I gave this talk on Monday 29th June 2020, as the inaugural installment of the Mathematics Graduate Seminar series at Victoria University of Wellington.
Abstract: Many fundamental results in Ramsey theory concern the structure of certain semigroups (sets with an associative binary operation). These include the Hales–Jewett theorem, the Graham–Rothschild theorem, Gowers’ \mathrm{FIN}_k theorem, and Hindman’s theorem. In this talk, we will discuss and aim to understand these fundamental results. Time permitting, we will see a common generalisation of many of these theorems to the setting of arbitrary layered semigroups, as introduced by Farah, Hindman and McLeod, and further generalised in ongoing work of Barrett and Lupini.