# Mathematical Foundations of Topological Data Analysis; Theory & Applications

#### Funding

Self-funded PhD students only

MPHY4410219

#### Faculty

School of Mathematics and Physics

#### Closing date

Applications accepted all year round

This 3-year self-funded 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

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.

## Funding

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).

#### 2019/2020 entry

Home/EU/CI full-time students: £4,327 p/a*

Home/EU/CI part-time students: £2,164 p/a*

International full-time students: £15,900 p/a*

International part-time students: £7,950 p/a*

#### By Publication Fees 2019/2020

Members of staff: £1,610 p/a*

External candidates: £4,327 p/a*

*All fees are subject to annual increase

#### 2020/2021 entry

Home/EU/CI full-time students: £4,327 p/a**

Home/EU/CI part-time students: £2,164 p/a**

International full-time students: £16,400 p/a*

International part-time students: £8,200 p/a*

#### By Publication Fees 2020/2021

Members of staff: £1,680 p/a**

External candidates: £4,327 p/a*

*All fees are subject to annual increase
**This is the 2019/20 UK Research and Innovation (UKRI) maximum studentship fee; this fee will increase to the 2020/21 UKRI maximum studentship fee when UKRI announces this rate in Spring 2020.

## Entry requirements

### Entry requirements

• 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

When you are ready to apply, you can use our online application form. 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.

Apply now

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