Ruth A. Reynolds | Searcharxiv

arXiv2026

Generative AI performance in core undergraduate mathematics: a curriculum-level case study

Generative artificial intelligence (GenAI) tools such as OpenAI's ChatGPT are transforming the educational landscape, prompting reconsideration of traditional assessment practices. In parallel, universities are exploring alternatives to in-person, closed-book examinations, raising concerns about academic integrity and pedagogical alignment in uninvigilated settings. This study systematically investigates the performance of GenAI on typical mathematics questions from across a first-year mathematics curriculum. Adopting an empirical approach and utilising current examination questions as a proxy for course content, we generate, transcribe, and blind-mark GenAI submissions to eight undergraduate mathematics assessments, spanning the entirety of the first-year curriculum. By combining independent GenAI responses to individual questions, we enable a meaningful evaluation of GenAI performance, both at the level of modules and across the first-year curriculum. We find that GenAI attainment is at the level of a first-class degree, though current performance can vary between modules. Further, we find that GenAI performance is remarkably consistent when viewed across the entire curriculum, significantly more so than that of students in invigilated examinations. Our findings evidence the pressing need for redesigning assessments in mathematics in the era of generative artificial intelligence.

cs.CY↗

arXiv2020

Idealizers in the Second Weyl Algebra

Given a right ideal $I$ in a ring $R$, the idealizer of $I$ in $R$ is the largest subring of $R$ in which $I$ becomes a two-sided ideal. In this paper we consider idealizers in the second Weyl algebra $A_2$, which is the ring of differential operators on $\mathbb{k}[x,y]$ (in characteristic $0$). Specifically, let $f$ be a polynomial in $x$ and $y$ which defines an irreducible curve whose singularities are all cusps. We show that the idealizer of the right ideal $fA_2$ in $A_2$ is always left and right noetherian, extending the work of McCaffrey.

math.RA↗

arXiv2018

Idealisers in Skew Group Rings

Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. We obtain necessary and sufficient conditions for when this idealiser subring is left and right noetherian. We also give an example of these conditions in practice which translates to an interesting number theoretic problem.

math.RA↗