# Take 10 with... Jeroen Schillewaert

Dr Jeroen Schillewaert from the Department of Mathematics gives us 10 minutes of his time to discuss his research on geometry and algebraic groups, and the applications of his work in the transmission of secure information.

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

Geometry and groups.

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

The bulk of my research is in the area of geometry and symmetry. Dating back to the ancient Greeks, who studied Platonic solids, this area of study is still very active today.

I study the geometries corresponding to algebraic groups, which can be subdivided into two classes. The first class are the classical groups, which have been well understood since the end of the 19th century, including the rotation group, the Lorentz group (used in special relativity) and other groups used in both classical and quantum mechanics.

In the second class are exceptional algebraic groups, which are less well understood. Jacques Tits (Abel Prize 2008) pioneered a geometric approach to these groups starting in the 1950s. Building on his work, the main goal of a long-term project is to understand E8, the largest and most elusive of these groups. The E8 group is very important in pure mathematics, connecting algebra and analysis, and it also plays a role in modern theoretical physics.

A lot of my earlier work focussed on combinatorial aspects of this theory. In particular, I have worked on finite geometries - geometric systems that only have a finite number of points. I have also ventured into extremal combinatorics, which studies the sizes of sets of objects satisfying certain restrictions.

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

Depending on what stage of a project I am at, there will be either 1) a lot of emphasis on reading background material (start), 2) being mostly stuck, in front of a whiteboard (middle), or 3) many hours in front of the computer carefully writing down the results. During the write-up stage I end up back in stages 2) and 1) as I overlooked or did not fully understand something. As I often have a couple of projects running concurrently these stages are not clearly separated.

Understanding and solving a problem which I and others could not solve before, especially if this is the result of a short argument due to a clear improvement of conceptual understanding, rather than hours of tedious case-by-case calculation and analysis. This is most enjoyable, as it is rare.

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

At the start of my PhD, I was working with one of my advisers and a postdoc. We were all going to explain our approach to a problem, and my adviser went first. He wrote for 15-20 minutes on the blackboard, and when he finished writing, my adviser and I expected to have a discussion about his idea. Instead, the postdoc immediately erased everything my adviser wrote and started to explain his own idea, which he deemed superior. When it was my turn, I decided to give the postdoc a ‘cookie from his own dough’, at which point he understood that this is not done. Our next joint meetings were a lot more interactive!

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

In the first instance by working harder and longer. In the second instance, by revising the entire strategy - instead of trying to push through a wall, I walk around it. I have also had to abandon a couple of projects, which I may revisit if new insights into the obstructions become available.

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

I have learned over the years to use failed attempts as a means of generating new research questions. A particular obstruction is often an interesting phenomenon in itself.

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

I don’t like the word impact, as the word is often confused with (perceived) immediate applicability. For instance, the study of abstract fields, in particular finite fields, in the 19th century would have been viewed as having zero impact, but is now probably the most widely used concept in practice with the emergence of the personal computer.

So, I'll refer instead to the applicability of my work. As I mentioned above, understanding E8 better is very beneficial in view of its central importance in mathematics and modern theoretical physics. Some recent joint work led to the construction of the first very regular expander graphs, which are important for large networks. Aspects of my PhD in finite geometry found applications in coding theory, which is used to correct potential transmission errors when sending messages, as well as cryptography, for the encryption and decryption of valuable information.

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

I recently collaborated with Distinguished Professor Marston Conder and world leader Professor Alexander Lubotzky from the Hebrew University in Jerusalem (our 2019 Michael Erceg Senior Visiting Fellow) on the project mentioned above on expander graphs.

The attraction of international visitors of the highest quality is crucially important to our thriving mathematics department. Visiting colleagues and attending international conferences is the mathematician's equivalent of spending time in the lab. This is where ideas are tested, new things are tried out and creativity reaches its peak. It can never be replaced by e-mail and online meetings.

I have held positions in five different countries on three continents, resulting in a strong and diverse network. This is beneficial scientifically as well as socially.

Locally I have benefitted from many discussions with colleagues and students, in particular during my reading groups on spaces of non-positive curvature and arithmetic groups.

These interactions have improved my understanding of topics in groups, geometry and topology, which was very useful in two recent papers.

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

Don’t waste time and energy waiting and speculating about the outcome of paper submissions, job applications etc. Accept the things you cannot change. Reach out to more people at conferences, rather than staying in your own small bubble of people you already know.