Hello my name is Elizabeth Mansfield and I'm a Professor of mathematics here at the University of Kent and I'm here to talk to you about my practice as a mathematician and what it feels like and a little bit about why I do it.
I've thought for a very long time that mathematics is a human art form, it's a very ancient art form, it has its roots in surveying and astronomy. To do mathematics it feels really in between being an intuitive conceptual artist and a magician because when you find a new solution to a problem or it just there's such a tremendous joy about the just fit of it and it feels almost magical and one of the things about mathematics is that it's not just beautiful it's also useful. We're used to thinking of poetry as matching between an inner emotional reality and the outer reality of the language and how we communicate. Mathematics communicates between an inner conceptual reality and our physical world. There is of course pure mathematics that communicates between two different layers of conceptual realities but I'm an applied mathematician and I prefer, that's a little too recursive for me to do pure mathematics.
Alright, so I want to give an analogy about what I mean by conceptual art form and so my first, and I've given you one little analogy which is to poetry and my second analogy is really to painting and drawing. So I've written here for you what are my colours, so the usual boring stereotype of mathematics is that it's logical, its exact, it's what robots do. This is so far from my reality as to be laughable, this is the black and white colour is this logical exact stuff and some people really like that but I prefer I have to live in a vivid mathematical universe and I think geometrically, analytically, visually, I mainly think visually I think myself but my colleagues have other strengths. I think approximately heuristically, algorithmically, dialectically, inductively, probabilistically, algebraically and I combine my colours just like an artist would. I think algorithmically about my approximations and I think approximately about my geometries and so on. So on this slide I've drawn for you a picture of myself and a little snapshot of what's in my conceptual space in which I quite often live. And these are the analogues of my shapes lines and textures. So the actual content that you see here are the solutions to every problem I've ever studied or come up with myself and the most deeply embedded ones and the most joyful ones are the solutions that I've come up with myself even if these are only minor modifications of of high school problems or even primary school problems it doesn't really matter there's a real joy there coming up with something yourself and this is really how to learn to become a mathematician is to learn to solve nearby but slightly different problems to the ones you're given in school. So what we have here you can see I've drawn for you, you might be able to see the box with the triangle in it, the two one root three triangle that is the trigonometry area and that is the part that's connected to the very ancient problem of surveying. And the amazing thing about this is that even though this theory has such ancient roots, nevertheless it's continually expanded and enlarged to address new and modern problems and and the amazing thing is that the subject matter carries over to such a huge variety of other things like nonlinear order waves optical fibres a lot of the same mathematics is used. You can also see what is root 2 that was a huge philosophical problem in ancient Greece and now we just get on with it, we just find it and so on, approximate theories more modern theories to do with the structure of spaces, so for example how could you tell the difference between a torus and the sphere if you were only given the Atlas and you were not given the actual shape of the whole thing. So the new artworks are my new solutions and I write about them in my mathematical papers and I present them in conferences and I do feel like a magician presenting my solutions. There's always this point in the talk where you see the expert in the audience or the jaw drops, that's fun, that that's really fun.
Okay, so what we have here is on the left we have my colleague Dr Joe Watkins who runs our outreach programme and our ambassadors programme and my third- year honours student Rachael Wyman was an ambassador and so she learned all about the mathematics of juggling as part of this outreach programme but she wanted to describe some theory of mathematical musical tilings in her project because she's also a musician and it didn't take her very long to realise that actually when she looked at the abstract structure of the theory she was actually looking at two theories which were exactly the same even though one of them was juggling and one of them was musical tiling and of course the whole point is to have fun and so what she did was she worked out the juggling pattern which corresponded to the Wallace & Gromit theme tune. What we have is that the musical notes, the first few bars of the Wallace & Gromit theme tune, and then the second row is the second few bars realised as a musical tiling. Above that we have the notation for the corresponding juggling pattern. Now you can see two things straightaway one of them is that there are seven balls and seven voices, of voices. It could be instruments, the voices could be different instruments or different pitches, it depends on how you realise it and the number, so you can see that you throw one ball at a time and that you sound one note at a time. Beyond that it takes a little bit of effort to see that the patterns within the patterns actually do correspond. Juggling is a physical thing where we're restricted to the fact that we only have two hands and it has to be physically realisable by the juggler and similarly there are patterns within the musical tiling which make it musical. So hopefully you can come along to the outreach if you want to hear more about that.
So what do I do? Well, I'm really interested in symmetry and in the physical implications of the symmetries in the world that we have. Now normally when we talk about symmetry, we talk, we show the symmetries of a cube or the symmetries of the snowflake and these are very pretty but these aren't the symmetries that really float my boat. What I'm interested in are much more subtle symmetries, so what I've drawn for you here is a picture we have an amalgam of different kinds of experiments, we have a molecular vibration, we have interacting magnetically electromagnetic waves and we have a nonlinear water way of experiment. And these experiments share the following symmetries: it doesn't matter whether I conduct my experiment on the Monday the Tuesday or the Wednesday, I get the same result. It doesn't matter whether I conduct my experiment in Sydney or London, I get the same result. And it doesn't matter which way I wrote which way my apparatus is facing north, south, east, or west. These are very subtle symmetries and they are to do with the fact that the physical experiment is a little bit invariant under how I put co-ordinates, so this is a mathematical abstraction but if our world didn't have these symmetries it would indeed be extremely confusing. Nevertheless so these symmetries are there and the mathematical consequence for these symmetries for systems which satisfy at least action principle and these physical these systems satisfying at least action principle are most of them, they are physically the most important ones and the mathematical consequence of the symmetries our conservation laws, they're very famous: conservation of energy and momentum, and the mathematical result is due to they're one of the most famous women mathematicians of all time Emmy Noether and we will be celebrating soon the centenary of this very famous paper. These conservation laws are very important to embed into numerical simulations of what you're doing. It's not hard to imagine that that the numerical simulation means to embed the physics if you're an engineer or a physicist, but even for you as a consumer of entertainment looking at CGI simulations in movies and in video games if what you're seeing does not incorporate the physical laws you will know that it is dodgy, you will see it straightaway. So what is the creative leap that I have to make? So this slide visualises for you the problem, the problem that I have to solve. Mathematically the symmetries are formulated as smooth actions on a smooth space, so that's the topmost graphic but if I discretise in order to put it onto a computer I have to discretise that space and then I have to think about where has my smooth group action gone, well it's disappeared but it's still there because if you take the view that physical reality is really discrete and that the smooth is really the approximation that it should be there and, lo and behold, you can find it and it is there and that was the creative leap. And I've been exploring this with my colleagues and my students for some time and I have in my penultimate slide a graphic drawn by my PhD student, Michele Zadra, showing you the result of one of our new numerical methods. On the left is the new method and on the right is the old method and this I hope you can see that the new method has far greater resolution in the spiral part of the solution curve to the problem. These particular drawings were an extension of a paper I wrote with a former PhD student of mine Tania Goncalves who's now in Brazil and I suppose this is one of the most amazing things about mathematics is how cross-cultural it is and how it really is a deeply human art form, which cuts across geography, ethnicity, gender and I think I thoroughly enjoy being a research mathematician. I get to travel the world and I get to be an artist with a little touch of magic. I hope you've enjoyed my talk. Thank you for coming.