| Résume | Many physical processes or phenomena — such as eroded coastlines, snowflakes, fluid turbulence, human brain networks or blood vessels — exhibit self-similarity, characterised by the repetition of similar patterns across different scales of resolution. A measure of the complexity of a self-similar phenomenon is its fractal dimension, which extends the topological notion of dimension beyond integers. Estimating the fractal dimension of data is an important and difficult problem in data analysis, with applications ranging from image compression to cancer detection.
In the past 20 years methods from the field of Topological Data Analysis (in particular, persistent Betti numbers and persistent homology), have successfully been used to develop estimations of fractal dimensions for metric spaces that have been shown to outperform existing methods. At the same time, in a distinct line of work, magnitude, a cardinality-like invariant of metric spaces introduced by Tom Leinster around 2010, has shown to be related to the Minkowski dimension of compact subsets of Euclidean space. More recently, through a connection established between the two fields in my work, there have been new attempts to estimate fractal dimensions through the notion of persistent magnitude introduced by Govc-Hepworth.
In this talk, I will first give an overview of the different approaches in TDA and magnitude theory for estimating fractal dimensions. I will then discuss the connection between the two fields and introduce persistent magnitude dimensions. Finally, I will explain how these methods have the potential of helping address problems difficult to solve with more traditional approaches, such as the definition and estimation of fractal dimensions of relational models of data given by networks, hypergraphs or simplicial complexes.
The talk is partly based on joint work with Sara Kališnik, Miguel O’Malley, as well as Rayna Andreeva, Haydée Contreras-Peruyero, Sanjukta Krishnagopal, Maria Antonietta Pascali, Elizabeth Thompson. |
