# Mathematician Peter Koymans wins KWG PhD prize

Leiden PhD student Peter Koymans has been declared ‘best mathematics PhD student’ by the Royal Dutch Mathematical Society (KWG). He received the prize at the Dutch Mathematical Congress (NMC) on 24 and 25 April. With his talk about his research into the Cohen-Lenstra conjecture, Koymans left eleven other candidates behind him.

## Different disciplines

Koymans studied mathematics and computer science at Eindhoven University of Technology. He came to Leiden for his master's degree to further deepen his knowledge of theoretical mathematics, in particular number theory. Here he received a PhD position under the supervision of Jan-Hendrik Evertse and Peter Stevenhagen. 'I started my research in the diophantine approximation (see box), but later became more and more interested in the Cohen-Lenstra conjecture. 'These are actually two completely different disciplines in number theory, but both are very interesting.' Koymans is grateful to his promoters: 'They helped me a lot in the preparation of this presentation'.

### Diophantic approximation

Koymans: 'The mathematician Diophantus of Alexandria lived in the third century. He was particularly interested in solving equations in which all variables are integers. The best known such equation is "Fermat's last conjecture" which deals with whole solutions of X^n + Y^n = Z^n. Diophantine approximation is about approximating real numbers through fractions, and the main application is to solve Diophantine equations.'

## Factoring

In his current PhD research, Koymans is therefore concerned with the Cohen-Lenstra conjecture. He explains that one of the most important mathematical theorems describes that each positive integer can be written as a product of prime numbers. This is called the factoring of a number. Koymans: 'Mathematicians are also very interested in the factorization properties of more general structures, called "number rings". This has applications in geometry, number theory and cryptography. The Cohen-Lenstra conjectures predict the factorization properties of these number rings.'

## Progress at last

A total of twelve PhD students was in the running for the prize. During the congress, the four best candidates were allowed to give a ten-minute talk to the mathematicians present. The jury led by Gerald van der Geer (University of Amsterdam) chose Koymans as the winner.

In his talk, entitled 'Factoring in number rings', Koymans went deeper into his research on the Cohen-Lenstra conjecture. These conjectures were originally formulated in 1984', he says. 'Since then, very little progress has actually been made in this field. Together with fellow PhD students Djordjo Milovic and Carlo Pagano, I have been able to prove a number of these conjectures. The jury therefore specifically referred to 'solving longstanding open problems' in their commentary. It was a lot of fun to win! Koymans wins a trophy and an amount of 1000 euros. I will probably use it to go out to dinner a few times. The rest will go to my savings account.'

More information about the NMC: https://nmc2019.nl/

More information about the KWG prize: https://www.wiskgenoot.nl/index.php?page=28&sid=2