Pseudorandom majorants over number fields with applications in arithmetic geometry

A Diophantine equation, named after the ancient Hellenistic mathematician Diophantus of Alexandria, is a polynomial equation in which all the coefficients are integers (whole numbers) or rational numbers (fractions). The most fundamental question, given a Diophantine equation, is whether it has a solution, that is a collection of integers or rational numbers which satisfy this equation. To decide whether a given Diophantine equation has a solution can be extremely hard, in spite of extensive mathematical machinery that was developed over centuries to attack these questions. A famous example is Fermat's Last Theorem. Despite the relative simplicity of its statement that for any integer n greater than two, the sum of two positive nth powers can not be an nth power, a proof has eluded the efforts of mathematicians for more than 350 years. It has spawned numerous new developments and was finally completed by Andrew Wiles at the end of the 20th century.

Equations define not just number theoretic, but also geometric objects. A particularly successful approach, developed in the 20th century, tries to investigate solutions to Diophantine equations via the corresponding geometric objects. The modern study of Diophantine equations using these geometric techniques is called arithmetic (or Diophantine) geometry.

Another branch of number theory, in which UK mathematicians play a world leading role, is called additive combinatorics. One of the aims of this discipline is to understand subsets of the integers by decomposing them into structured and random looking parts, with the main challenge arising from the fact that this is usually not a clean dichotomy, but rather a full spectrum.

Extremely fruitful connections between these two fields were initiated very recently by applying certain results and techniques from additive combinatorics to questions in arithmetic geometry, thus expanding our knowledge of Diophantine equations significantly. The central aim of this project is to enhance the impact of these techniques by making them available in a much wider context that is natural in arithmetic geometry.

This project will strengthen young ties between two important areas of pure mathematics in which the UK occupies a leading position and which have existed in separation until recently. As such, it will have immediate positive impact on UK and international researchers working in these areas, and further on pure mathematics as a whole.

Lying at the heart of pure mathematics, it is likely that the results and methods of this project will go through several steps of refinement and adaptation by other researchers before entering areas closer to industrial applications. Impact could arise through computer science, where both pseudorandomness and number fields have applications and are therefore approached with interest.

This kind of impact is hard to predict, but its likelihood can be increased by efficient dissemination of the results. The research conducted in this project will be published in high-impact general mathematics journals, which will allow it to reach a large and diverse audience. In addition, it will be posted on arXiv.org and thus be available to everyone free of charge. I will present the research at numerous seminars, colloquia, workshops and conferences with various audiences, and will make myself available for discussion with researchers from all fields.

The fundamental nature of pure mathematics often leads to the typical situation that topics which were first studied purely out of their own interest are later found to be central sources of innovation with large impact on society. At present, this can be observed at the example of number theory, which was for a long time thought of as having no practical use, but now plays a central role in information security and underlies, in form of public key cryptography, every electronic transaction. I am well equipped to identify potential cryptographic applications of the methods developed in this project. Manchester, with its recently established Heilbronn Institute for Mathematical Research North, offers an ideal environment for the further development of such ideas.