Topological Data Analysis

My research interests lie in applying pure mathematics to develop theoretical groundwork for new methods and applications in the field of Topological Data Analysis.

In recent years acquisition and storage of large data sets has increased dramatically. These often high dimensional and noisy data sets present analytic challenges. Topological Data Analysis approaches this challenge through the lens of Algebraic Topology and provides geometric insights into the shape of the data set. Topological Data Analysis has wide ranging practical applications in fields including oncology, neuroscience and protein analysis.
I was inspired to pursue research in this area following a course in Computational Algebraic Topology which introduced Persistent Homology, Discrete Morse Theory and Contextuality. I particularly enjoyed the part of this course that followed the paper "Categorification of persistent homology" by Bubenik and Scott. The categorification of filtered topological spaces and the subsequent definition of the interleaving distance results in straightforward stability proofs that only need appeal to functoriality, and thus apply in great generality.

Recent publications in Topological Data Analysis have worked towards extending theory beyond the well understood setting of single parameter filtrations with a field as the base domain. Examples of more complex situations that have been studied include:

Permitting multiparameter filtrations
This generalisation loses the structure theorem for finitely generated modules over a PID, and so new invariants to analyse multiparameter persistent homology are developed
Admitting a ring as the base domain
This generalisation results in persistent homology groups with torsion and consequently a new pseudo-distance to compare homology groups is constructed
Endowing the original space with a sheaf
This technique greatly generalises ordinary persistent cohomology since the ordinary case is merely the cohomology of the constant sheaf

These developments require theory from various areas of pure mathematics including Category Theory, Homology Theory, Algebra and Sheaf Theory. Within my research I aim to explore and develop the mathematical theory behind more complicated settings akin to the above examples, and where appropriate, I intend to construct algorithms to efficiently implement these new analytic methods.

This project falls within the EPSRC Mathematical Sciences, Geometry and Topology research area. EPSRC states that as a part of its strategic focus for Geometry and Topology research, it wishes to support "research focused on overarching government priorities including but not limited to data science". My project clearly falls in this area of strategic focus, since Topological Data Analysis "has the potential to play a transformative role in data analytics". Moreover my project serves several of EPSRC's outcomes and ambitions for Geometry and Topology research:
C1: Enable a competitive, data-driven economy
C3: Deliver Intelligent Technologies and systems
C5: Design for an inclusive, innovative and confident digital society