BME Fractal Geometry Seminar

The study of orthogonal projections of sets and measures is an important topic in geometric measure theory. About two decades ago, Falconer and Howroyd established the projection theorems for the packing, and upper and lower box dimensions. They showed that for every Borel set \(A\) in \(\mathbb{R}^d\), each of the packing, upper box and lower box dimensions of the projection \(P_V(A)\) of \(A\) takes a constant value, which can be expressed as a certain dimension profile, for almost all \(k\)-dimensional subspaces \(V\). However, these dimension profiles are defined indirectly and very difficult to compute. Recently, we have succeeded in obtaining the precise values of these dimension profiles for homogeneous Gatzouras-Lalley carpets, which exhibit remarkable phase transitions. The talk is based on joint work with Dejun Feng and Károly Simon.

In this talk, I will survey some of my results related to unrectifiability. These include irregular/purely unrectifiable 1-sets on the graphs of continuous functions like the Takagi, the Weierstrass-Cellerier and the typical (in the sense of Baire) continuous function. Moreover, I will also discuss the fact that the fractal percolation is almost surely purely α-unrectifiable for all α > α0. This latter result is from a joint paper with Esa Järvenpää, Maarit Järvenpää, Tamás Keleti and Tuomas Pöyhtäri. This lecture is based on my slides prepared for the Second Winter School in Geometric Measure Theory: Rectifiability vs. Pure Unrectifiability, Westlake University, Hangzhou, China.

TBA