Take 10 with... Matthew Conder

Dr Matthew Conder, Research Fellow in the School of Mathematics, gives us 10 minutes of his time to discuss his research into linear groups using algebraic and geometric techniques.

1. Describe your research topic to us in 10 words or less.

Studying linear groups using algebraic and geometric techniques.

2. Now explain it in everyday terms!

A group is an abstract algebraic structure used to model and analyse the symmetries of an object, and a linear group is a group consisting of matrices. My research involves investigating the interactions between the algebra and geometry associated with linear groups, and using this to develop computational methods of studying these groups.

3. Describe some of your day-to-day research activities.

I start on a research project by first reading the relevant literature. Then I will often be found with a pen and paper, or working on a whiteboard with colleagues, trying to develop new ideas or techniques for solving a problem. Sometimes I will use computational algebra software to help me identify patterns which can inform my approach to the problem. Once a project is finished, I will spend some time writing and editing the resulting research paper.

What I enjoy most about my research (and mathematics in general) is the moment when everything falls into place and I manage to solve a problem I am working on. Often it can take some time to get to this point, and so it is always a great feeling!

5. Tell us something that has surprised or amused you in the course of your research.

When I was a PhD student, I got very excited because I thought I had developed new formulae for groups which act on combinatorial structures known as trees. I then found a paper from the 80s which proved these formulae and so I felt disappointed. However, in further computational investigations, I actually discovered a missing case in this paper which had gone unnoticed for over 30 years!

6. How have you approached any challenges you’ve faced in your research?

Often I will get stuck working on a research problem, and I will keep trying different approaches but nothing seems to work. I then find that taking a step back and taking some time to work on something else will help with this. Coming back a few days or weeks later to look at the problem again with a fresh perspective has almost always resulted in new progress.

7. What questions have emerged as a result of your recent work?

A lot of questions have emerged from my recent work, which are generally of the form “can I also apply these techniques in higher dimensions?” or “do similar methods still work if I remove these requirements?”

8. What kind of impact do you hope your research will have?

The main goal of my research is to further our knowledge about linear groups and make it easier to study them using computational methods. I hope that this will result in new research problems to collaborate on, and that it will attract students to study and research this area of mathematics.

9. If you collaborate across the faculty or University, or even outside the University, who do you work with and how does it benefit your research?

I regularly collaborate with Dr Jeroen Schillewaert. This has been extremely beneficial, as when one of us gets stuck, the other will often find a way to fix the issue. We have made a lot of progress together on research problems, and certainly much more progress than we would have made individually!

10. What one piece of advice would you give your younger, less experienced research self?

Collaborate more! Take every opportunity to network and discuss research with your colleagues.