# Maarten Markering - Fascinated by maths problems

What do you do if you discover during your bachelor’s research that the problem you are working on has already been resolved by someone else? Maths student Maarten Markering didn’t let that deter him; instead he completed and even improved the evidence. He is one of the four nominees for the Young Star Award 2020.

## Problems and evidence

Maarten enjoys talking about maths and his research. ‘As a maths student, you spend the whole day resolving problems. I often spend days or even weeks trying to fathom out a difficult question. Sometimes it simply doesn’t work out, which is very frustrating. But when I eventually find the evidence I need, it can made me happy for days on end.’ He is motivated by the intrinsic beauty of maths. ‘During my studies, I discovered just how rich the world of maths can be.’ Maarten is a fan of probability theory, but he’s convinced that every aspect of maths is filled with good arguments and surprising evidence.

## Friendship in a model

For his bachelor’s thesis, Maarten studied complex networks, which are playing an increasingly important role in the modern world. ‘Think of such things as our social network, the trading networks between countries or the network of international air traffic,’ he says. ‘These are complex paradigms that we can describe in mathematical terms as points joined together by lines. In a social network, the points are people and  two points are connected if these people are friends.’

These intricate networks are too complex to describe in precise terms. This is why mathematicians use probability theory models where the points are connected with one another randomly. Not all models distinguish between different points and lines. Maarten explains: ‘Let’s take the example of the social  network of students at this faculty. In the model, each pair of students has the same chance of becoming friends. So, according to the model, it’s equally likely that a maths student will be friends with someone from biology as with another maths student. That’s not what happens in reality. My thesis is about a model that does distinguish between different points and lines.

## Unexpected deviation

Maarten proved a proposition that describes the probability of unlikely occurrences in the model. ‘With this proposition you can, for example, calculate how great the likelihood is that the average number of friends of first-year students at our faculty is a lot lower than this model would lead you to expect (the chance of a deviation). If this chance is really low, but yet the number of friends turns out to be a lot lower than expected , there has to be another explanation for it: for example, that there is a pandemic raging throughout the country, which means everyone is having to stay at home!’

'I often spend days or even weeks trying to fathom out a difficult question. When I eventually find the evidence I need, it can made me happy for days on end.’

## Not the first

Halfway through the thesis Maarten and his supervisor Frank den Hollander discovered that other mathematicians had just solved ‘their problem’. Maarten: ‘Obviously, that was a big disappointment; we had hoped to be the first. However, the evidence of our “competitors” was not absolutely complete and it only applied under strict conditions. In my thesis, I give the full evidence under milder conditions.’  Maarten has also improved the evidence on a number of points. For example, by simplifying an important function. He himself regards it as a good result that has a lot of potential applications. ‘This summer another article was published that used that first set of evidence. In that article there were a number of things they couldn’t prove because the function is so complex, but I’ve simplified it!’ Maarten has now written his own results up in an article, which he has submitted for publication.

## Cambridge and Leiden

Maarten was due to leave for Cambridge in October 2020 to do a Master’s in Mathematics, but because of the coronavirus, he decided to stay another year in Leiden. ‘I’m taking some master’s subjects here, but most of my time is spent on research. I’m writing a scientific article together with Frank, which is about complex networks, like my thesis. For this article, I’ve resolved a particular open problem about probability-theoretical models.’

Maarten hopes he will be able to go to Cambridge in the coming academic year, if all goes well. After that, he’s intending to start a PhD. He doesn’t have any firm plans for after that. ‘I find maths research fascinating, so I’d like to carry on in the scientific world.’