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.”