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.