Abstract: In a Euclidean space, a (crystallographic) root system is special set of nonzero vectors, called roots. Each root determines a reflection in the Euclidean space, and these reflections generate a finite group, called the Weyl group. I will show some interesting facts about root systems and Weyl groups. In particular, I will talk about classification of root systems, and group actions of (extended) Weyl groups on the Euclidean space.

Abstract: Given a foliated smooth manifold, you'd like to do analysis/topology on the space of leaves. Bad news: this quotient is poorly-behaved! Instead, you can work with a smooth proxy called the holonomy groupoid (which defines the same stack). Using this gadget, Connes defined things like the orientation class and fundamental class of the leaf space. Trying to do similar things with singular foliations, one encounters weird and interesting phenomena such as "continuous holonomy". We'll explore these ideas by concentrating on a special class of examples.