Jonas Azzam
SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Jonas Azzam Abstract. Rectifiable sets are measure theoretic generalizations of smooth manifolds. Rather than being parametrized locally by smooth chart-maps, these sets are parametrized on measurable subsets by "almost" differentiable maps. These arise in subjects like complex analysis and singular integrals, as some classical results originally shown for smooth sets also hold for the larger class of rectifiable sets. In some applications, more quantitative information about the multiscale behavior of a rectifiable set is needed. For example, while a C^{1}-curve (which is rectifiable) looks flatter and flatter flat as we zoom in on any point, the "Analyst's Traveling Salesman Theorem" of Peter Jones quantifies how often such a curve is not approximately flat. In fact, he characterizes exactly when an arbitrary set may be contained in a curve of finite length. In this talk, I will give an introduction to the study of (quantitative) rectifiability, its applications, and some recent higher dimensional generalizations of Jones' theorem. |