News

Saturday, 26 March 2016

D is for Diophantine Equations - the latest in the Oxford Mathematics Alphabet

diophantine equation is an algebraic equation, or system of equations, in several unknowns and with integer (or rational) coefficients, which one seeks to solve in integers (or rational numbers). The study of such equations goes back to antiquity. Their name derives from the mathematician Diophantus of Alexandria, who wrote a treatise on the subject, entitled Arithmetica.

The most famous example of a diophantine equation appears in Fermat’s Last Theorem. This is the statement, asserted by Fermat in 1637 without proof, that the diophantine equation has no solutions in whole numbers when n is at least 3, other than the 'trivial solutions' which arise when XYZ = 0. The study of this equation stimulated many developments in number theory. A proof of the theorem was finally given by Andrew Wiles in 1995.

The basic question one would like to answer is: does a given system of equations have solutions? And if it does have solutions, how can we find or describe them? While the Fermat equation has no (non-trivial) solutions, similar equations (for example ) do have non-trivial solutions. One of the problems on Hilbert’s famous list from 1900 was to give an algorithm to decide whether a given system of diophantine equations has a solution in whole numbers. In effect this is asking whether the solvability can be checked by a computer programme. Work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson, culminating in 1970, showed that there is no such algorithm. It is still unknown whether the corresponding problem for rational solutions is decidable, even for plane cubic curves. This last problem is connected with one of the Millennium Problems of the Clay Mathematics Institute (with a million dollar prize): the Birch Swinnerton Dyer Conjecture. 

To find out more about diophantine problems read Professor Jonathan Pila's latest addition to our Oxford Mathematics Alphabet.

Tuesday, 15 March 2016

Andrew Wiles awarded the Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to Sir Andrew J. Wiles (62), University of Oxford, “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”

The President of the Norwegian Academy of Science and Letters, Ole M. Sejersted, announced the winner of the 2016 Abel Prize at the Academy in Oslo today, 15 March. Andrew J. Wiles will receive the Abel Prize from H.R.H. Crown Prince Haakon at an award ceremony in Oslo on 24 May.

The Abel Prize recognizes contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of NOK 6,000,000 (about EUR 600,000 or USD 700,000).

Andrew J. Wiles is one of very few mathematicians – if not the only one – whose proof of a theorem has made international headline news. In 1994 he cracked Fermat’s Last Theorem, which at the time was the most famous, and long-running, unsolved problem in the subject’s history.

Wiles’ proof was not only the high point of his career – and an epochal moment for mathematics – but also the culmination of a remarkable personal journey that began three decades earlier. In 1963, when he was a ten-year-old boy growing up in Cambridge, England, Wiles found a copy of a book on Fermat’s Last Theorem in his local library. Wiles recalls that he was intrigued by the problem that he as a young boy could understand, and yet it had remained unsolved for three hundred years. “I knew from that moment that I would never let it go,” he said. “I had to solve it.”

The Abel Committee says: “Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem.”

Wednesday, 9 March 2016

Comparing the social structure of different cities

People make a city. Each city is as unique as the combination of its inhabitants. Currently, cities are generally categorised by size, but research by Oxford Mathematicians Peter Grindrod and Tamsin Lee on the social networks of different cities shows that City A, which is twice the size of City B, may not necessarily be accurately represented as an amalgamation of two City Bs.

The researchers use Twitter data from ten different UK cities, showing reciprocal tweets within each city. By defining cities in terms of these social network structures, they break each city into its comprising modular communities. Next, they build virtual cities from the actual cities. For example, Bristol has 74 communities. Randomly sampling (with replacement) from these communities 145 times builds a virtual city the same size as Manchester - but made up of modular communities actually observed in Bristol. How much does our virtual Manchester network resemble the true Manchester network? The answer is very closely. So if one was trying to spread a message via Twitter through Manchester, or make other social interventions, it may prove beneficial to test the same activity in Bristol first.

However, sampling the Bristol communities to create a virtual city the same size as Leeds, which is smaller than Manchester, does not create a network of similar structure to the 'real' Leeds. This highlights that the relationship between social structures of cities is not immediately obvious, and requires further analysis. Furthermore, this relationship is not symmetrical: a virtual city created by randomly sampling 74 communities from the Leeds network, does in fact resemble the true Bristol social network. So Bristol could learn from Leeds but not vice versa.

In summary, we may sometimes replicate one city using the communities from another. However, some cities have a very diverse range of communities, making them difficult to replicate - Leeds is a good example of this. Perhaps cities can be put into classes where those cities in the same class are socially similar and so any experience of social phenomena or reactions to interventions in one such city may be relevant to another.

Tuesday, 8 March 2016

The Mirzakhani Society - Oxford Mathematics' society for female undergraduates

27% of mathematics undergraduates in Oxford are female. We would like the figure to be higher and we are putting a lot of resource in to making it so. However, it is also important that current female and non-binary Oxford mathematicians feel they have time and space to discuss and share experiences that may be specific to them.

The Mirzakhani Society is the society for women studying maths at Oxford, named after Maryam Mirzakhanithe first woman to win a Fields Medal (Maryam met the society in September 2015 on her visit to Oxford to collect her Clay Mathematics Institute Research Award).  With over 100 active members, it holds relaxed weekly ‘Sip and Solve’ meetings (aided by high-quality baking), and socials and talks. In a University where your immediate and regular contact is often limited to other members of your college, it is an invaluable way of broadening contacts and providing a support network. The society is open to both undergraduates and postgraduates, and is central in encouraging more women to take a fourth year (undergraduates currently can choose between the three and four year mathematics courses). Find out more about the society on their Facebook page

On Saturday 27 February 2016, the society (pictured) met up with their Cambridge University counterparts, the Emmy Noether Society, sharing experience of gender equality in the universities. Three speakers gave their perspectives: Anne Davis, a Professor of Mathematical Physics and the University Gender Equality Champion for STEMM subjects at Cambridge: Perla Sousi, a Lecturer in the Statistics Laboratory at Cambridge; and Christie Marr, Deputy Director of the Isaac Newton Institute. Thanks to the London Mathematical Society for funding the trip.

Tuesday, 23 February 2016

C is for Calabi-Yau manifolds - the latest in the Oxford Mathematics Alphabet

Model II 1a

Calabi-Yau manifolds have become a topic of study in both mathematics and physics, dissolving the boundaries between the two subjects.

A manifold is a type of geometrical space where each small region looks like normal Euclidean space. For example, an ant on the surface of the Earth sees its world as flat, rather than the curved surface of the sphere. Calabi-Yau manifolds are complex manifolds, that is, they can be disassembled into patches which look like flat complex space. What makes them so special is that these patches can only be joined together by the complex analogue of a rotation.

Proving a conjecture of Eugenio Calabi, Shing-Tung Yau has shown that Calabi-Yau manifolds have a property which is very interesting to physics. Einstein's equations show that spacetime curves according to the distribution of energy and momentum. But what if space is all empty? By Yau's theorem, not only is flat space a solution but so are Calabi-Yau manifolds. Furthermore, for this reason, Calabi-Yau spaces are possible candidates for the shape of extra spatial dimensions in String Theory.

Find out more from Oxford Mathematician Dr Andreas Braun in this latest instalment of our Oxford Mathematics Alphabet.

 

Monday, 22 February 2016

Well behaved cities - what all cities have in common

How are people, infrastructure and economic activity organised and interrelated? It is an intractable problem with ever-changing infinite factors of history, geography, economy and culture playing their part. But a paper by Oxford Mathematician Hyejin Youn and colleagues suggests “a mathematical function common to all cities.”

Think of the city as an ecosystem, types of businesses as species interacting in that system. Ecosystems in the natural world often share common patterns in distributions of species. That got the researchers thinking. Maybe the same consistency arises in the city too. Only instead of the food web, it’s people and money and businesses that require one another. We usually think of cities as unique. London is very different from Moscow. But, it turns out, what governs the distribution of their resources stays the same across the board.

The team analysed more than 32 million establishments in U.S. metro regions. An establishment, the unit of analysis of their study, indicates “a single physical location where business is conducted”. When the team measured relative sizes of business types (e.g. agriculture, finance, and manufacturing) in each and every city, and compared these distributions among cities, the universal law is found: despite widely different mixes of types of businesses and across different-sized cities, the shape of these distributions was completely universal. Cities have their own underlying dynamics. It doesn’t matter where they are, how old they are and who is in charge.

This underlying pattern allowed researchers to build a stochastic model. As cities grow, the total number of establishments is linearly proportional to its population size (more people, more businesses). When an establishment is created it differentiates from any existing types with a probability which determines how diversified a city is given its size. This probability turns out to be inversely proportional to city size: the more businesses, the harder it is to differentiate them from existing businesses. This process, with further research, displays an open-ended, never-ending, albeit slowing, diversification of businesses in a statistically predictable way, constituting a human eco-system.

For a fuller explanation of the work also see articles in Forbes and Next Cities.

Friday, 19 February 2016

Are big-city transportation systems too complex for human minds?

Many of us know the feeling of standing in front of a subway map in a strange city, baffled by the multi-coloured web staring back at us and seemingly unable to plot a route from point A to point B. Now, a team of physicists and mathematicians has attempted to quantify this confusion and find out whether there is a point at which navigating a route through a complex urban transport system exceeds our cognitive limits.

After analysing the world’s 15 largest metropolitan transport networks, the researchers estimated that the information limit for planning a trip is around 8 bits. (A ‘bit’ is binary digit – the most basic unit of information.)

Additionally, similar to the ‘Dunbar number’, which estimates a limit to the size of an individual’s friendship circle, this cognitive limit for transportation suggests that maps should not consist of more than 250 connection points to be easily readable.

Using journeys with exactly two connections as their basis (that is, visiting four stations in total), the researchers found that navigating transport networks in major cities – including London – can come perilously close to exceeding humans’ cognitive powers.

And when further interchanges or other modes of transport – such as buses or trams – are added to the mix, the complexity of networks can rise well above the 8-bit threshold. The researchers demonstrated this using the multimodal transportation networks from New York City, Tokyo, and Paris.

Mason Porter, Professor of Nonlinear and Complex Systems in the Mathematical Institute at the University of Oxford, said: ‘Human cognitive capacity is limited, and cities and their transportation networks have grown to the point where they have reached a level of complexity that is beyond human processing capability to navigate around them. In particular, the search for a simplest path becomes inefficient when multiple modes of transport are involved and when a transportation system has too many interconnections.’

Professor Porter added: ‘There are so many distractions on these transport maps that it becomes like a game of Where’s Waldo? [Where’s Wally?]

‘Put simply, the maps we currently have need to be rethought and redesigned in many cases. Journey-planner apps of course help, but the maps themselves need to be redesigned.

‘We hope that our paper will encourage more experimental investigations on cognitive limits in navigation in cities.’

The research – a collaboration between the University of Oxford, Institut de Physique Théorique at CEA-Saclay, and Centre d’Analyse et de Mathématique Sociales at EHESS Paris – is published in the journal Science Advances.

Monday, 15 February 2016

Hitchin70 - a celebration of one of Oxford's most influential mathematicians

In celebration of Nigel Hitchin's 70th birthday and in honour of his contributions to mathematics, a group of his former students and his colleague Frances Kirwan, in partnership with the Clay Mathematics Institute, are organising a conference in September 2016. It will begin in Aarhus with a workshop on differential geometry and quantization and end in Madrid with a workshop on Higgs bundles and generalized geometry, with a meeting in Oxford in between aimed at a general audience of geometers.

The three components of the conference are:

Hitchin70: Differential Geometry and Quantization, QGM, Aarhus, 5-8 Sept. 2016

Hitchin70: Mathematical Institute, Oxford, 9-11 Sept. 2016

Hitchin70: Celebrating 30 years of Higgs bundles and 15 years of generalized geometry, Residencia la Cristalera, Miraflores de la Sierra (Madrid), 12-16 Sept. 2016

More information, including registration, can be found at http://projects.au.dk/hitchin70/

The confirmed speakers at the Oxford component of Hitchin70, which is supported by the London Mathematical Society are:

Sasha Beilinson
Fedor Bogomolov
Philip Candelas
Bill Goldman
Klaus Hulek
Maxim Kontsevich
Marta Mazzoco
Shigefumi Mori
Shing-Tung Yau

A poster can be downloaded here

Nigel Hitchin is one of the most influential figures in the field of differential and algebraic geometry and its relations with the equations of mathematical physics. He has made fundamental contributions, opening entire new areas of research in fields as varied as spin geometry, instanton and monopole equations, twistor theory, symplectic geometry of moduli spaces, integrables systems, Higgs bundles, Einstein metrics, hyperkähler geometry, Frobenius manifolds, Painlevé equations, special Lagrangian geometry and mirror symmetry, generalized geometry and beyond. He is the Savilian Professor of Geometry at University of Oxford and was previously the Rouse Ball Professor of Mathematics at Cambridge University. He is a Fellow of the Royal Society and has been the President of the London Mathematical Society.

Tuesday, 2 February 2016

The universal structure of language

Semantics is the study of meaning as expressed through language, and it provides indirect access to an underlying level of conceptual structure. However, to what degree this conceptual structure is universal or is due to cultural histories, or to the environment inhabited by a speech community, is still controversial. Meaning is notoriously difficult to measure, let alone parameterise, for quantitative comparative studies.

Using cross-linguistic dictionaries across languages carefully selected as an unbiased sample reflecting the diversity of human languages, Oxford Mathematician Hyejin Youn and colleagues provide an empirical measure of semantic relatedness between concepts. Their analysis uncovers a universal structure underlying the sampled vocabulary across language groups independent of their evolutionary phylogenetic (evolutionary) relations, their speakers’ culture, and their geographic environment.

Monday, 1 February 2016

Mathematical theories of consciousness

How a complex dynamic network such as the human brain gives rise to consciousness has yet to be established by science. A popular view among many neuroscientists is that, through a variety of learning paradigms, the brain builds relationships and in the context of these relationships a brain state acquires meaning in the form of the relational content of the corresponding experience. Indeed, whilst it is very difficult to explain why a colour looks the way it does, it is easy to see that consciousness is awash with relationships and associations; for example, red has a stronger relationship to orange than to green, relationships between points in our field of view give rise to geometry, some smells are similar whilst others are very different, and there’s an enormity of other relationships involving many senses such as between the sound of someone’s name, their visual appearance and the timbre of their voice. Moreover, relationships in various forms are also ubiquitous in mathematical structures and have given rise to whole areas of investigation such as graph theory.

 

It’s perhaps surprising then that mathematicians haven’t rushed to provide a mathematical theory for how the brain defines the relational content of consciousness, and consequently there are few mathematical theories of consciousness. However, in November 2015, Oxford Mathematician Dr Jonathan Mason had a research article published in the journal Complexity that provides an approach to this challenge. The theory stems from information theory and introduces a version of Shannon entropy that includes relationships as parameters. The resulting entropy value (referred to as Float entropy) is a measure of the expected amount of information required to specify the state of the system beyond what is already known about the system from the relationship parameters. It turns out that, for non-random systems, certain choices for the relationship parameters are isolated from the rest in the sense that they give much lower float entropy values and, hence, the system defines relationships. One outcome of the work suggests that many relationships are determined in a mutually dependent way such as the relationships between colours and those that give the geometry of the field of view.

 

One other mathematical theory of consciousness is called Integrated Information Theory. Its initial development prioritised quantifying the level of consciousness of a system and, consequently, it is ill-suited for determining relational content. However, under further development, the two theories may turn out to be somewhat complementary.

 

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