For a fixed background manifold $M$ and parameter-space $X$, the associated configuration space is the space of $n$-point subsets of $M$ with parameters drawn from $X$ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space.
It is a classical result that the sequence of unordered configuration spaces, as $n$ increases, is homologically stable: for each $k$ the degree-$k$ homology is eventually independent of $n$. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability.
The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations.
If time permits, I will also say something about homological stability with twisted coefficients.