**Rohit Nagpal**

University of Chicago, USA

July 6, 2017

**Representation stability and FI-modules**:
We often encounter sequences of representations of a family of groups. For example, the cohomology of ordered configurations of $n$ distinct points on a manifold is a representation of the symmetric group $S_n$ Similarly, the homology of the congruence subgroup of level $m$ inside $GL_n(Z)$ is a representation of $GL_n(Z/mZ)$. As $n$ grows to infinity the two examples
above become, in a sense, stable as representations. Stable representations can be thought of as finitely generated objects in a
suitable functor category. This point of view is due to
Church-Ellenberg-Farb who introduced and studied such a category called
FI-modules. We provide an introduction to FI-modules and explain what it entails about the examples above.