Comparing transformations of topological spaces

Louis Hainaut will receive his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with professor Benson Farb at the University of Chicago.

Algebraic topology uses algebra to measure topological properties and has proven to be effective in classifying topological objects by topological invariants. Invariants are quantities that do not change when topological objects undergo continuous deformation without breaking, such as stretching or twisting. Topological objects are the spaces where geometry takes place and are blind to quantitative geometry like angles, lengths, areas, and volumes. 

A basic topological invariant is homology. The idea of homology was developed for the analysis of the holes in geometric objects. It originates from Euler's polyhedron formula, which was named after the Swiss mathematician Leonhard Euler. In the mid-eighteenth century, he proved that the sum of the number of vertices and sides of a polyhedron, such as a cube or a tetrahedron, is always equal to the number of edges plus two. 

The concept of homology developed over the next few centuries, eventually proving to have extremely broad applications in mathematics. Of particular interest for the planned project are configuration spaces and studies of the relationship between transformations of topological spaces to themselves and transformations of their related homology groups.

Photo: Dan Petersen