Georgia Institute of Technology College of Sciences

Fractal sets arise naturally when classical notions of length, area, and smoothness are no longer adequate. A central theme in geometric measure theory is to understand how the dimension and geometry of a fractal set are reflected in its projections. Even some of the simplest questions of this type lead to surprisingly difficult problems connecting geometric measure theory with harmonic analysis, combinatorics, and number theory.

One classical example is the Favard length problem, also known as Buffon's needle problem. Given a planar set, its Favard length is the average length of its orthogonal projections. A theorem of Besicovitch implies that a purely unrectifiable set of finite length has Favard length zero. For self-similar sets obtained as limits of finite Cantor constructions, however, a much more quantitative question remains: how quickly does the Favard length of the finite approximations tend to zero? Despite the simple formulation, determining the correct decay rate remains a major open problem.

I will describe the geometry behind the classical problem, some of the main ideas that have led to progress, and what happens when linear projections are replaced by nonlinear families. This leads naturally to a nonlinear analogue of Buffon's needle problem—a Buffon circle problem—and to new questions about unions of circles, nonlinear projections, and the geometry of fractal sets. I will discuss recent results in this direction and the new phenomena that appear when lines are replaced by curves.