New Ways to Compare Geometric Objects

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

Topology is a relatively new area of mathematics, but can be said to originate from the famous puzzle about the seven bridges of Königsberg: was it possible to walk through the city without passing the same bridge more than once? The negative answer, which came as early as 1736, led to new mathematical developments.

Central objects of study in topology include manifolds, geometrical objects that have important applications in physics and other natural sciences, as well as pure mathematics. The aim of the current project is to apply methods from algebraic topology to studying the symmetries of manifolds, leading to better ways of classifying manifolds.

Such classification has been a driving force for the development of algebraic topology in the 20th century. The underlying ideas revolved around understanding the properties of manifolds when they are deformed but not torn apart. If such deformation can transform one manifold into the other, the manifolds are said to belong in the same class. In how many different ways can this be done? The manifolds’ symmetries may answer this question.

The challenge is that studying symmetries is very complicated, even for easily classifiable two-dimensional surfaces. The aim of the project is to introduce new methods for analyzing manifold symmetries in higher dimensions. Some progress has already been made, but it encompasses only one special class of manifolds, so the goal is to extend the results to the general case.