Mathematics: Current Research
The School of Mathematics at Bristol is one of the strongest and most productive centres for mathematical research and teaching in the UK. The School’s research portfolio covers a wide range of areas in pure and applied mathematics and statistics with a strong tradition of interdisciplinary work on important problems.
1 Chaos Theory Explains Origin of New Moons (Professor S Wiggins)
The ability to understand how small bodies such as moons switch from orbiting the Sun to orbiting a planet has long remained one of the outstanding problems of planetary science. A paper published in Nature in 2003 shows how this problem has been resolved using chaos theory, enabling scientists to predict where astronomers might search for new moons orbiting the giant planets.
In the last couple of years many small moons have been found orbiting the giant planets in our Solar System. For example, Jupiter now has 60 moons in total and Saturn more than 30. Astronomers believe that understanding the nature of these moons can reveal important clues about the early history of the planets. Such insights into understanding our own Solar System will help us understand how other solar systems came into being, and whether they might be favourable to life.
The moons can be divided into two groups - regular and irregular. Regular moons have a roughly circular orbit around their planet and are believed to have been formed there during the early history of the Solar System. Irregular moons have an orbit that is highly elliptical, orbiting the planet at a distance of many millions of miles. These are believed to have originally encircled the Sun and to have been subsequently “captured” by the planet they now orbit. The discovery of these new moons has shaken cherished ways of understanding the Solar System. In particular, the outstanding problem of satellite capture - the mechanism by which bodies switch from an orbit around the Sun to an orbit around the planet. Secondary to this was the problem of why some moons have prograde orbits - revolving in the same direction as the planet - while the vast majority have retrograde orbits.
Professor Stephen Wiggins and Dr Andrew Burbanks, along with Professor David Farrelly and Sergey Astakhov, theoretical chemists at Utah State University, were using chaos theory to understand the mechanics of chemical reactions. They realised that the approach they had been using in chemistry might also be applied to the problem of “capture”. Furthermore, they thought that if they could solve the capture problem it might give them some insight into their chemistry problems.
Using the mathematical equations they developed to explain the capture mechanism, the Bristol and Utah research groups present an explanation which not only agrees well with the observed locations of the known irregular moons, but also predicts new regions where moons could be located. The ability to predict where new moons might be found should make life much easier for astronomers who face the daunting task of searching huge regions of space for them.
The joint UK/US research team also showed that the moons initially captured into prograde orbits of moons are not only chaotic, but that they have a tendency to approach the region very close to the planet. This means that they have a greater chance of being eliminated by collisions with the inner giant moons or the planet, thereby explaining the far larger number of retrograde moons, especially around Jupiter. This work shows that chaos-assisted capture may be a necessary, and quite general, predecessor of certain types of orderly and stable satellite orbits.
2 Evolutionary Game Theory (Professor J McNamara)
Evolutionary game theory models often ignore differences between individuals. Professor John McNamara has shown that such differences are not always innocuous noise, but can fundamentally change the nature of a game. Differences promote the need to have extensive interactions to find out about others, so changing the game strategy set and hence the game outcome. These ideas apply to a whole range of phenomena considered by behavioural ecologists and render their modelling predictions unrealistic at best. The direction of evolution in cooperation/competitive situations can depend on the variation in the cooperativeness trait. Differences promote choosiness, and the co-evolution of choosiness and cooperativeness can lead to high levels of cooperation when repeated interactions with the same partner are possible. Finally, differences in personality promote social sensitivity; and once individuals are socially sensitive, this changes the selection pressure on those observed, leading to more cooperative behaviour.
3 How to know less than nothing (Professor A Winter)
Even the most ignorant cannot know less than nothing. After all, negative knowledge makes no sense. But, although this may be true in the everyday world, it has been discovered that negative knowledge does exist in the quantum world of very small things. This discovery, that quantum knowledge can be negative, was made by Professor Andreas Winter and his colleagues from Gdansk and Cambridge, and was published in Nature in 2005.
What could negative knowledge possibly mean? It is as if, after someone imparts information, the recipient knows less than before the conversation. Such strange situations can occur because what it means to know something is very different in the quantum world. Negative knowledge (or more precisely - “negative information”) turns out to be precisely the right amount to cancel the fact that we know too much.
In the quantum world, there are things that just cannot be known: for example, both the exact position and momentum of a small particle. There are also situations where someone knows more than everything. This is known as quantum “entanglement”, and when two people share entanglement, there can be negative information.
While all this might appear to be very mysterious, the idea of negative information can be put on a rigorous scientific footing. Information can be quantified in terms of how much stuff needs to sent before someone gets to know something. In the case of negative quantum information, someone can get to know something without sending any quantum particles. In fact, they will gain the potential to learn more quantum information in the future.
Negative information is due to exotic features of quantum information theory, an exciting new area of physics which includes such phenomena as quantum teleportation and quantum computation. While classical information theory deals with subjects such as classical communication and computation, quantum information replaces classical “bits” by quantum “qubits”, quantum particles like electrons or atoms. While classical bits can only be in the state 0 or 1, qubits can be both in the 0 or 1 state at the same time. By understanding that quantum information can be negative, researchers hope to gain deeper insights into phenomena such as quantum teleportation and computation, as well as the very structure of the quantum world.