This week we are lucky to have Nalini Joshi giving the SUMS talk. Pizza and drinks available afterwards. Abstract: Nonlinear integrable systems arise in applied mathematics and many areas of modern physics, including models of quantum gravity. Recently, it has been discovered that integrable systems (such as the Painleve equations) arise from symmetry groups and translations on lattices. These are more familiar to experts in algebra but I will explain, through examples, how this amazing development gives rise to properties that all applied mathematicians should be able to master. The results also turn out to be interesting for algebraists.