Loop Spaces – a new way to solve old problems 

Kaj Börjeson will present his doctoral thesis in mathematics at Stockholm University in 2017. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Nathalie Wahl at the Department of Mathematical Sciences, University of Copenhagen, Denmark. 

His research project consists of two parts, each of which studies algebraic structures; these structures can help describe spaces, which are difficult to visualize. The project’s main theme revolves around algebraic methods that were originally developed for computations in quantum field theories. 

The first part deals with free loop spaces. Such spaces can be formed from all closed curves in a geometric space. An example of such a space could be rubber bands stretched around a sphere. In a free loop space, these bands can be placed in any direction on the surface of the sphere.  

Infinite dimensional loop spaces can be studied with help of algebraic topology, a branch of mathematics with applications spanning from physics and computer science to health sciences and economics. Algebraic topology can be used to describe particle motions in quantum mechanics, as well as to create three-dimensional images in computer tomography. 

The goal of the second part of the project is to find general properties of special functions, differential operators, which are invariant even under a change of coordinate system. Studying phenomena independent of the choices made in advance, such as pre-chosen coordinate systems, is of great interest to both mathematicians and physicists. 

In summary, the goal of the research project is to create and study new algebraic structures and use them to solve outstanding problems in topology, geometry, and physics

Photo: Stockholm University