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Faculty of Mathematics

 

The Faculty is delighted to welcome Ailsa Keating as a new Lecturer in the Department of Pure Mathematics and Mathematical Statistics. Keating is an expert in symplectic geometry and at DPMMS joins a thriving research group in her field.

That is one of the joys of being a mathematician: you are part of a community that is progressing together. So life is full of surprises. Ailsa Keating

What is symplectic geometry?

Geometry is often considered the friendly face of mathematics. When we learn about it at school we start with triangles, circles and other shapes we can easily imagine and draw pictures of. But as you can imagine, the cutting edge research done by experts like Keating goes a long way beyond those simple objects.

In modern geometry more abstract shapes called manifolds take centre stage. Viewed from up close, a manifold looks like the ordinary Euclidean space we are used to from school, but its overall structure could be much more complex. A simple example of a manifold is a sphere. A small creature walking around on the sphere (for example a human walking on the Earth) will experience it as the flat two-dimensional plane. In general a manifold can take all sorts of weird and wonderful shapes — and since mathematicians have a notion of Euclidean space in any dimension (not just one, two and three) they also have a notion of manifolds in any dimension.

In her work Keating considers manifolds that come with additional information. "One example of an extra structure you could put on a manifold is a notion of distance between any two points. Or you could have a notion of angles," she explains. "Symplectic manifolds are equipped with more complicated structures, which generalise the notion of area, and come up naturally, for example in classical mechanics."

Some of the problems Keating thinks about can be understood by abstracting from situations we are used to. When you arrange two mirrors at an angle and place an object between them, the number of reflections you see depends on the angle between the mirrors. When the angle is a rational multiple of $\pi$ such as a right angle $\pi/2$, you see finitely many reflections, their number depending on exactly what rational multiple of $\pi$ you are dealing with. But when the angle between the mirrors is an irrational multiple of $\pi$ you see infinitely many reflections. This is what sometimes happens in bathrooms and lifts: two opposing mirrors are not exactly parallel, but form a very small irrational angle, which creates a fascinating infinite series of reflections.

"Both of these situations involve something called reflection groups and you have gone from something that was finite to something that was infinite," says Keating. "What I was thinking about this morning are reflection groups in higher dimensions, not just our three dimensions. I am trying to understand what the reflection group looks like in that case."

Connections

Symplectic geometry is a relatively young area, so it is able to draw on tools and advances from all sorts of other areas within mathematics: analysis and algebra, and obviously other areas of geometry.

A particularly beautiful connection takes us back into physics, but not to the classical mechanics that inspired some classes of symplectic manifolds. String theory is a highly mathematical framework that has been constructed to describe all the fundamental forces and particles of nature. Manifolds also appear within that theory and in their study of them theoretical physicists came across a particular type of duality, called mirror symmetry, though it is unrelated to the kind of reflection symmetry you find with an actual mirror.

"Mirror symmetry gives you a correspondence between certain kinds of symplectic objects and certain kinds of algebraic objects," explains Keating. You can then use results from either side and prove results on the other side." The field of mirror symmetry is a great example of the on-going cross-fertilisation between mathematics and theoretical physics. You can find out more about it in this lecture by David Tong, Professor of Theoretical Physics at the Department of Applied Mathematics and Theoretical Physics here at Cambridge.

Keating really enjoys this crossing of the boundaries within and outside maths. "A lot of the fun of being a mathematician is that lots of other people come up with really exciting things. That is actually one of the joys of being a mathematician: you are part of a community that is progressing together. So life is full of surprises."

Why pure?

If you are not a pure mathematician yourself, you might be asking yourself a question that is often put to pure mathematicians: "what is it good for"? Symplectic geometry does have applications outside of mathematics. It informs knot theory, which has some applications in biology. Symplectic geometry also has applications in modern classical mechanics problems, for example it is used to study low energy transfer orbits for space probes. But applications are not what motivates Keating.

"I find the maths I study aesthetically beautiful and really fun intellectually," she says. "But also really frustrating sometimes. I like the fact that even though maths progress is much slower compared with other academic disciplines, it is atemporal. Unlike, for instance, biology, where lots of theory gets rewritten every couple of years."

"Most people in pure maths, from time to time, have a crisis of conscience where they wonder why it is they are doing what they are doing, because of the applications angle. On the other hand, if you look at all the pure maths that we knew circa 1920 [you will find that it] has now been applied, including things that were applied fairly recently where no one really expected applications."

"So I think from a history of science angle we are doing just fine on the applications front. It's just that the applications will be for someone else to work on — probably not quite in our lifetimes."

Home coming

Keating isn't new to Cambridge: she was an undergraduate at Clare College between 2005 and 2009 and has been a Junior Research Fellow at Trinity College since 2014. Had someone asked the undergraduate in 2005 where she saw herself in thirteen years' time, back in Cambridge as a maths lecturer wouldn't have featured as an answer. "When I was an undergraduate, I didn't dream of becoming a mathematician," she says. "But then, during my first term of Part III [of the Mathematical Tripos], the courses were really, really fun."

Keating's enjoyment of maths during Part III inspired her to do a PhD at the Massachusetts Institute of Technology in the US. Her subsequent career evolved along similar lines. "At every juncture my decision has been, 'this is fun, I am going to keep going', rather than career planning," she says. "So I feel very lucky."

As a lecturer she can now pass her positive experience as a student on to the next generation — and herself reap the benefits of working at a world leading research institution. "What I didn't realise as an undergraduate is that this is a really nice mathematical community," she says. "It's a stimulating environment for research."