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The rolling of dice in a casino, Heisenberg's uncertainty, the meaning of consciousness. All are explored as Marcus takes us on a personal journey into the realms of the scientific unknown. Are we forever incapable of understanding all of the world around us or is it perhaps just a question of language, not having the right words to describe what we see?

What is a network and how can you use mathematics to unravel the relationships between a variety of different things? How can this understanding then be applied to a range of different settings?

In this Oxford Sparks podcast Oxford Mathematician Mason Porter studies how things are connected using mathematics. He builds up models of these connections to represent them as networks. But what are the basic components of a network? In the podcast Mason describes how from social networks to transport systems to locating a lost umbrella, the mathematics of networks can be used to address a range of apparently unconnected problems and how organisations around the world are using them to penetrate their ever-growing mass of data.

Congratulations to Oxford Mathematicians Martin Bridson, Marcus du Sautoy and Artur Ekert who have been elected Fellows of the Royal Society. Martin is Whitehead Professor of Pure Mathematics, a Fellow of Magdalen College and Head of the Mathematical Institute in Oxford. He has been elected for his many distinguished contributions to group theory and topology. Marcus is Charles Simonyi Professor for the Public Understanding of Science and a Fellow of New College and has been elected for his outstanding achievements in promoting the understanding of science and mathematics to a global audience and for eminent research that has completely transformed the study of zeta functions of groups. Artur is Professor of Quantum Physics at the Mathematical Institute and a Fellow of Merton College.  Artur has been elected FRS for his work on quantum physics, quantum computation and cryptography.

Appearing everywhere from state-of-the-art cryptosystems to the proof of Fermat's Last Theorem, elliptic curves play an important role in modern society and are the subject of much research in number theory today. Jennifer Balakrishnan, a researcher working in number theory, explains more in the latest in our Oxford Mathematics Alphabet.

Oxford Mathematician Rob Style has been awarded the 2016 Adhesion Society Young Scientist Award, sponsored by the Adhesion and Sealant Council, for his fundamental contributions to our understanding of the coupling of surfaces tension to elastic deformation.  Rob researches the mechanics of very soft solids like gels and rubber, in particular investigating why they don’t obey the same rules as hard materials that are more traditionally used by engineers.

Oxford Mathematician Jake Taylor King has won the Lee Segel Prize for Best Student Paper for his paper 'From birds to bacteria: Generalised velocity jump processes with resting states.' Jake worked on his research with Professor Jon Chapman. The prize is awarded annually by the Society for Mathematical Biology. One of Jake's co-authors on the paper, Gabs Rosser, previously also studied Mathematics at Oxford in the Wolfson Centre for Mathematical Biology.

Oxford Mathematician Linus Schumacher has won the prestigious Reinhart Heinrich Doctoral Thesis Award. The award is presented annually to the student submitting the best doctoral thesis in any area of Mathematical and Theoretical Biology. 

In the judges' view "Linus' thesis is an outstanding example of how mathematical modelling and analysis that is kept close to the experimental system can contribute efficiently to advance the understanding of complex biological questions. The roles of cellular heterogeneity, microenvironmental cues and cell-to-cell interactions, which are common themes in the study of biomedical systems, are skillfully dissected and analysed in relevant experimental model systems, leading to significant advances in the current understanding of said systems."

The judges concluded: "the modelling aims to derive generic, theoretical insights from specific, biological questions. The work has led to a number of excellent publications."

The Society for Industrial and Applied Mathematics (SIAM) has announced that Professors Xunyu Zhou and Endre Suli from Oxford Mathematics are among its newly elected Fellows for 2016.

SIAM exists to ensure the strongest interactions between mathematics and other scientific and technological communities through membership activities, publication of journals and books, and conferences.

A 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.