Together with colleagues at the Warwick Mathematics Institute, we have two fantastic fully funded collaborative projects for a Fall 2025 start as part of the HetSys CDT at Warwick, with different cross-disciplinary teams and a common theme of multi-physics interfacial fluid dynamics (links complete with embarrassing videos!):
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3-descent on genus 2 Jacobians using visibility
Abstract
We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur. These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group. Joint work in progress with Tom Fisher.
James has recently started as the department’s Industry Engagement Officer. He acts as the primary point of contact for day-to-day interactions between the department and our partners in business, industry and government. As well as nurturing existing relationships, he is looking to expand the portfolio of companies we work with. Currently, James is working on pulling together industrial projects for the MMSC MSc students, but is more than happy to have a chat about knowledge exchange more generally.