Configuration Spaces on Trees with Loops

Given a general graph, we can construct discretized models for its n​​-point configuration space that are cubical complexes. The model constructed by A. Abrams in his 2000 PhD thesis is the most well-known, but in 2001 Świątkowski constructed a lesser-known model whose dimension stabilizes as the number of points increases. In recent work with Daniel Lütgehetmann, we have considered a Świą​tkowski-style discretized model for configurations with sinks, where multiple points are allowed to occupy certain vertices of the graph. In my talk, I will discuss these various constructions and sketch the techniques we used to prove torsion-freeness and representation stability for the homology of configurations on trees with loops.