An introduction to effective fractal dimension by Benjamin Koch

Today's theory seminar will be given by Benjamin Koch, who will give as an introduction to effective fractal dimension. 

Abstract:

The purpose of this talk is to give an overview of effective fractal dimension, which is an application of computability theory to geometric measure theory. I will start with an overview of two of the most important notions of dimension in classical measure theory, the Hausdorff and packing dimensions. Following this will be the formulation of so-called effective dimensions for points in R^n via Kolmogorov complexity. The highlight of the study of effective dimension thus far has been the “point-to-set principle”, which connects the effective dimension of points in R^n to the classical dimension of the subsets in which they reside. I will discuss a handful of the results that have come from this connection, which include an alternate proof of the Kakeya conjecture in R^2, an improved lower bound on the Hausdorff dimension of Furstenburg sets (which generalize Kakeya sets in R^2), and a generalization of an upper bound on the Hausdorff dimension of intersections and products of subsets of R^n