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.

One of the key tools to study surfaces of finite-type is the curve graph. Masur and Minsky showed that the curve graph is both infinite diameter and Gromov hyperbolic. Additionally, Masur and Minsky showed the curve graph's utility by using it to study the geometry of the mapping class group for surfaces of finite-type. Unfortunately, for surfaces of infinite-type the curve graph has diameter 2. In this talk, we introduce the grand arc graph and show that for large collections of infinite-type surfaces, the grand arc graph has infinite diameter and is Gromov hyperbolic. This work is joint with Assaf Bar-Natan.

> Solitons (or solitary waves) are waves that travel at a constant speed while maintaining their shape, due to the perfect balance of nonlinear and dispersive effects. These were first described in 1834 by J.S. Russell as he travelled on horseback beside a canal, pursuing a wave for almost two miles. However, solitons are more ubiquitous in nature from rogue waves that can overturn boats, to being responsible for transmitting signals in the brain. They are well studied in different models like the Korteweg-de Vries equation first introduced in 1877 using various methods such as inverse scattering. On the other hand machine learning techniques, in particular physics informed neural networks (PINNs) are much newer methods, only formally introduced in the last few years. They rely on the governing physical laws and employ modern optimisation techniques to find solutions to the underlying equations. In this talk, we discuss the preliminary work bridging the two fields together. We examine how well machine learning methods can capture solitons as well as their nonlinear interactions and where the new methods may come up short.