Jack Westbrook

MSc in Pure Mathematics, Imperial College London

Email: jsw225 [at] ic [dot] ac [dot] uk

I study mathematics because, in a world where truth is often shaped by perception, the clarity and internal consistency found in mathematics offers something both beautiful and deeply compelling.

My research interests include algebraic geometry, number theory, commutative algebra, category theory, and formal proof verification. I am particularly drawn to arithmetic local-to-global principles and the rich interplay between number theory, algebraic geometry, group theory, and complex analysis.

Outside of mathematics, I enjoy writing fiction and philosophy, over-sharing elliptic curves, finite fields, p-adic numbers, and number-theoretic results with unsuspecting family and friends, playing 1|0 chess, and pursuing independent cryptographic implementation projects.

Research

On Deformation of Perfectoid Purity in Gorenstein Domains — with Baily, Dovgodko, and Simpson.
We investigate when perfectoid purity, a mixed-characteristic analog of F-purity, deforms from a quotient \( R/pR \) to a complete Gorenstein local ring \( R \). This work establishes a precise connection between vanishing of local cohomology in absolute integral closures and deformation of splinters. [arXiv:2504.02966]
Some Applications of the Brenner–Monsky Quartic — with Dovgodko and Simpson. In preparation, 2025.
We construct a two-parameter family of Segre products involving Monsky’s quartics with varying Hilbert–Kunz multiplicities and present a new counterexample to localization of tight closure in the mildly singular case. [Preprint (outdated)]
Examples of Lie Algebras with Specified Newton Polygons — with Alwan, K. Huang, T. Huang, and Stovall. In preparation, 2025.
We determine all minimal Newton polygons on \( \mathbb{R}^n \) for \( n \leq 13 \), and classify degree configurations that produce outermost extremal points for Newton polygons associated with iterated Lie brackets.

Other Research & Projects

The Rising Sea Solutions — A comprehensive, self-contained write-up of exercises and results from Vakil’s The Rising Sea. [GitHub]
The Arithmetic of Elliptic Curves Solutions - A collection of proofs in remarks, propositions, theorems, examples as well as solutions to exercises in Silverman's The Arithmetic of Elliptic Curves. [Github]
Describing a Hyperbolic Surface: From Lengths and Twists to Matrices — UW–Madison MxM Undergraduate Research Seminar, Fall 2022.
We constructed explicit generators in \( \mathrm{PSL}(2, \mathbb{R}) \) for Fuchsian groups representing hyperbolic surfaces (e.g. pairs of pants), using twist and length parameters. Future directions include Fenchel–Nielsen coordinates. [program link]
Segre & Veronese Algorithms in Macaulay2 — Code for computing presentations of Veronese subrings and Segre products of graded rings. [GitHub] [GitHub]