Mathematical Foundations of Topological Data Analysis Theory and Applications

Faculty

School of Mathematics and Physics

This 3-year self-funded PhD will be based in the School of Mathematics and Physics and will be supervised by Dr James Burridge.

Topological Data Analysis (TDA) is an emerging and highly successful approach to Big Data problems. TDA represents a unique intersection point of pure mathematics with applied mathematics; the underlying idea is to employ topological techniques in the analysis of large quantities of data.

One of the main difficulties with real-world data is that its collection introduces various types of 'noise'. Distinguishing between true features inherent to the phenomenon in question and misleading features from noise contamination can be challenging.

Topology, by design, is blind to matters of scale or dimensionality. Its tools aim to extract geometric features which are robust, in comparison to scale or dimensionality distortions. TDA seeks to exploit the inherent noise-blindness of topology by employing topological techniques to data analysis.

The mathematical tools used in current TDA applications are all based on tools of algebraic topology developed over the past century. However, topological tools were not created with applications (such as data analysis) in mind, and are never given in terms of a metric.

However, because the starting point of any data analysis investigation is a specification of a particular metric space, this represents a clash of ideologies: topology is the art of doing geometry without distances while TDA is the extraction of geometric features from distance information.

Despite this, topological techniques can work well in many situations – from brain research to commercial pharmaceutical applications – when they have been suitably adapted to the metric situation. This situation is an indication that the topological formalism is not optimally aligned to the needs of TDA.

It's therefore desirable to investigate alternative formalisms of topology with the aim of facilitating a smoother transition of topological tools to TDA – thereby enlarging the scope of applicability of TDA, while at the same time adjusting the classical topological methodologies to allow insights from TDA to filter back to topology.

The aim of this project is to identify and develop a new formalism for topology under which the clash between topology and TDA ceases to exist, and within which cross-fertilisation between the two fields becomes natural.

In recent work such a formalism for topology was identified and is being developed. The advantages of the formalism to the foundations of topology are demonstrated while more advanced features are being tested. Alongside the topological gains, applications to the foundations of TDA are emerging as well.

Fees and funding

Visit the research subject area page for fees and funding information for this project.

Funding availability: Self-funded PhD students only. 

PhD full-time and part-time courses are eligible for the UK Government Doctoral Loan (UK and EU students only).

Bench fees

Some PhD projects may include additional fees – known as bench fees – for equipment and other consumables, and these will be added to your standard tuition fee. Speak to the supervisory team during your interview about any additional fees you may have to pay. Please note, bench fees are not eligible for discounts and are non-refundable.