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Features: Faculty Insights


Ivan Smith, Professor of Geometry at the Department for Pure Mathematics and Mathematical Statistics (DPMMS), has been elected as a Fellow of the Royal Society. Founded in 1660, the Royal Society is the UK's national academy of sciences and the oldest science academy in continuous existence.

This year 80 researchers, innovators and communicators from around the world have been elected as Fellows of the Royal Society for their substantial contribution to the advancement of science. As one of them, Ivan Smith joins the ranks of previous fellows Stephen Hawking, Isaac Newton, Charles Darwin, and Albert Einstein.

One thing that is very nice about this topic is that the methods of proof go via connections to other subjects. It's a confluence of many ideas. Ivan Smith

"I am surprised, delighted and hugely honoured to be elected a Fellow of the Royal Society," Smith says. "I've been very fortunate to work in a rapidly advancing field, learning it alongside many inspirational and generous collaborators, who should definitely share this recognition."

All about area

The field Smith refers to is symplectic topology, which involves objects known as symplectic manifolds. Generally, a manifold is a space that locally looks like the ordinary Euclidean space we are used to from school geometry, but globally may have a very different structure. An example is our planet. Because we are too small to detect the Earth's curvature, the local patch of Earth we are standing on appears to us as a flat Euclidean plane. Seen as a whole, however, the Earth is anything but flat: it's a sphere. Spheres are examples of two-dimensional manifolds.

When working with a manifold it's natural to want to impose a geometric structure on it — to define ways of measuring lengths and angles, for example. This gives rise to an area of maths called Riemannian geometry, named after the 19th century mathematician Bernhard Riemann.

Symplectic topology focuses on the geometry of area, rather than length and angles. A symplectic manifold, even if it has more dimensions than we can visualise, comes with a clearly defined, generalised notion of area.

From dynamics to geometry

Symplectic topology was to a large part inspired by a version of mechanics formulated in 1833 by the mathematician and physicist William Rowan Hamilton. While Hamiltonian mechanics can describe the same physical phenomena as the versions developed by Newton and Lagrange before him, it has certain advantages over both, which is why it attracted people's interest.

Like Newtonian and Lagrangian mechanics, Hamiltonian mechanics can be used to describe the behaviour of dynamical systems. Imagine, for example, a marble rolling around a circular track.To describe the state of this system at a given moment in time, you need to know the marble's current position on the track and its speed, or, alternatively, its momentum. Momentum is the product of an object's speed and its mass which, loosely speaking, measures how hard it would be to bring the object to a stand-still.

To keep track of both position and momentum at once, you might visualise the marble's journey on a vertical cylinder instead of a circle. The marble's journey around the track is represented by a path winding around the girth of the cylinder. Changes in momentum are reflected by the path spiralling up the cylinder (when the marble accelerates) or down it (when it slows down).

In fact, the cylinder represents all the possible states of the system — all possible combinations of position and momentum — at the same time. Each state corresponds to a point on the cylinder's surface and vice versa. It is what mathematicians call the phase space of our dynamical system. Since a cylinder is a geometric shape, in fact it's a simple example of a manifold, we have now entered the world of geometry.

Going symplectic

Hamilton's equations of motion now induce what you might call a flow on the cylinder, which tracks how the states evolve over time as dictated by the equations. A patch on the cylinder's surface is moved around by this flow in a way that reflects how the states that sit within the patch evolve. And while the patch may change shape as it moves along, Hamilton's equations imply that its area will always remain the same.

Something similar is true for all sorts of dynamical systems, even ones that are much more complicated than our example. The phase space of such a system forms a manifold. You may need more than one coordinate to describe the positions and momenta of objects in the system (in three-dimensional space, for example, you need three coordinates each) so the manifold won't necessarily be a two-dimensional surface like the cylinder. However, it is possible to define a measure of size with reference to the position and momentum pair which generalises the notion of area in the two-dimensional case. This measure of size is the generalised notion of area mentioned above, and it is preserved by the flow induced by the dynamical system.

Beyond Hamilton

The shape of the symplectic manifold that embodies the phase space of a Hamiltonian system can tell you a lot about the possible behaviours of the system itself, and that's where the interest in symplectic manifolds originated. But since symplectic manifolds can also be defined and studied without reference to Hamiltonian systems, this now presents a field of research in its own right. It lies at the intersection of various other disciplines, drawing on algebraic geometry (which investigates geometric shapes defined by polynomial equations), topology, and representation theory (find out more here). There are also deep connections to theoretical physics through mirror symmetry, a notion that comes up in string theory.

One thing that has interested Smith are the symmetries of symplectic manifolds and he has done important work in this area. "[Together with Nick Sheridan I] proved that for some [symplectic manifolds] the [group of symmetries] is much larger than you might have expected, in particular much larger than it would be if you didn't keep account of the symplectic structure," Smith told us at the International Congress of Mathematicians in 2018, where he was an invited speaker.

"One thing that is very nice about this [topic] is that the methods of proof go via connections to other subjects, in particular algebraic geometry which is usually a more rigid subject. The bridge between this differential and algebraic geometry has its origins in mirror symmetry. So it's a confluence of many ideas."

Apart from proving that symmetry groups of symplectic spaces can have infinite complexity, and establishing new cases of mirror symmetry, Smith and his collaborators have also proved important results about the dynamics of mathematical billiards and even the structure of knots. His work has previously been recognised with a London Mathematical Society Whitehead Prize (2007), the Adams Prize (2013), and a Clay Senior Scholarship (2022).

See here for an interview with Ivan Smith conducted at the International Congress of Mathematicians in 2018.

For a report on research in symplectic geometry and singularity theory conducted by faculty member Ailsa Keating, see this article.