In an interview with Rolling Stone Magazine in 1965, Bob Dylan was pushed to define himself: Do you think of yourself primarily as a singer or a poet? To which, Dylan famously replied: Oh, I think of myself more as a song and dance man, y’know. Dylan’s attitude to pigeonholing resonates with many applied mathematicians. I lack the coolness factor of Dylan, but if pushed about defining what kind of mathematician I am, I would say: Oh, I think myself more as an equation and matrix guy, y’know.

One of the greatest strengths of applied mathematics is that it has established itself by defying simple categorisation. Applied mathematics, be it an art, a craft, or a discipline, is not bound to a particular scientific application, a particular mathematical universe, or a well-defined university department. The drawback is that applied mathematics usually gets no mega-funding or the limelight associated with big scientific breakthroughs. But its biggest advantage is that it can insert itself into all scientific disciplines and easily reinvent itself by moving fluidly from one field to the next, guided only by methods, theory, and applications: it is all equations and matrices. Many applied mathematicians see new challenges as an opportunity to expand their mathematical horizons, and in our rapidly changing modern new society such new challenges abound. Here are three of these.

Major scientific efforts are required for major society challenges. These include fighting climate change, optimising new renewable energy sources, developing new medical treatments, and understanding the brain. Traditionally, applied mathematicians involved with these collaborative efforts were considered a useful but small cog in a huge scientific machine, but it is now appreciated that quality science requires clever modelling, state-of-the-art numerical methods, and fundamental theoretical insights from simplified models. This is the realm of applied mathematics, and accordingly our role in these endeavours is bound to increase. By the end of the day, we may not get the fame, but we’ll certainly have the fun.

A second relatively recent development of applied mathematics is the theory of networks. Networks represent connections between multiple physical or virtual entities. They are found in information theory (web links, social connections), biological systems (gene regulatory networks, metabolic networks, evolutionary trees), and physical systems (axon connections, electric grid). Regardless of their origin, these networks share common mathematical features. Their analyses span many different fields of study, and network theory has now established tentacular connections to various parts of pure and applied mathematics, a network of its own.

For about five years there has been much excitement about BIG DATA. The initial hope was that one could go straight into data and use empirical methods to unravel the mysteries of the universe. Quite the opposite is happening. The success of many methods has shed a bright light on the need to understand the underlying mathematical structure of both data and methods. The subject now presents a rich field of study that brings all mathematical sciences together, including statistics and computer science.

These examples share a common thread that highlights a new trend in mathematical and scientific discoveries: beyond inter-, multi-, and supra-disciplinarity, we live in a post-disciplinary world. Things have changed, and Oxford University with its collegiate system, and the Mathematical Institute with its collegial atmosphere, are particularly well equipped to thrive in this new scientific world. But despite all the hype, we’re also fully aware that there’s nothing wrong with the old world, the old problems, or the old conjectures. We have an intellectual responsibility to promote and cherish these areas of knowledge defined by the great thinkers, past and present, especially if they are believed to be useless or irrelevant.

Bob Dylan in the same interview, foresaw yet another possible application of mathematics: What would you call your music? His reply: I like to think of it more in terms of vision music – it's mathematical music.

Alain Goriely, Professor of Mathematical Modelling, Oxford Mathematics, University of Oxford.

The caption next to Bob above is a scan of Alain’s brain. Bob sang: “My feet are so tired, my brain is so wired”. But will collaborative applied mathematics untangle the mystery of the author’s brain, mathematical or otherwise?

Alain's 'Applied Mathematics: A Very Short Introduction' will be published by OUP later in the Year. You can also watch his lecture via the Oxford Mathematics YouTube page.