Singularity theory of plane curves and its applications

We review the classification of singularities of smooth functions from the perspective of applications in the physical sciences, restricting ourselves to functions of a real parameter t onto the plane (x,y). Singularities arise when the derivatives of x and y with respect to the parameter vanish. Near singularities the curves have a universal unfolding, described by a finite number of parameters. We emphasize the scaling properties near singularities, characterized by similarity exponents, as well as scaling functions, which describe the shape. We discuss how singularity theory can be used to find and/or classify singularities found in science and engineering, in particular as described by partial differential equations (PDE's). In the process, we point to limitations of the method, and indicate directions of future work.