Difference algebraic groups

Summary

Difference algebraic groups can be described as subgroups of the general linear group defined by algebraic difference equations in the matrix entries. They occur naturally as Galois groups in certain Galois theories and have applications to arithmetic problems. The main goal of this project is to further develop the very rich and beautiful theory of these groups and to expand their areas of application.

Algebraic groups can be described as subgroups of the general linear group defined by algebraic equations in the matrix entries. They play an important role in algebraic geometry and representation theory and have a well-developed structure theory. This project is at the crossroads of the theory of algebraic groups and difference algebra, i.e., the systematic study of difference equations from an algebraic point of view.

A substantial body of results is already available for difference algebraic groups defined by ordinary difference equations. See e.g., [1]. A main theme of this project is to generalize these results to partial difference equations.

This project is ideal for students with a good background and interest in algebraic geometry and commutative algebra.

The successful applicant will benefit from large and vibrant research groups in algebra and logic running several weekly seminars.

[1] Finiteness properties of affine difference algebraic groups, International Mathematics Research Notices 2022(1), 506-555, 2022

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.