The types of spaces where data resides - graphs, meshes, grids, manifolds - are becoming increasingly varied and heterogeneous. Therefore, translating ideas, models, and theoretical results between different domains is becoming more and more challenging. Nonetheless, two fundamental principles unite all these settings. The first states that data is localised, meaning that data is associated with some regions of the underlying space. The second says that data is relational, and this relational structure reflects how the various regions of the space overlap.

It is natural to formalise these axioms using algebraic topology. The "space'' in question is a topological space - a set with a neighbourhood structure - and the data attached to its neighbourhoods are algebraic objects like vector spaces. Since graphs, manifolds and everything in between is a topological space, we adopt this mathematical viewpoint to smoothly transition between domains, improve our theoretical understanding of existent models and design new ones, including for spaces that are yet to be explored in Machine Learning. Guided by this perspective, this work introduces Topological Deep Learning, a research programme studying (deep) models performing inference on data glued to a topological space.

This thesis includes four research works expanding upon the directions outlined above. The first work proposes Message Passing Simplicial Networks (MPSNs), a family of models operating on simplicial complexes, a higher-dimensional generalisation of graphs coming from algebraic topology. We study the symmetries these models must satisfy, the topological invariants that describe their behaviour, and how they can learn representations based on discrete differential forms. The second work takes this generalisation further to cell complexes, a class of spaces that also subsume simplicial complexes. We show their additional flexibility benefits molecular applications, where the resulting models outperform prior art on molecular property prediction tasks. The third work proposes a general topological framework for constructing graph coarsening (aka pooling) operators in a way that naturally generalises existing pooling approaches in computer vision. We show that this framework can be used to construct graph-based hierarchical models and visualise attributed graphs. Finally, the last work introduces a new perspective on graph models based on sheaf theory, a subfield of algebraic topology. Sheaves, which are mathematical data structures that naturally store the data attached to a topological space and its relational structure, faithfully realise the axiomatic principles of Topological Deep Learning. We show that sheaf structures on graphs are intimately connected with the asymptotic behaviour of message passing graph models and exploit these connections to design new sheaf-based convolutional architectures. We demonstrate that these models can cope with the challenges of oversmoothing and heterophilic graphs, which affect many existent graph models.

Overall, this thesis introduces a novel topological perspective on deep learning for structured data, whose ramifications establish many new connections with algebraic topology.