Get to know Dr Martin Kerin who recently joined our Department of Mathematical Sciences.

I obtained BSc and MSc degrees at Maynooth University in Ireland, followed by a PhD in 2008 from the University of Pennsylvania in the USA. From 2008 to 2019, I was a postdoc at the University of Muenster in Germany, before taking up a permanent position at the University of Galway in Ireland. I joined Durham University in 2022.

I am a mathematician with interests in geometry and topology. One important goal of these fields is to study the “shape” of abstract mathematical objects called manifolds. Most of my research focuses on trying to understand manifolds which are non-negatively curved, meaning that the space locally looks somewhat like a piece of a sphere or flat space in every direction you look. It is often helpful and easier to assume that the manifolds being studied have a lot of symmetries, so I am also interested in finding ways to make progress in situations where less symmetry is assumed.

I joined the university in November 2022. I am currently an Assistant Professor in Geometry in the Department of Mathematical Sciences.

#### What are the projects/research you are currently working on?

I am currently trying to understand the topological obstructions to being able to decompose a manifold into two simple pieces called disk bundles. If such a decomposition exists, we call the manifold a double disk bundle. There are many interesting examples of double disk bundles. For instance, exotic spheres can always be described in this way.  An exotic sphere is a strange object that is topologically, but not smoothly, equivalent to a standard sphere sitting in Euclidean space. By exploiting a double disk-bundle decomposition, my collaborators and I were able to show in 2020 that all exotic spheres in dimension seven admit non-negative curvature, solving a question that had been open for about 60 years.

One hope in my work is that by understanding double disk bundles better, we might be able to say more about the geometry of exotic spheres in other dimensions.

Double disk bundles are, on the one hand, sufficiently simple to allow explicit computations and, on the other, complicated enough to yield a large family of interesting objects with the potential to carry desirable geometric properties. They appear frequently in the mathematical literature, both in constructions of examples and as consequences of geometric assumptions, but they have received little attention in their own right. I hope that my research changes this and lays the groundwork for further developments and new discoveries.

In particular, exotic spheres have intrigued mathematicians ever since their discovery in 1956 and the quest to understand them from various viewpoints has driven many of the most important developments in mathematics over the last decades. Any progress in understanding how their geometry resembles or differs from that of the standard sphere is, almost by definition, of general mathematical interest.

#### What is the scientific and societal relevance of your research?

Research in pure mathematics is typically not conducted with applications or societal impact in mind, and mathematicians are often not remotely interested in finding or exploring applications of their work. That being said, seen through a certain lens, my particular branch of geometry can be viewed as the ideal, abstract model which inspires new approaches and developments in more applied fields such as data analysis. There are many people exploring these ideas.

#### What are your plans for future research/study?

The more I learn about the geometry and topology of double disk bundles, the more interesting questions that arise. I hope that someday my collaborators and I will be able to answer at least some of them!

#### Any interesting hobbies/passion outside of work?

I enjoy playing football, but nowadays don’t get to do so very often. I like to relax by spending time my wife and young son. I also like to travel and have managed to visit all continents other than Antarctica.