Algebraic geometry over arbitrary (in particular, non-algebraically closed) fields

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The project will begin with two distinct but related strands. Both strands are concerned with studying structures which arise in a traditional algebraic geometry setting over arbitrary (in particular, non-algebraically closed) fields.Strand 1: In applications, one is usually working over the field real numbers, and often over the real positive numbers which are not even a field. Linear actions of the real positive torus on ODE systems naturally occur in dimensional analysis, can be exploited to achieve further model reduction, and sometimes explain structural non-identifiability. Other types of symmetries are generally much more challenging to detect and exploit, but may be attainable in special cases. For example, chemical reaction networks are encoded by a graph and could be a good place to start. Building on the work of supervisor Dufresne and collaborators, the aim for this strand of the project is exploit more general symmetries to obtain results about model reduction and/or structural identifiability.Strand 2: Algebraic groups. By working from a more modern functorial point of view, one can develop much of the theory of algebraic groups over arbitrary fields; from this point of view a key class of objects are the pseudo-reductive groups, which play a similar role to reductive groups in the more classical theory. This strand of the project aims to answer some fundamental structural questions about pseudo-reductive groups and their representation theory, building on work of the supervisor Bate and coauthors.