**AWM Emmy Noether Lecture**

January 2012

Boston, MA

**Abstract.** Emmy Noether's
famous theorem connects “conservation laws" with symmetries, and so
perhaps the first thing that a speaker should do in a talk bearing her name is
to explain that the relation between that theorem and the research area of
hyperbolic conservation laws is not very close. That done, I will describe the
main features of conservation law theory, beginning with the role of weak
solutions (which break symmetry), and elements of the theory in a single space
variable, which is now in a reasonably satisfactory state, and concluding with
an outline of the current state of the theory for multidimensional conservation
laws. Here there are a few results, most of them very recent.