New insights into the geometry of highly symmetric spaces

Simon Cooper received his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Fred Diamond, King’s College London, United Kingdom.

The field of arithmetic geometry tries to answer questions about numbers (arithmetic), by studying the underlying structure of the problem (geometry). The first part often seems deceptively simple, while the task of understanding the underlying structure is very difficult. 

A famous example is Fermat’s last theorem, which dates from 1637. French amateur mathematician Pierre de Fermat claimed that there are no positive integer solutions to the equation xⁿ + yⁿ = zⁿ for n greater than 2. It took over 350 years for this to be proven by the British mathematician Andrew Wiles, who used modern mathematical machinery, including significant parts of the Langlands programme. First formulated in the late 1960s by the Canadian mathematician Robert Langlands, the programme consists of a large set of conjectures that tie together several areas of theoretical mathematics: number theory, harmonic analysis and algebraic geometry.  

In this project Simon Cooper aims to study Shimura varieties, which are highly symmetric spaces that play a vital role in the Langlands programme. The extra symmetries make their geometry unusually accessible.  Of particular interest are the tautological rings of Shimura varieties, which contain a lot of refined information about the varieties’ geometry. The aim of the project is to move closer to a complete description of the tautological rings, to better interpret important geometrical features of the Shimura varieties.

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