In classical statistical physics, a phase transition is understood by studying the geometry (the zero-set) of an associated polynomial (the partition function). In this thesis, we will show that one can exploit this notion of phase transitions algorithmically, and conversely exploit the analysis of algorithms to understand phase transitions.

As applications, we give efficient deterministic approximation algorithms (FPTAS) for counting $q$-colorings, and for computing the partition function of the Ising model.