Features: Faculty Insights

## Ailsa Keating, from the Department of Pure Mathematics and Mathematical Statistics (DPMMS), has been awarded a prestigious Starting Grant from the European Research Council (ERC) to further her research into the symmetries of symplectic spaces.

As well as enabling Keating to devote more time to her research, the grant will also give her time and funding to employ and mentor a PhD student and two postdoctoral researchers to join her in exploring this fascinating area of mathematics.

"I'm very excited about the mathematics. And it's a really exciting opportunity to hire students and postdocs with the time to make sure you do a good job of looking after these mentees." Ailsa Keating

### Symplectic symmetries

We are all familiar with some simple types of symmetries such as reflections or rotations – transformations that leave objects essentially unchanged. Mathematically we can extend this notion of symmetry to a transformation, or map, from an object to itself. Then a map can be thought of as a symmetry if it preserves a structure that you are interested in, such as keeping the area of a local region unchanged.

"For example, imagine the surface of a donut, which is called a torus," says Keating. "You can cut it open along a particular curve and twist both edges of the cut by 180 degrees and glue it back together. That map has put a twist in your donut, called a Dehn twist about that curve, but you can do that in a way that preserves local area. And in my world that's a valid symmetry."

The world Keating works in is called symplectic geometry. She works with manifolds, mathematical objects that look locally like the ordinary flat space we are used to from school. Our torus is a simple example of a manifold. (You can read an accessible introduction to symplectic geometry in Keating's previous interview.) But in higher dimensional manifolds Keating is interested in symmetries that preserve a generalisation of the notion of local area called a symplectic form.

"All symmetries of tori can be obtained by doing a series of Dehn twists in different curves," says Keating. "This is true more generally for symmetries of any two-dimensional surface. Unlike in the 2D case, in higher dimensions we know very little about possible symmetries."

The main source of symmetries here is a higher-dimensional analogue of a Dehn twist, which leads to two broad flavours of questions: Can you find, in a systematic way, new examples of symmetries beyond Dehn twists? And when do Dehn twists give all possible symmetries? In her research with this grant Keating is trying to find answers to both, including finding subclasses of spaces for which it's possible to try to answer the completist question of finding all possible symmetries.

### A powerful perspective

Another exciting part of the work Keating will explore is the connections between her area of symplectic geometry and questions in other areas of mathematics. One of these connections is found through mirror symmetry: here the term symmetry refers to a correspondence between two different areas of geometry, which originated in theoretical physics.

"There are families of pairs of spaces that on one side belong to the symplectic world, and on the other side they belong to the algebraic geometry world. Parts of algebraic geometry are more developed than the corresponding parts of symplectic geometry, so the current state of the theory is much richer. But mirror symmetry tells us that these areas are equally rich."

Mirror symmetry pairs up symplectic characteristics on one side with algebraic characteristics on the other side. Keating proposes to construct a piece that was missing from this mirror correspondence. Natural well-understood symmetries exist on the algebraic geometry side of the mirror and Keating hopes to construct their symplectic incarnations. "We don't know what these natural symmetries correspond to on the other side of the mirror."

Keating believes that the combined insight across these mathematical areas, combined through mirror symmetry, provides something greater than the sum of the individual parts. "It's a powerful perspective in modern geometry because of the way the understandings of the two spaces complement each other and shed light on each other."

### Time for research and time for researchers

This is the first round of ERC Starting Grants to be awarded under the EU’s new programme, Horizon Europe. "Letting young talent thrive in Europe and go after their most innovative ideas - this is the best investment in our future, not least with the ever-growing competition globally," says Professor Maria Leptin, President of the European Research Council. "I am thrilled to see these new ERC Starting Grant winners ready to cut new ground and set up their own teams."

The UK government is in the process of formalising the UK's association with Horizon Europe, and has guaranteed that the funding will be provided by the UKRI in the meantime. Anne Ferguson-Smith, Pro-Vice Chancellor and Arthur Balfour Professor of Genetics at the University of Cambridge, emphasised the importance of these research connections between Cambridge and the wider European research community: "The recognition of the excellence of our emerging talent through an increase in the number of ERC Starting grants awarded to Cambridge this year emphasises the importance of our continued research connectivity with Horizon Europe."

With this five year grant Keating has the opportunity to devote more time to research, particularly by reducing her teaching load. The grant also provides her with funding for a PhD student and two postdoctoral researcher positions to join her in her research area and gives her time for mentoring these new researchers. "I’m very excited about the mathematics," she says. "And it's a really exciting opportunity to be able to hire students and postdocs to jobs that have good conditions and to free up time to make sure you do a good job of looking after these mentees."

Collaboration with colleagues working in this vibrant area in DPMMS has been an important contribution to Keating's research, such as connections to work of Mark Gross and Ivan Smith. The research of these and other colleagues at DPMMS has also been facilitated by grants from ERC and EPSRC (Engineering and Physical Sciences Research Council). Keating says: "My research fits in with the intellectual ecosystem of the Department."

Some quotes have been taken from a University of Cambridge news story.