8 October 2019

# ‘We can use physics predictions to create new mathematics’

Shortly before the summer holiday, Martijn Kool received a Vidi grant from NWO for his research into vector bundles on curved spaces. It’s a fairly abstract subject, but he thrives in abstraction. “I’m not primarily motivated by concrete applications. My main drive is the question: does it exist mathematically?” Nevertheless, Kool connects his research to fields outside of mathematics. “For example, in Theoretical Physics you can find curved spaces near black holes, where the four-dimensional space – space and time together – is curved.”

When asked what a vector bundle on a curved space is, Martijn Kool walks to the wall of his office and draws a circle on the chalkboard. “This is an example of a curved space. Imagine that this circle is located on the floor, and that there’s a line pointing straight up from every point on the circle: that would give you a cylinder. We call that a ‘vector bundle’. You can also swist the line while going around, which makes a Möbius strip.”

## Algebraic geometry

Such shapes can often be described using algebraic geometry. “Algebraic equations are actually the simplest type of equations, like the Pythagorean theorem”, Kool explains. “I use those types of equations in geometry in order to describe curved spaces, like circles, spheres, and donuts. You can only describe a limited collection of curved spaces using algebraic equations. However, due to their simplicity, you can study them in much greater depth.”

Algebraic geometry has some surprising applications, like black holes and cryptography.

## Ancient, but still relevant

Algebraic geometry was studied by the ancient Greeks, which makes it a very old field of mathematics. “And yet it’s still extremely relevant today”, says Kool. “It’s one of the most active fields in modern mathematics, because it has some surprising applications, like black holes and cryptography.” In his research, Kool intentionally looks for connections with other fields. “One part of my Vidi project refers back to research into quantum theory from the 1990s. I want to prove certain geometric predictions that were made back then.”

Is the chalkboard on the wall behind him still essential today? “I write out almost all the finished proofs that I’ll publish using pen and paper. But we also use the computer a lot today for conducting experiments and finding patterns.”

## Stacking boxes

Another part of his Vidi project has a link to combinatorics. Kool gestures to the corner of his office. “In how many ways can you stack a number of boxes in that corner? If you only have two boxes, then there are three ways: next to each other against one wall, next to each other against the other wall, or one on top of the other. If you have four boxes, then there are 13 different ways. And how does it work if you stack boxes in more than three dimensions? That’s an old problem from combinatorics, for which researchers haven’t been able to find an elegant answer. I aim to shine a new light on this problem with my Vidi research.”

## Creating new math

How does Kool make these connections? “It depends. Sometimes I’m doing some calculations, and I come across a familiar problem. Or I purposely try to find old problems, which I attempt to solve with my current research.” Although his motivation lies mainly in the field of geometry, he actively looks for collaborations outside of his own field. “I think it’s fascinating to make those connections.” Recently, he organised a joint conference for geometers and physicists. “We don’t always understand each other, but we draw a lot of inspiration from each other’s work. We can often use their physics predictions to create new mathematics.”