Analysis Seminar: Nonlinear Projections and Quantitative Rectifiability

Abstract: From the delicate geometry found in a snowflake to the intricate patterns of a coastal shoreline, nature holds infinite patterns and scales. The world is not easily described using mere lines and cones, and classic Euclidean geometry falls short. The notion of fractals gives us a language and a set of tools to understand more complex phenomena. The modern application of fractals spans both pure and applied mathematics - from the study of lung vasculature to surprising constructions of counter examples.

There are many classical results relating the geometry, dimension, and measure of a fractal set to the structure of its orthogonal projections. Among them, the Favard length problem, also known as Buffon’s needle problem (after Count Buffon), concerns the average length of the linear projections of a subset of the plane. This fascinating and difficult problem lies in the intersection of harmonic analysis, combinatorics, and number theory. In more detail, let $K = \bigcap_{n=1}^\infty K_n$ be a self-similar set in the plane, constructed as a limit of Cantor iterations $K_n$. Assuming that $K$ has finite length and is purely unrectifiable (so that its intersection with any Lipschitz graph has zero length), a classic theorem of Besicovitch implies that the Favard length of $K$ vanishes. It is an open problem to determine the exact rate of decay. Substantial progress has only been achieved in recent years. In this talk, we survey these developments with emphasis on main ideas and present new developments in a nonlinear setting.

Host: Alan Chang