The Allen-Cahn equation is a fundamental model for phase transitions, originally in alloys between metals. It is governed by a separation parameter; in the limiting case the phases separate completely, and the interface between them may be expected to be of minimal area ("minimal surface").

We will discuss recent progress in the understanding of stable solutions (i.e. those "observable in nature") to the Allen-Cahn equation. The focus will be on the connections (and lack thereof) with minimal surface theory and two influential conjectures of De Giorgi and Yau.