A quantification of the Besicovitch projection theorem and its generalizations

Abstract: The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a randomly dropped line intersects such a set E is equal to zero. This shows us that the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. This leads us to discuss generalized Besicovitch theorems and the ways in which we can quantify such results by building upon the work of Tao, Volberg, and others. This talk covers joint work with Laura Cladek and Krystal Taylor.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.