H.B. Keller Colloquium

An important and fascinating feature of many large, complex systems arising in mathematics, physics, and elsewhere is that they undergo phase transitions. Mathematically, phase transitions typically manifest themselves in the following way: Suppose that we have some system with a large number of individual components, possibly arranged according to some spatial structure. Each of these components interacts with its neighbours in some way that is governed by some parameters (e.g. the temperature) which we are free to vary continuously. A phase transition occurs when varying this parameter by a small amount through some special value, known as the critical value, leads to a stark, qualitative change to the behavior of the system on a macroscopic level. While we are all familiar with the solid/liquid/gas phase transitions from our everyday lives, phase transitions also occur in many other large systems that have nothing to do with physics, such as in the formation of traffic jams, in the average-case computational complexity of optimization problems, and in the (recently very topical) study of how a novel disease will spread through a population. In each case, understanding when, how, and why the system undergoes a phase transition is of central importance in both theory and practice. Moreover, the basic mathematical principles underlying the occurrence of such phase transitions have much in common across these diverse situations, and the study of phase transitions has come to be recognised as a rich source of deep and beautiful pure mathematics that is of interest beyond and complementary to its practical origins.

The simplest models of phase transitions are mean-field, meaning that every particle interacts in the same way with every other particle – there is no geometry. In this talk, I will give an introductory overview of what happens when one introduces geometry to these models, and in particular to the dimension-dependence of critical phenomena. Time permitting, I will also briefly describe some recent progress on these questions.