The group ring of a group $G$ is characterized by a universal property: it is initial among rings containing $G$ in their groups of units. Despite this simple characterization, determining the exact unit group of a group ring is a surprisingly hard problem with implications for a wide range of mathematics via algebraic $K$-theory. To illustrate this, I'll describe how the unit groups---and thus the $K$-groups---of integral group rings control the behavior of cobordisms between high-dimensional manifolds, and the consequences this has for geometric topology. If time permits, I'll describe my own work towards understanding the $K$-theory of group rings over finite fields.

Note special day, time and place