Pure Maths Colloquium: Ariel Rapaport

A compact $K\subset\mathbb{R}^{2}$ is called self-affine if there exist affine contractions $\varphi_{1},...,\varphi_{m}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ such that $K=\cup_{i=1}^{m}\varphi_{i}(K)$. In the 1980s, Kenneth Falconer introduced a value $s$, called the affinity dimension, which is the ‘‘expected’’ value for the dimension of $K$. I will discuss a recent project with Mike Hochman, building on our joint work with Balázs Bárány, in which we establish the equality $\dim K=s$ under mild assumptions.