The University of Bath is inviting applications for the following PhD project within the Department of Mathematical Sciences under the supervision of Dr Daniel Loughran https://researchportal.bath.ac.uk/en/persons/daniel-loughran
A Diophantine equation is a polynomial equation where one seeks solutions in the integers or the rational numbers. Modern research mathematicians study such problems through the guise of rational points on varieties, in order to emphasise the geometric nature of the problem.
Basic questions are: is there a rational point? Are there infinitely many rational points? Can one obtain a finer quantitative description of the distribution of rational points? This project will make progress on understanding such problems for various classes of varieties (Fano varieties, norm equations, conic bundles, K3 surfaces).
To check whether a variety has a rational point, one first checks whether there is a real point and a p-adic point for all primes p. If this criterion is sufficient one says that the Hasse principle holds. In general the Hasse principle can fail, and this project will study such failures using the Brauer-Manin obstruction.
As for distribution of rational points, there is a conjecture of Manin which predicts the precise asymptotic behaviour of the number of rational points of bounded height. The project will try to make progress on this conjecture in some special cases.
In a similar philosophical vein, we could study the distribution of number fields of bounded discriminant which satisfy certain properties of algebraic interest. For example, studying the number of number fields of bounded discriminant which admit a normal integral basis. This complements Bhargava’s work on counting number fields, for which he was awarded the Fields medal.
Depending on the interests and background of the student, the project could involve a range of techniques from analytic number theory, algebraic number theory, and algebraic geometry.
Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in a relevant subject. A master’s level qualification would also be advantageous. The applicant should have an interest in number theory and algebraic geometry.
Informal enquiries are welcomed and should be directed to Dr Daniel Loughran, [email protected]
Formal applications should be made via the University of Bath’s online application form for a PhD in Mathematics: https://samis.bath.ac.uk/urd/sits.urd/run/siw_ipp_lgn.login?process=siw_ipp_app&code1=RDUMA-FP03&code2=0014
More information about applying for a PhD at Bath may be found here: http://www.bath.ac.uk/guides/how-to-apply-for-doctoral-study/
Anticipated start date: 28 September 2020.
Note: Applications may close earlier than the advertised deadline if a suitable applicant is found and therefore early application is recommended.
UK and EU students who have been ordinarily resident in the UK since September 2017 will be considered for an EPSRC DTP studentship. Funding will cover UK/EU tuition fees, maintenance at the UKRI Doctoral Stipend rate (£15,009 per annum, 2019/20 rate) and a training support grant of £1,000 per annum for 3.5 years.
For more information on eligibility: View Website.
Poonen, Bjorn. Rational points on varieties. Graduate Studies in Mathematics, 186. American Mathematical Society, Providence, RI, 2017.
Browning, Timothy D. Quantitative arithmetic of projective varieties. Progress in Mathematics, 277. Birkhäuser Verlag, Basel, 2009.