# Take 10 with... Florian Lehner

Florian Lehner, Department of Mathematics, gives us 10 minutes of his time to discuss his research on large scale tree structures, collaborations, and advice for early-career researchers.

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

Large scale tree structure of infinite combinatorial objects.

#### 2. Now explain it in everyday terms!

When confronted with a complicated problem, it often makes sense to split it into simpler sub-problems, solve those, and then combine the solutions to get a solution for the original problem. This approach works particularly well if the solutions of the sub-problems do not depend on one another, but it is still feasible if there is limited interdependency. Tree structure gives a way of decomposing infinitely large objects into finite parts with limited interactions between them.

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

I always spend part of my day reading research papers. Often these are directly related to a question that I am working on, but sometimes I also read papers because I think that they solve an interesting problem. When I am actively thinking about a research question, I spend a lot of time drawing sketches and diagrams to find ways to simplify the problem, or view the problem from a different angle. Once I am convinced that I have a solution, I write up all the details – sometimes I only find out at this stage that I have missed an important detail, which means I need to go back to the previous stage.

Usually I spend most of the time working on a research problem either being stuck or making slow progress through tedious calculations, but sometimes a simple new idea which can be explained in a few minutes replaces pages of calculations or solves a decades-old open problem. Understanding such an idea always feels great, no matter if myself or a collaborator comes up with it, or if I find it in a research paper.

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

It still puzzles me how unreasonably efficient random constructions can be. For some problems no known (deterministic) construction is better than simply picking a random solution. I experienced this first hand during my PhD, when I was able to show that while a random construction did not quite solve a conjecture I was working on, it came very close (and I got a nice publication out of it).

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

Usually when I get stuck I try to change something. Depending on how stuck I am, this can range from taking a few steps back from the whiteboard or having a coffee to changing the premise of the research project by adding an assumption, or working on a different research problem for a few weeks.

#### 7. What questions have emerged as a result?

Adding an assumption to a research problem often leads to a number of new questions: Can we solve the problem under the new assumption? Does our assumption have an interpretation in a broader context? How does it relate to other open problems? What happens if we strengthen or weaken our assumption? Do we get similar results or perhaps even stronger results?

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

I don’t think that the impact of fundamental research should be measured in terms of real world applications. While most technological advances are based on maths in one way or another, it often takes decades to turn a mathematical idea into an application. Instead, I hope to discover interesting and beautiful mathematical truths, and to convince people that mathematics is beautiful and fun.

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

In my opinion, one of the most important aspects of collaboration is being able to discuss ideas on an informal level. This often helps me determine whether a half-baked idea is worth following up on, and to put it into a formal definition if it is. Most of my current collaborators are based in Europe and, despite the time difference, I try to talk to them regularly because email does not allow for the same kind of informal interaction.

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

Understanding the big picture is often more valuable than being formally correct, especially in the early stages of a research project. While it is important to consider every small detail in the final write-up, obsessing over details too early can be a waste of time and slow down the progress of the whole project.