Mathematical Foundations of Topological Data Analysis; Theory & Applications
Self-funded PhD students only
School of Mathematics and Physics
Applications accepted all year round
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
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.
- 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.
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
Please contact Dr Ittay Weiss (firstname.lastname@example.org) 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 self-funded PhD opportunity you must quote project code MPHY4410219 when applying.