The $C^*$ -algebras associated to Hausdorff \'{e}tale groupoids have been produced many interesting results. Allowing for non-Hausdorff groupoids introduces non-trivial obstacles to extending these results.

We survey results providing sufficient conditions for a non-Hausdorff groupoid $C^*$-algebra to be purely infinite simple and discuss work towards a necessary condition.

Based on joint work with Lisa Orloff Clark and Astrid an Huef.

Bio: Ilija Tolich is a New Zealand mathematician. He completed his PhD from the University of Otago in 2017 under the supervision of Astrd an Huef and Iain Raeburn. He is currently a Postdoctoral Fellow at Victoria University of Wellington working on non-Hausdorff groupoids.