Families of groups such as symmetric groups or general linear groups, and
families of spaces such as con guration spaces or certain moduli spaces of manifolds,
display a stability phenomenon in their homology. I'll describe this phenomenon and give an idea about how such theorems are proved.

Reading List:

First three sections of
 O. Randall-Willliams and N. Wahl, \Homological stability for autormorphism
groups", Advances in Mathematics, Vol. 318 (2017), pp. 534 - 626.

Symmetric groups, braid groups, certain mapping class groups and general linear groups are examples of families of groups known to display a stability phenomenon in their homology. In my talks, I will give an answer to the following questions: What do these examples have in common? When should one expect that a family of groups satisfies homological stability? and how does one check that such a family does indeed stabilize