Terry Harris
University of Wisconsin
Title
Some continuum incidence problems connected to Fourier analysis and wave equations
Abstract
In the first part of this talk, I will give a high-level overview of some continuum incidence problems from fractal geometry for which progress has been made through Fourier analysis. These problems include the Hausdorff dimension of projections of fractal sets, intersections of fractal sets with planes, and the dual versions of such problems on the Hausdorff dimension of sets containing many lines or curves. An example (due to T. Wolff) is that a planar set containing a circle of every possible radius must have maximal Hausdorff dimension (i.e. 2). No familiarity with technical terms (e.g. Hausdorff dimension) will be assumed.
In the second part of the talk, I will discuss some recent joint work on the circular Furstenberg problem. This problem asks the question of how "large'' a planar set must be if it contains "large'' subsets of "many'' circles (in a way that will be made precise during the talk). This is based on joint with J. Green, Y. Ou, K. Ren, and S. Tammen.