### Mathematics as a Conceptual Art Form | Professor Elizabeth Louise Mansfield | Think Kent

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 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 communicate between two different layers of

conceptual realities but I’m a 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,

so the number of balls equals the number of voices.

It could be instruments, the voices could be different instruments or different

pitches, it depends on how you realise it and the 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.

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