Mathematics Colloquium: Rectifiability and density of measures

Speaker:  Jeremy Tyson, University of Illinois and NSF

Title:  Rectifiability and density of measures

Abstract:  Rectifiable sets are the analogs of smooth submanifolds within geometric measure theory. 1-rectifiable sets in the plane were originally studied by Besicovitch in the 1920s under the name `regular set’. A detailed study of rectifiable sets in Euclidean spaces of all dimensions was undertaken in the mid-20th century by Federer, who also introduced the modern terminology. Euclidean rectifiable sets admit a wealth of equivalent characterizations, each of which captures an infinitesimal notion of regularity with a measure-theoretic flavor. One of the most intriguing of these characterizations involves the almost everywhere existence of a suitable measure density. The full characterization of Euclidean rectifiability via the density of measures is a celebrated 1987 theorem of David Preiss. Preiss’ proof introduced a wealth of new tools and a novel perspective on the tangential structure of Radon measures in Euclidean space. In the modern era, rectifiability has been fruitfully extended to nonsmooth environments, including sub-Riemannian manifolds, metric spaces, and infinite-dimensional Banach spaces. This talk will survey the history of the subject, highlighting some of the important applications across analysis and geometry, and culminating in recent work on the relationship between rectifiability and measure density in the sub-Riemannian Heisenberg group. No prior background in geometric measure theory will be assumed.

Time:  Friday, November 18, 2022, 3:30pm – 4:20pm

Place:  Exploratory Hall, room 4106

Zoom and In-person