Supervisor: Dr Ivan Tomasic
Project description:
Several years ago, I formulated an ambitious long-term research programme of developing Relative Algebraic Geometry over an arbitrary base topos, with a view toward applications in difference and differential algebra, as outlined in a 200 page sketch I publicised in 2020. This opens a vast new area for research, because every chapter of classical algebraic geometry can be investigated in the relative setting, yielding novel and highly applicable results, especially in difference algebra and equivariant algebraic geometry.
This PhD project will form part of my established programme, and the student will work on the development of etale cohomology of relative schemes, alongside myself, Noohi, my two PDRAs and my current three PhD students. At the start, the student will learn the methods of categorial logic and algebraic geometry required to formalise and make precise the initial steps of the theory as outlined in my sketch. Upon gaining independence, the student will apply relative methods to derive results on etale cohomology in the equivariant and the difference setting. At the adventurous end of the project, we will prove the difference analogue of the celebrated Weil conjectures. Such a result has just become attainable using the proposed methods and the combined expertise of our team.
Further information:
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Entry requirements
Fees and funding