Finding the correct measure for abstract geometric objects

Rolf Andreasson will receive his doctoral degree in mathematics from Chalmers University of Technology and the University of Gothenburg in 2026. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Cristiano Spotti at Aarhus University, Danmark.

Algebraic geometry is the study of varieties, which are the geometric equivalents of the solutions to algebraic equations. A simple variety is a circle with radius r, given by the solutions to the equation x²+y²=r². However, varieties can be considerably more abstract geometric objects. One way of organising them is to assign to each object a point in a moduli space, that parametrizes all objects of the same type and where similar objects are close to each other.

One tool for investigating varieties is to study their metrics – ways of measuring distance in space. One well-known metric is the Kähler–Einstein metric, which can also be used to define a natural metric on the moduli space: the Weil–Petersson metric. However, in most cases neither the Kähler–Einstein nor the Weil–Petersson metric can be described explicitly.

The project will develop a method for constructing an explicit approximation of the Weil–Petersson metric, inspired by a previous approximation of the Kähler–Einstein metric. The method originates in statistical mechanics, the study of large systems of interacting particles. As the number of particles increases, the Kähler–Einstein metric emerges from the system’s collective behaviour.

The aim is to show how the Weil–Petersson metric can also arise from the same particle system. The partition function, a fundamental concept in statistical mechanics, plays a central role. This function gives rise to a new and more explicit metric on the moduli space. When the number of particles increases, there is reason to believe that this metric approximates the Weil–Petersson metric, thus providing a new way of understanding it.

Photo: Setta Aspström