Rational points and arithmetic statistics

The PhD project will require a mix of tools from algebraic geometry, algebraic number theory, Galois theory, and analytic number theory. It will work on some of the programme grant objectives, which concern applications of the theory of rational points on varieties to problems in arithmetic statistics, including some of the following topics:

• Distributions of number fields and class groups
• Local-global principles, including the Brauer-Manin obstruction to the Hasse principle and weak approximation
• Existence and distribution of rational points on varieties and stacks
• The inverse Galois problem

As part of the PhD project you will be fully integrated within the programme grant activities, which will include collaborative retreats, summer schools, study groups, conferences, and opportunities for secondments at the partner universities.

Further details on the scientific aspects can be found in the survey paper from the references. For further details on the programme grant, please see the webpage: https://sites.google.com/view/unifying-arithmetic-statistics/

Funding is currently available for at least 3 years.

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or good Upper Second Class UK Honours degree (or the equivalent) in a relevant subject. A master’s level qualification would also be advantageous.

Non-UK applicants must meet the programme’s English language requirement by the application deadline.

Enquiries and Applications:

Informal enquiries are encouraged and should be directed to Prof Daniel Loughran, dtl32@bath.ac.uk.

Formal applications should be submitted via the University of Bath’s online application form for a PhD in Mathematics prior to the closing date of this advert.

IMPORTANT:

When completing the application form:

1. In the Funding your studies section, select ‘University-LURS’ as the studentship for which you are applying.
2. In the Your PhD project section, quote the project title of this project and the name of the lead supervisor in the appropriate boxes.

Failure to complete these two steps will cause delays in processing your application and may cause you to miss the deadline.

More information about applying for a PhD at Bath may be found on our website.

PLEASE BE AWARE: Applications for this project may close earlier than the advertised deadline if a suitable candidate is found. We therefore recommend that you contact the lead supervisor prior to applying and submit your formal application as early as possible.

Equality, Diversity and Inclusion:

We value a diverse research environment and aim to be an inclusive university, where difference is celebrated and respected. We welcome and encourage applications from under-represented groups.

If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.

Funding Notes

Candidates may be considered for a University of Bath studentship tenable for a minimum of 3 years. Funding covers tuition fees, a stipend (£21,805 p/a in 2026/7) and access to a training support budget.

Studentships are open to both Home and International students; however, International applicants should note that funding does NOT cover the cost of moving to the UK, obtaining a student visa (View Website) or payment of the UK healthcare surcharge (View Website).

References

1. Daniel Loughran, Tim Santens, The leading constant in Malle's conjecture, https://arxiv.org/abs/2606.04983
2. Daniel Loughran, Tim Santens, Malle's conjecture and Brauer groups of stacks, https://arxiv.org/abs/2412.04196
3. Daniel Loughran, Ross Paterson, Lower bounds for A4-quartic fields, https://arxiv.org/abs/2510.05248