Dr Oliver Janzer: combinatorics and the Premier League

Dr Oliver Janzer explains combinatorics in our latest Q&A with Junior Research Fellows. 

Could you describe the difference between pure and applied mathematics?

Applied mathematics mainly deals with questions which arise in “real life” such as in physics and engineering. On the other hand, in pure mathematics we consider theoretical problems, and our results are usually judged based on their applications in mathematics itself. Also, pure mathematics provides the foundation for other subjects and it is common that methods developed for tackling very theoretical problems become useful in applied mathematics, computer science or physics.

What is combinatorics?

Combinatorics is a broad area in mathematics, mainly focusing on questions which involve counting or arrangements in finite sets. For example, designing the schedule of the Premier League is a combinatorial problem.

A characteristic of the field is that (unlike in other areas of mathematics) questions are often easy to state, but very difficult to solve. For example, the following question may sound like a simple puzzle but is one of the most notorious unsolved problems in Combinatorics. What is the maximum possible number of people in a group where among any five people, at least one pair shakes hands, but not all the pairs do?

My research is mostly on the subfield of Combinatorics called Graph Theory. Graphs are essentially networks in which some pairs are “connected” and some pairs are not. For example, Facebook can be regarded as a graph in which two people who know each other are considered connected and two people who don’t know each other are considered not connected. The problem about handshakes is also a Graph Theory question. In our research, we focus on studying graphs from a theoretical perspective: instead of analysing specific networks, we prove general theorems which apply to any network.

What new questions will you be exploring?

One of the topics I will be researching can be stated in a much simplified form as follows. In a group of n people, at most how many can shake hands such that there are no three people who all shake hands with each other. When n=10, the answer to this question is 25. To see that 25 handshakes are indeed possible, imagine that out of the 10 people in the group, 5 are women and 5 are men, and that all women shake hands with all men, but women don’t shake hands with women and men don’t shake hands with men. Then there were 5×5=25 handshakes, but there are no three people who all shook hands with each other. Of course, to solve the above question completely, one would also need to prove that more than 25 handshakes are not possible.

What are you looking forward to most about your Fellowship?

I’m excited to return to Trinity, my alma mater, where I spent seven unforgettable years. Importantly, Cambridge and Trinity have several world-class professors, as well as excellent postdocs and PhD students doing research in Combinatorics. I’m really looking forward to starting collaborations with them. I am also excited about the teaching prospects: giving my own lecture course and supervising the very bright Trinity undergraduates.

Do you have other hobbies/interests?

I enjoy being active physically as well as mentally. I like running and I’m looking forward to (re)joining Trinity’s tennis team.