Utrecht University gets ERC Consolidator Grant

With a European Consolidator Grant of nearly two million euros, Martijn Kool, a mathematician at Utrecht University, will investigate a completely unexplored area in algebraic geometry: counting surfaces on Calabi-Yau fourfolds. It is very fundamental research that is valuable for the development of mathematics, but also for other areas, such as string theory.


The circle in purple presents a solution to Apollonius’s Problem (source: wikipedia)
The research of Associate Professor Martijn Kool is part of enumerative geometry. This is a subfield of mathematics which essentially involves the description of shapes: algebraic geometry. Enumerative geometry was already practiced by the Ancient Greeks, and revolves around counting solutions to geometric questions. A well-known example of such a geometric question is Apollonius’s Problem: If you draw three random circles in a plane, how many circles can you then draw that touch all three circles? Apollonius’s Problem may seem abstract, but it plays an important role in current GPS techniques.

Logical sequel
Kool will work on such geometric issues for Calabi-Yau fourfolds. These are special, four-dimensional objects. It is a logical sequel to research on counting curves on Calabi-Yau threefolds, which is ongoing but has already borne fruit. “It is a truly successful story,” Kool says. “It contributed to the breakthrough of string theory and the discovery of mirror symmetry.”

I find it fascinating to do research at the edge of what we know

Martijn Kool, Associate Professor
Millenium Prize Problems
Martijn Kool
Martijn Kool
In enumerative geometry, the limit of knowledge starts at fourfolds. What comes next is completely open. Coincidently, physics research in the field of string theory is also partly about fourfolds. “My research will not answer specific questions about string theory directly, but it will provide tools to move forward,” Kool said. “Through fourfolds, there is a deep interaction between the two areas of research.” And even in mathematics itself, researchers get stuck on fourfolds, for example in the case of the Hodge Conjecture. This conjecture is one of the seven renowned Millenium Prize problems set in the year 2000, which were never solved. The number two on the list, Poincaré’s Conjecture, has since been unraveled. But the remaining six are still shrouded in mystery twenty-three years later.

Surprising connections
Kool likes the fact that his research has connections to other areas. “It’s very fundamental,” he says. “But I expect this project to create several new bridges between yet unrelated fields.” Incidentally, those unexpected connections are by no means his only motivation, and it is equally important to him that enumerative geometry continues to be developed. “I find it fascinating to do research at the edge of what we know, where the new geometry begins,” he says. “I can lose myself completely in this abstract world.”