Blundon Lecture (Mathematics): Louigi Addario-Berry

Title: Random graphs and random trees
Abstract:  One of the most active areas of research in probability theory concerns phase transitions: systems that change state when a parameter crosses a certain threshold. Melting and boiling points are common real-world examples of phase transitions. The ubiquity of phase transitions in real-world systems has spurred mathematicians to try to find tractable mathematical models which provably exhibit phase transitions.
For systems possessing a phase transition, it is common to study the system's behaviour when the parameter is at or near the threshold (the so-called critical behaviour of the system). Frequently, around the threshold, fascinating self-similar or fractal structures emerge, at least conjecturally. I will give a high-level introduction to the subject, then zoom in on two settings in which it is possible to prove mathematically rigorous results: those of random graphs and random trees.