The polaron describes a charged particle moving in a polarizable medium. Due to the interaction with the medium, the particle appears to be heavier that it would be outside the medium. This new apparent mass is called the effective mass of the polaron. In 1948, Landau and Pekar argued that the effective mass of the polaron at strong coupling should behave as the fourth power of the coupling constant. To show this mathematically has been quite a prominent open problem until recently. In the last five years, there has been tremendous progress, and the problem is now almost solved. The mathematical tools for this progress are mostly probabilistic, via the path integral formulation. In this talk, I will introduce the polaron problem mathematically, and give an overview over the methods that have made its solution possible.
