In this talk, I will discuss recent work of Robert Bryant and Phillip Griffiths in which exterior differential systems are used to study the existence and geometry of conservation laws for PDE systems in general, and in particular for systems consisting of a single, second-order parabolic equation for one function of two independent variables. Bryant and Griffiths were able to give a complete analysis of conservation laws for such equations, and in particular, they showed that any such equation with at least four independent conservation laws is locally equivalent to a linear equation.
In my work, I consider second-order parabolic equations for one function of three independent variables. Here things turn out to be more complicated; for example, I will give an example of a non-linearizable equation which has an infinite-dimensional space of conservation laws. I will describe my progress toward classifying those equations of this type which have conservation laws and, if time permits, I will discuss some of the techniques involved.