The key to uniting global and local properties

Associate Professor Alexander Berglund will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, Stockholm University.

Even in ancient times, people believed that the Earth was round. Locally, it seems as flat as a sheet of paper but, if you look globally, the Earth is a sphere. Many mathematical principles are about determining global properties in terms of local ones, and this is the purpose of the planned project in algebraic topology.

In general, topology is a form of geometry that only considers the shapes of the objects; from a topologist’s perspective, a triangle, a circle and a rectangle are equivalent manifolds. A central part of algebraic topology has been developed to describe manifolds and their transformations, but manifolds can be complicated, so one solution is to approximate a manifold with an algebraic model.

A simple example is using a few points to approximate a road. Instead of drawing the whole road, you draw one point on the road, then two, three, etc. If you use many points you will have a good approximation of the road. This means that if you want to study the space of roads, you can use algebraic methods to study the simpler space of many points.

When a similar method is applied to special families of higher dimensional smooth manifolds, important invariants can be defined locally in terms of how the individual manifolds in the family are curved. They also provide a global invariant for the whole family. All the global invariants for smooth families of manifolds generate a tautological ring, which thus originates in the local invariants. Tautological rings for surfaces have been extensively studied, but tautological rings for higher dimensional manifolds are, so far, a relatively unexplored area. This project is based on a new idea for approximating smooth families of manifolds, and thus explores tautological rings for the higher dimensional smooth manifolds.