The CIM Colloquium is a Mathematical Sciences event that takes place once a year, right before the annual council of the CIM Associates. This year the lecture will be delivered by Ricardo Campos (University of Toulouse), at 2 pm on April 2, 2026, in hybrid format, with the in-person talk held in Sala Sousa Pinto, Departamento de Matemática da Universidade de Aveiro.
The topology of configuration spaces of points
Abstract:
How can we place one space inside another without it intersecting itself? This fundamental question lies at the heart of the study of embedding spaces and generalises familiar concepts like knot theory. Modern topology has revealed that to understand such embedding spaces, we must first understand a seemingly simpler scenario: the movement of a finite number of distinct points on a manifold. This leads us to the study of configuration spaces.
In this talk, we will explore the rich topological landscape of configuration spaces from the perspective of homotopy theory. While the geometric premise of "particles moving without colliding" is very intuitive, the resulting spaces are notoriously difficult to study. We will discuss the inherent topological problems that arise, primarily driven by the intricate nature of the "collision locus" where points approach one another.
Finally, after outlining these geometric challenges, we will transition to modern topological solutions. We will see how, by studying these spaces through the lens of rational homotopy theory, we can construct algebraic models that allow us to encode and compute their topological properties.