## Talk 1: Mateusz Olechnowicz - 1:00 pm

Pure Mathematics Department, University of Waterloo

### "Group schemes"

The Mordell-Weil theorem says that if $E$ is an elliptic curve over $\mathbf{Q}$, then the abelian group $E(\mathbf{Q})$ is finitely generated. Mazur's theorem then pinpoints the torsion part of $E(\mathbf{Q})$ as one of fifteen finite groups. Central to Mazur's proof is the notion of a group scheme, which is a group object in a category of schemes. In this talk we will define group objects and give examples in various categories. We will also discuss affine groups and see how they can be realized as group schemes."

## Talk 2: Ritvik Ramkumar - 2:30 pm

Pure Mathematics Department, University of Waterloo