Mathematical Conversations

In broad terms, a phase transition is a variation in the qualitative behavior of a system under changes of some parameter. For instance, as the temperature is changed, water goes through a gaseous, a liquid, and several solid phases, each of which have drastically different physical properties. In some cases, an ordered phase appears, in which the degrees of freedom of the system is drastically reduced. For example, an ice cube can be moved as a whole and rotated, whereas a glass of water can be stirred in (near-)infinitely different ways. This is what is usually called "symmetry breaking" in this context. In this conversation, I plan to discuss some elements of our mathematical understanding of phase transitions, and bring up some of the mathematical models in which symmetry breaking phase transitions can be shown to occur. In many cases, however, symmetry breaking is quite difficult to approach mathematically, in particular when the broken symmetry is a continuous group (as in the liquid/solid transition), and I also plan on touching upon this subject.