have often hinted at some new physics previously not considered in the simplified models.

Identifying this new physics has allowed researchers to further refine their mathematical models and so improve the agreement between model and reality.

If, as many believe, the Navier-Stokes equations do posses singular solutions then perhaps the next Millennium Prize will go to the person that discovers just what new physics is required to remove the singularity.

Then nature can, as all fluid mechanists already do, come to delight in the equations handed down to us by Claude-Louis Navier and George Gabriel Stokes.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Jim Denier, University of Adelaide

How fluids move has fascinated researchers since the birth of science.

Among the seven problems in mathematics put forward by the Clay Mathematics Institute in 2000 is one that relates in a fundamental way to our understanding of the physical world we live in.

It’s the Navier-Stokes existence and uniqueness problem, based on equations written down in the 19th century.

The solution of this prize problem would have a profound impact on our understanding of the behaviour of fluids which, of course, are ubiquitous in nature. Air and water are the most recognisable fluids; how they move and behave has fascinated scientists and mathematicians since the birth of science.

But what are the so-called Navier-Stokes equations? What do they describe?

The equations

In order to understand the Navier-Stokes equations and their derivation we need considerable mathematical training and also a sound understanding of basic physics.

Without that, we must draw upon some very simple basics and talk in terms of broad generalities – but that should be sufficient to give the reader a sense of how we arrive at these fundamental equations, and the importance of the questions.

From this point, I’ll refer to the Navier-Stokes equations as “the equations”.

The equations governing the motion of a fluid are most simply described as a statement of Newton’s Second Law of Motion as it applies to the movement of a mass of fluid (whether that be air, water or a more exotic fluid). Newton’s second law states that:

Mass x Acceleration = Force acting on a body

For a fluid the “mass” is the mass of the fluid body; the “acceleration” is the acceleration of a particular fluid particle; the “forces acting on the body” are the total forces acting on our fluid.

Without going into full details, it’s possible to state here that Newton’s Second Law produces a system of differential equations relating rates of change of fluid velocity to the forces acting on the fluid. We require one other physical constraint to be applied on our fluid, which can be most simply stated as:

Mass is conserved! – i.e. fluid neither appears nor disappears from our system.

The solution

Having a sense of what the Navier-Stokes equations are allows us to discuss why the Millennium Prize solution is so important. The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equations. The second focuses on whether these solutions are bounded (remain finite).

It’s not possible to give a precise mathematical description of these two components so I’ll try to place the two parts of the problem in a physical context.

1) For a mathematical model, however complicated, to represent the physical world we are trying to understand, the model must first have solutions.

At first glance, this seems a slightly strange statement – why study equations if we are not sure they have solutions? In practice we know many solutions that provide excellent agreement with many physically relevant and important fluid flows.

But these solutions are approximations to the solutions of the full Navier-Stokes equations (the approximation comes about because there is, usually, no simple mathematical formulae available – we must resort to solving the equations on a computer using numerical approximations).

Although we are very confident that our (approximate) solutions are correct, a formal mathematical proof of the existence of solutions is lacking. That provides the first part of the Millennium Prize challenge.

2) The second part asks whether the solutions of the Navier-Stokes equations can become singular (or grow without limit).

Again, a lot of mathematics is required to explain this. But we can examine why this is an important question.

There is an old saying that “nature abhors a vacuum”. This has a modern parallel in the assertion by physicist Stephen Hawking, while referring to black holes, that “nature abhors a naked singularity”. Singularity, in this case, refers to the point at which the gravitational forces – pulling objects towards a black hole – appear (according to our current theories) to become infinite.

In the context of the Navier-Stokes equations, and our belief that they describe the movement of fluids under a wide range of conditions, a singularity would indicate we might have missed some important, as yet unknown, physics. Why? Because mathematics doesn’t deal in infinites.

The history of fluid mechanics is peppered with solutions of simplified versions of the Navier-Stokes equations that yield singular solutions. In such cases, the singular solutions have often hinted at some new physics previously not considered in the simplified models.

Identifying this new physics has allowed researchers to further refine their mathematical models and so improve the agreement between model and reality.

If, as many believe, the Navier-Stokes equations do posses singular solutions then perhaps the next Millennium Prize will go to the person that discovers just what new physics is required to remove the singularity.

Then nature can, as all fluid mechanists already do, come to delight in the equations handed down to us by Claude-Louis Navier and George Gabriel Stokes.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jim Denier*