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

该项目将从两个不同但相关的部分开始开始。这两条链都涉及研究在任意（特别是非代数封闭）域上的传统代数几何设置中出现的结构。链1：在应用中，人们通常研究域上的真实的数，并且经常研究甚至不是域的真实的正数。常微分方程系统上真实的正环面的线性作用自然发生在量纲分析中，可以用来实现进一步的模型简化，有时还可以解释结构的不可识别性。其他类型的对称性通常更难以检测和利用，但在特殊情况下可能是可以实现的。例如，化学反应网络由图形编码，可能是一个很好的起点。在导师Dufresne和合作者的工作基础上，该项目的目的是利用更一般的对称性来获得关于模型简化和/或结构可识别性的结果。从更现代的函子观点出发，我们可以发展任意域上的代数群理论;从这个观点出发，一类关键的对象是伪约化群，它们在更经典的理论中扮演着与约化群类似的角色。该项目的这一部分旨在回答关于伪约化群及其表示理论的一些基本结构问题，建立在导师Bate和合著者的工作基础上。

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.