This manuscript forms part of my “Habilitation à Diriger des Recherches” (HDR) and provides a synthetic, contextualized overview of my recent research. Rather than presenting new findings, the work focuses on organizing and reflecting on my contributions over the past several years.
At its core, the manuscript is concerned with topological spaces – especially manifolds – and their homotopical properties. The investigation of these spaces is carried out using algebraic tools, including familiar structures such as groups and rings, as well as more specialized concepts like operads and their algebras. While the foundation of this manuscript is in topology, it also draws on ideas from homological algebra, category theory, and mathematical physics.
Much of the research presented here is motivated by a desire to understand configuration spaces: collections of distinct points within a given space. Although their definition is simple, configuration spaces exhibit intricate algebraic structures that become apparent when one attempts to compute their invariants. Understanding these structures is a central theme of the manuscript.