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

该项目将从两个不同但相关的方面开始。这两个分支都涉及研究在任意（特别是非代数封闭）领域的传统代数几何设置中出现的结构。Strand 1：在应用程序中，人们通常处理实数字段，并且经常处理甚至不是字段的实数正数。实正环面在ODE系统上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约简，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，化学反应网络是由图形编码的，这可能是一个很好的起点。在导师Dufresne和合作者的工作基础上，这个项目的目标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。第二部分：代数群。从更现代的泛函观点出发，我们可以发展任意域上代数群的许多理论；从这个观点来看，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相似的角色。这个项目的目的是回答一些关于伪约化群及其表示理论的基本结构问题，建立在导师Bate和合作者的工作基础上。

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.