Topological data analysis

Topological data analysis captures connectivity, loops and multiscale organization that can be obscured by assuming data occupy ordinary Euclidean space. The workshop connects this mathematical viewpoint with the need for interpretable, robust learning systems. Mathematicians, computer scientists and researchers in chemistry, biology and materials science will examine theory, computational methods and scientific applications. Discussions link mathematical frameworks, efficient algorithms and practical datasets, with the goal of developing reliable, scalable methods and connecting early-career

Michael Hill explains how topological data analysis extracts structural information from large datasets. Ideas about connectedness, holes and shape, traditionally associated with pure mathematics, provide tools for recognising patterns as the scale of analysis changes. The discussion explores applications including breast-cancer research and questions about fair voting.

Heather Harrington explains topological data analysis and the construction of barcodes that summarize the shape of a dataset across scales. She describes how these tools reveal structure in complex information and how they can support studies of proteins and diseases such as cancer. The interview was recorded at the European Congress of Mathematics.

…(SCH) and healthy controls (HC).We use topological data analysis methods, namely persistent homology using the weight rank clique filtration on brain connectivity matrices obtained from a probabilistic fibre-tracking algorithm run on the Centre for Biomedical Research Excellence (COBRE) dataset (N = 44 SCH, N = 44 HC) to investigate group differences. We show that some of the connectivity structures differ in strength re

…Here, we apply topological data analysis to simultaneous recordings of hundreds of grid cells and show that joint activity of grid cells within a module lies on a toroidal manifold. Each position of the animal in its physical environment corresponds to a single location on the torus, and each grid cell is preferentially active within a single “field” on the torus. Toroidal firing positions p