Math 595 Section GMT (Geometric Measure Theory)
Geometric measure theory considers the structure of Borel sets and Borel measures in metric spaces. In addition to its intrinsic interest, it has been a valuable tool for problems arising from real and complex analysis, harmonic analysis, PDE, and other fields. For instance, rectifiability criteria and metric curvature conditions played a key role in Tolsa's resolution of the longstanding Painlevé problem on removable sets for bounded analytic functions.
The classical setting for geometric measure theory is finite-dimensional Euclidean spaces. Contemporary trends in analysis and geometry in metric measure spaces, including analysis on fractals, motivate extensions of the subject to non-Riemannian and nonsmooth spaces. Sub-Riemannian spaces, particularly, the sub-Riemannian Heisenberg group, are an important testing ground and model for the general theory.
In the first part of the course we will give an extended review of Euclidean geometric measure theory. Major topics to be covered include Hausdorff measure and dimension, density theorems, energy and capacity methods, almost sure dimension distortion theorems, and tangent measures. Rectifiable sets and measures provide a rich measure-theoretic generalization of smooth differential submanifolds and their volume measures. We will give a short introduction to this important topic.
In the second part of the course we will discuss ongoing efforts to extend this theory into non-Riemannian and general metric spaces. Motivation for such efforts will be indicated. We will emphasize notions of rectifiability and almost sure dimension distortion theorems in sub-Riemannian spaces.