The link between mathematics and kirigami

The British-Lebanese mathematician Sir Michael Atiyah spoke at the Heidelberg Laureate Forum on 24^th September.  In a 45 minute talk he claimed to have found a “simple proof” to the Riemann hypothesis, a problem that has remained unsolved since 1859.  Correct proof to support the hypothesis, labelled by the Clay Mathematics Institute as one of the seven “Millenium Prize Problems”, could have huge implications for the majority of modern day cryptography, including cryptocurrencies like Bitcoin.

The hypothesis concerns prime numbers and the ability to find the number of primes smaller than any given integer, N.  The hypothesis relies on the Riemann zeta function and its return of zero.  It is known that zero is returned when a negative integer is used in the function, these are known as trivial zeros.  This is also the case with any complex number whose real part is ½, known as non-trivial zeros.  The non-trivial zeros have varying imaginary units but consistent ½ values, allowing for their calculation.

However, the continuing problem is the lack of proof that complex numbers with a ½ real value are the only form of non-trivial zeros.  Until now this has been assumed to be true and has provided the basis for modern cryptography.  This is due to the property of prime numbers where calculating the product of two primes is simple but finding the two primes used when the only information given is the result is very difficult.  This allows for one-way functions that cannot be easily inversed by those that are not the intended recipient.

But proof of the hypothesis may lead to a connection being observed between prime numbers and this could be exploited to counteract contemporary forms of coding.  This could lead to hash algorithms being easily hacked and bitcoin being mined at an exponentially faster rate. 

Atiyah’s claim of proof is currently met with scepticism but if correct his work could have an enormous impact.  Computer Science and Mathematics applicants can develop their understanding of the Riemann hypothesis and its underpinning of cryptography and its other applications.

How many primes can you name? In March this year mathematician Robert Langlands won the Abel Prize for research showing how concepts from different branches of mathematics all share links to prime numbers. To analyse these numbers mathematicians have to sift through numbers using mathematical filters, eliminating all non-primes. This search has its origins in antiquity; Euclid wrote in 300 BC that “a prime number is that which is measured by the unit alone”, and it was he who proved that the number of primes is infinite. However, it was probably Eratosthenes who first developed the sifting process, which filters out all multiples of 2, 3, 5, and 7—the first 4 primes.

A notable figure in the early history of the study of primes is John Pell, whose urge to categorise and collect useful numbers led him to identify and publish the primes up to 100,000 in the early 1700s. A century later, others had found the primes up to 1 million. As more and more primes were found, the process was made easier by the invention of sliders and stencils to quickly eliminate multiples. However, it was Carl Friedrich Gauss who decided to actually analyse prime numbers, looking for interesting patterns. He found, for example, that the higher he counted, the fewer prime numbers there were. More recently it has been found that, with the exception of 2 and 5, all prime numbers end in 1,3,7, or 9.

Langlands’ research, which has been described as “revolutionary”, is founded on the work of previous mathematicians, in particular Gauss. In the late 18^th century he formulated a law of reciprocity whereby certain types of primes share defining characteristics; for example, primes that are the sum of two squares also leave a remainder of 1 when divided by 4. Langlands built on this by proposing that prime numbers encoded in higher-degree equations than simply squares might be in a reciprocal relationship with the branch of mathematics known as harmonic analysis, which is often used in physics.

Applicants for Mathematics may wish to read Langland’s research and look into the contemporary questions in the study of prime numbers. Students wishing to study Physics could familiarise themselves with harmonic analysis and learn about how prime numbers are relevant to physics.

It has long been understood that dolphins are intelligent, social creatures, and that they have their own distinct language that humans can’t understand. Although we can distinguish the different sounds that dolphins make, and understand that each dolphin has a unique call, scientists face struggles when trying to study their communication, due to the difficulty in tracking which dolphin is making which sound and why. However, recently, psychologist and marine mammal scientist Diana Reiss and a group of biophysicists have built a ‘dolphin touchscreen’ in the form of a window into the wall of a pool at the National Aquarium in Baltimore. The researchers project interactive progammes onto it, and optical sensing technology can detect when the window is being touched by the dolphins. The project was inspired by an experiment Reiss conducted in the 1980s with an electronic keyboard with unique symbols on each key. Each key made a dolphinesque whistle when touched, with the idea that dolphins could use the keyboard to make requests of their handlers. When listening to recordings, Reiss noticed that the dolphins were mimicking the sounds made by the keyboard and combining with their own unique sounds.

One of the programmes the team have developed is a dolphin version of ‘whack-a-mole’. In the game, fish swim across the scream and disappear when touched. Within seconds of the screen turning on, the scientists witnessed the dolphin approaching the screen and touching the fish with his melon, or forehead. Motivated by this success and with the 1980s experiment in mind, the team are now developing an app similar to the keyboard. Alongside this the team will use microphones embedded in the walls to record the sounds, and multiple cameras to track the locations of the dolphins. The combination of audio and visual data the team will be able to trace the sounds back to a particular point in the pool and thus a specific dolphin. Data-mining algorithms will then be used to look for patterns in this information. 

Psychology, Biology and Veterinary students should explore our understanding of animal intelligence and consciousness and how technology is allowing us greater insight into this. Physics students can consider how such technology may help us communicate with potential extre-terrestrial life forms as we continue to pursue space exploration. Computer Scientists and Mathematicians can investigate the nature of the programmes and technology used to pursue this research.

The AlphaGo artificial intelligence program has defeated Ke Jie, the human champion of the game Go, in a series of three matches of designed to test its intelligence.

Developed by Alphabet Inc.’s Google’s DeepMind unit, AlphaGo is a computer programme designed to play the ancient Chinese board game. Go is one of the oldest and most complex in the world and involves placing either black or white stones to form territories on the board.

The co-founder and co-CEO of Deep Mind, Demis Hassabis has announced that AlphaGo’s recent triple victory is ‘the highest possible pinnacle’ that the competitive program could have possibly reached and therefore the program will now be retired. According to Mr Hassabis, the research team behind the A.I. program will now go on to use their algorithmical learnings on more complex projects, such as curing diseases, creating new types of materials and solving energy problems.

Many Go competitors are disappointed at this news that AlphaGo will no longer be playing games as they are eager to attempt to beat the machine. The data from the 50 online games that AlphaGo has played, however, will be shared with the Go community, so they can develop their own gameplay. After losing his final match to the computer, Ke Jie proclaimed that the ‘future belongs to A.I.’

Maths students should look at the algorithms used by the AlphaGo and Computer Science students would be wise to investigate zero-player games. Those going to study other logic based subjects should investigate the reasoning pattern employed by the game.

The US have slipped as a progressive nation for women – moving from 23^rd place to 45^th place this year in the WEF’s Global Gender Gap Report.

The report, established in 2006, compares the national, annual average income for men and women as one measure of equality. The US’s slip perhaps can be down to a chance in how the WEF recorded income; prior to this year, they measured incomes up to and including $40,000 as they believed that income above $40,000 doesn’t have a meaningful impact on someone’s quality of life. They now believe, however, that this threshold should be $75,000. Mathematics and Statistics students should consider how changes in the collection of data allow us to make meaningful year-by-year comparisons.

The fall of the US isn’t solely down to the change in measurement, however; women’s participation in the labour force has decrease, and “is stagnating among legislators, senior officials and managers.” HSPS and PPE applicants should consider how useful income is as a measure of gender equality.

Reaching the top of the table were the Nordic countries of Iceland, Finland, Norway and Sweden, which plays into the general conception of these countries as liberal and progressive. Geography students should consider perhaps the more surprising fifth place entry, Rwanda, and why they might have pay parity.

After performance enhancing drugs were almost eradicated from baseball in the early 2000s, far fewer people were hitting home runs – but this trend has reversed.

Research from Penn State University looked in to the trend of baseball players skewing increasingly heavier than in previous decades. Using self-reported heights and weights, researchers found that 70% of players between 1991 and 2015 had BMIs that classified them as overweight or obese, but before this, the average was between 30 and 40%.

Biological Sciences and Medicine applicants should note the flaw with using BMI as a measure of obesity, as by itself, you cannot determine the distribution of muscle versus fat. Nevertheless, it does make logical sense that a heavier weight behind a hit will lead to the ball being propelled further.

Physics students will be familiar with the formula k=(1/2)mv²  where m is the mass of the system and v is the velocity. In theory, a heavier batter will therefore hit the ball further than a smaller man with the same strength. However, there are many variables to consider including momentum and power which Natural Sciences (P) and Mathematics applicants would do well to investigate further.