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 unknown/moving interfaces, such as ice melting to water. In this context, it turns out that singularities do appear… sometimes. In a recent work with A. Figalli and J. Serra, we have proved for the first time that, while singularities may appear, they are actually extremely rare. Our precise theorem (whose proof is more than 100 pages long) completely solves a long-standing conjecture which had been open for almost half a century.
Our work has been published in Publ. Math. IHÉS, an extremely selective journal which publishes only around 10 papers every year (in all areas of mathematics).