Abstract: There are many natural sequences of moduli spaces in algebraic geometry whose homology approaches a "limit",
despite the fact that the spaces themselves have growing dimension. If these moduli spaces are defined over a field K,
this limiting homology carries an extra structure -- an action of the Galois group of K -- which is arithmetically interesting.
In joint work with Feng and Galatius, we compute this action (or rather a slight variant) in the case
of the moduli space of abelian varieties. I will explain the answer and why I find it interesting.
No familiarity with abelian varieties will be assumed -- I will emphasize topology over algebraic geometry.