Title : The Allen-Cahn equation and minimal surfaces

Abstract : The Allen-Cahn equation is a semilinear elliptic PDE which arose in the modelling of phase transitions in materials science. Solutions to the Allen-Cahn equation are critical points of an energy functional and are closely related to minimal surfaces and sets of finite perimeter. In recent years, geometric analysts have exploited this relationship to obtain new results about the existence and geometry of minimal surfaces in Riemannian manifolds. In this talk, I'll survey the basics of this connection, discussing the properties of solutions to Allen-Cahn, examples of solutions and methods for constructing solutions, the results linking the Allen-Cahn equation to minimal surface theory, and recent developments and open problems