Mathematical Foundations of Topological Data Analysis; Theory & Applications

Applications are invited for a fully-funded three year PhD to commence in October 2019.

The PhD will be based in the School of Mathematics and Physics and will be supervised by Dr Ittay Weiss and Dr James Burridge.

The work on this project will aim to:
-further enhance and develop the new mathematical foundations of topology
-investigate the flow of ideas between topology and TDA along this new bridge, with new applications of TDA in mind


Project description

Topological Data Analysis (TDA) is an emerging and highly successful approach to Big Data problems. 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 resulting from noise contamination can be very challenging. Topology, by design, is blind to matters of scale or dimensionality. Its tools are aiming 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.

TDA represents a unique intersection point of pure mathematics with applied mathematics. The mathematical tools used in current TDA applications are all based on tools of algebraic topology developed over the past century. However, the aim of the topological tools was not with applications in mind, certainly not to data analysis, and are never given in terms of a metric. The starting point of any data analysis investigation however is a specification of a very 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. Nonetheless, applying topological techniques, suitably adapted to the metric situation, works remarkably well in many situations, spanning cutting edge brain research to various commercial applications in, e.g., the pharmaceutical industry.

This situation is an indication that the topological formalism is not optimally aligned to the needs of TDA. It is 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 is to identify and develop a new formalism for topology under which the clash between topology and TDA ceases to exist and so that 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.


Entry Requirements

General admissions criteria
You’ll need a good first degree from an internationally recognised university (minimum upper second class
or equivalent, depending on your chosen course) or a Master’s degree in Mathematics or related subject. In exceptional cases, we may consider equivalent professional experience and/or Qualifications. English language proficiency at a minimum of IELTS band 6.5 with no component score below 6.0.

Specific candidate requirements
You should have a sufficient familiarity with formal proof based mathematics as gained by a degree or Master’s in mathematics with a substantial component of pure mathematics. You should be familiar with the basics of topology and with algebraic topology and/or category theory.

How to Apply
We’d encourage you to contact Dr Ittay Weiss ([Email Address Removed]) to discuss your interest before you apply, quoting the project code.

When you are ready to apply, you can use our online application form and select ‘Mathematics and Physics’ as the subject area. Make sure you submit a personal statement, proof of your degrees and grades, details of two referees, proof of your English language proficiency and an up-to-date CV. Our ‘How to Apply’ page offers further guidance on the PhD application process.


If you want to be considered for this funded PhD opportunity you must quote project code MPHY4410219 when applying.