Better approximations of abstract spaces 

Thomas Blom will receive his doctoral degree in mathematics from Stockholm University in 2023. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Jesper Grodal at the Department of Mathematical Sciences, University of Copenhagen in Denmark. 

Algebraic topology is a branch of mathematics in which abstract objects called topological spaces are studied using methods from algebra, and has many applications both inside and outside mathematics. For example, algebraic topology is used in modern physics’ string theory and in the theory of general relativity. At the same time, ideas from algebraic topology are utilised in many other areas of mathematics, such as topological data analysis, algebraic geometry and combinatorics. 

The planned project primarily deals with the theoretical side of algebraic topology. One part of the project is to further develop a method to approximate a space using simpler spaces. The intention is to advance the understanding of Goodwillie calculus, which is a technique for making such approximations that was developed by the American mathematician Tom Goodwillie in the 1990s. Within the project, Thomas Blom hopes to build upon Goodwillie calculus and further develop it in the context of equivariant homotopy theory, which is the study of spaces with symmetries.

Another part of the project is to apply Goodwillie calculus to the evasion path problem in topological data analysis, where the question is whether, in a room with movement sensors, it is possible to move from one point to another without being detected.