Fully Funded EPSRC and Swansea University PhD Scholarship: Effective Dimensions: Applications of Computability Theory to Geometric Measure Theory

Funding providers: EPSRC and Swansea University's Faculty of Science and Engineering

Subject areas: Computability theory, computable analysis, geometric measure theory, effective descriptive set theory

Project description: 

Recent works showed how tools from computability theory and computable analysis can be used to study the geometric properties of sets. In particular, the Point-to-Set Principle provides an explicit connection between the Hausdorff dimension of a set (a notion of fundamental importance in analysis and geometric measure theory) and the Kolmogorov complexity of its points (a computability-theoretical notion that measures the “(in)compressibility” of a string). This resulted in a new wave of results, both in computability theory and in other areas of mathematics, and some theorems in geometric measure theory have been proven, for the first time, using these tools.

The quest for point-to-set principles can be condensed into the question “If a set has a certain structural/topological property, what are the computability-theoretical characteristics of its points?”. The goal of this project is to strengthen the connection between computable analysis and “classical” mathematics, and exploring computability-theoretical notions affect the geometric structure of sets.

This project is naturally connected with several topics in theoretical computer science and mathematics, including of course computability theory and geometric measure theory, but also (effective) descriptive set theory, harmonic analysis, and algorithmic information theory. The ideal candidate would have a strong background in mathematics, theoretical computer science, or related topics. Prior exposure to (some of) the above topics is preferred but not required.

Eligibility

Candidates must hold an undergraduate degree at 2.1 level in Computer Science, Mathematics or a closely related discipline, or an appropriate master’s degree with a minimum overall grade at ‘Merit’ (or Non-UK equivalent as defined by Swansea University). If you are eligible to apply for the scholarship (i.e. a student who is eligible to pay the UK rate of tuition fees) but do not hold a UK degree, you can check our comparison entry requirements. Please note that you may need to provide evidence of your English Language proficiency.

English Language: IELTS 6.5 Overall (with no individual component below 6.0) or Swansea University recognised equivalent.

This scholarship is open to candidates of any nationality.

Please visit our website for more information on eligibility.

Funding

This scholarship covers the full cost of tuition fees and an annual stipend at UKRI rate (currently £18,622 for 2023/24).

Additional research expenses will also be available. 

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