Let G be a linear algebraic group defined over an algebraically closed field k, such as the complex numbers. This means that G can be identified with the vanishing locus of a set of polynomials on the entries of matrices in GL_n for some n. A good example is the group SL_n, since it is the vanishing locus of the polynomial det-1. A element is called unipotent if conjugate in GL_n to an upper triangular matrix with 1s along the diagonal.There are various cases where G is known to have finitely many conjugacy classes of unipotent elements. e.g. G reductive, or for certain of its parabolic subgroups. There should be tools available to classify these in general.Main Problem. Describe the linear algebraic groups with finitely many unipotent classes. EligibilityApplicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering related discipline. FundingAt Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers applying for competition and self-funded projects. For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.Before you applyWe strongly recommend that you contact the supervisor(s) for this project before you apply.How to applyApply online through our website: https://uom.link/pgr-apply-fap When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees. Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered. After you have applied you will be asked to upload the following supporting documents: Final Transcript and certificates of all awarded university level qualificationsInterim Transcript of any university level qualifications in progressCVContact details for two referees (please make sure that the contact email you provide is an official university/work email address as we may need to verify the reference)English Language certificate (if applicable)If you have any questions about making an application, please contact our admissions team by emailing [email protected]. Equality, diversity and inclusion is fundamental to the success of The University of Manchester, and is at the heart of all of our activities. We know that diversity strengthens our research community, leading to enhanced research creativity, productivity and quality, and societal and economic impact. We actively encourage applicants from diverse career paths and backgrounds and from all sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation and transgender status. We also support applications from those returning from a career break or other roles. We consider offering flexible study arrangements (including part-time: 50%, 60% or 80%, depending on the project/funder).