Research Interests

In topological data analysis, our data is often presented as a filtered shape. Morse theory describes how additions of new elements to the shape in the filtration induces topological changes; in particular, it distinguishes elements that are critical to the filtration from those that induce no change.  This information is presented as a Morse complex, which summarises the filtration without loss of topological fidelity. I am interested in the following aspects of Morse theory in TDA:

• Algebra and Computation.  How do we compute the optimal Morse complex? We show in this pre-print how this can be done algebraically and algorithmically. Joint work with Álvaro Torras Casas and Ulrich Pennig. 

• Generalisation to non-smooth Morse Theory. We show in this article how classical Morse lemmas generalise to distance functions to embedded surfaces, which differentiability fails at topological critical points. We also show how geometric configurations in surfaces induce topological critical points. Joint work with Anna Song and Anthea Monod.

• Reconstruction of Morse indices from finite samples. Given samples of a smooth function and its domain, can we infer the Morse (Conley) indices of its critical sets? We show in this article that finitely many local samples near critical sets suffices to reconstruct index pairs up to homotopy for computation of Morse indices. Joint work with Vidit Nanda. 

While we have an abundance of data in the form of graphs (social networks, molecules...), they present a unique data science challenge as it is computationally intensive to determine whether two graphs are isomorphic. As such, graphs that belong to different classes can be mis-classified to be in the same class, as our classification algorithms fail to distinguish them. I focus on using spectral features of graphs to enhance descriptive powers of graph classification or regression algorithms.

• Stable spectral features. In this pre-print, we show how spectral features of graphs can be packaged to  yield vertex features that are stable with respect to perturbations, and incorporate both eigenvalue and eigenvector information. Adding these features to graph neural networks can significantly improve their performance on benchmark graph regression problems. Joint work with Karamatou Yacoubou Djima. 

• Optimising Persistent Homology of Wavelets Signatures. In this article, we show how wavelet signatures of graphs can be optimised via back-propagation, so that the persistent homology of the filtered graph along the signature is a more informative graph feature. Joint work with Jacob Leygonie. 

Data can be presented on domains with non-trivial topology, such as tori, Klein bottles, and projective spaces. I am interested in developing data analysis methods that take the underlying topology of these domains into account. 

• Ambient vs local cycles. Persistent cycles in point samples can be generated by the ambient topology of the manifold. In this pre-print, we discuss how we can distinguish these `ambient cycles' from cycles that do not account for global topological features of the underlying manifold, provided the universal covering of the manifold is known.

• Generalised Klein Bottles. In this article, we define higher dimensional versions of Klein bottles by consider a wider family of symmetries on Euclidean space. We also provide means of parametrising functions and vector fields, which sets up the foundation for statistical methods on such spaces. 

To what extent can we recover properties of an embedded shape in Euclidean space? This is an ongoing topic of research. 

• Homology recovery. In our article on Conley index estimation, we also showed that the homology of subsets with positive reach (such as compact submanifolds) can be recovered with high confidence, given a dense enough sample relative to its reach. This generalises the classic result of Niyogi, Smale, and Weinberger. We also prove that nice intersections of sublevel sets of smooth functions on embedded manifolds have positive reach. 

• Approximate isometries of point clouds. In our pre-print on stable spectral features, we prove how these features can be used to cluster points that are related by approximate isomtries of the point cloud, using the spectral features obtained from diffusion map.

• Dimension estimation [Ongoing]. The dimension of a space is a topological invariant. In this ongoing work with James Binnie, John Harvey, Paweł Dłotko, and Jakub Malinowski, we perform a critical analysis of existing dimension estimation methods, and in particular evaluate the performance of dimensions estimators arising from persistent homology and magnitude.

• Characterising embeddings of S1 [Ongoing]. With Otto Sumray and James Binnie, we are investigating how embeddings of S1 can be characterised with a novel adaptation of persistent homology.

• Curvature estimation [Ongoing]. With John Harvey, we are investigating how topological features recovered from samples on manifolds can reliably estimate curvature.