in this talk we will investigate some topological properties of the space fk(m) of ordered k-tuples of distinct points in a manifold m. for a fixed manifold m, as k increases, we might expect the topology of the configuration spaces fk(m) to become increasingly complicated. church and others showed, however, that when m is a connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. in this talk i will explain these stability patterns, and how they generalize classical notions of homological stability proved by mcduff and segal in the 1970s. i will describe higher-order ``secondary stability'' phenomena established in recent work joint with jeremy miller. the project is inspired by work of galatius--kupers--randal-williams.

the moduli space of riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. we will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. the talk will mention joint works with alex eskin, simion filip, curtis mcmullen, maryam mirzakhani, and ronen mukamel.