On Saturday, I will be presenting some of my recent work, which extends Michel Lapidus’s theory of complex dimensions in \(\mathbb{R}^n\) to a larger class of homogeneous metric measure spaces. In particular, I will be discussing the complex dimensions of subsets of \(p\)-adic sets. In case they should become necessary: slides for Fractals 6.

**Abstract:** The higher dimensional theory of complex dimensions developed by Lapidus, Radunović, and Žubrinić provides a language for quantifying the oscillatory behaviour of the geometry of subsets of \(\mathbb{R}^n\). In this talk, we will describe how the theory can be extended to metric measure spaces that meet certain homogeneity conditions. We will provide examples from \(p\)-adic spaces and discuss the geometric information that can be recovered from the complex dimensions in these cases.