Understanding singularities in the Stefan problem

Partial Differential Equations (PDE) are a type of mathematical equations that are used in essentially all sciences and engineering. They are the language in which most physical laws are written.From the mathematical point of view, the most fundamental question in this context is to understand whether solutions to a given PDE may (or may not) develop singularities. For example, in the case of the PDEs that describe fluid mechanics, this is one of the Millenium Prize Problems in mathematics.During the last decades, there has been an increasing interest in understanding PDE problems that involve moving interfaces, such as ice melting to water. The oldest and most important example is the Stefan problem, a PDE which describes mathematically the melting of ice, and was first introduced in 1831. In this context, singularities do appear, but their understanding was still quite limited. In a recent groundbreaking work (J. Amer. Math. Soc. 2024), we have proved that, while singularities may appear, they are actually extremely rare. The precise result (whose proof is more than 80 pages long and had to develop new techniques in Geometric Measure Theory) completely solved a long-standing conjecture that had been open since the introduction of this model in the 19th century. It is the best result for the Stefan problem since the famous work of Caffarelli (Acta Math. 1977), one of the main results for which he got the Abel Prize.