Rebecca Sheppard, a recent graduate from our Department of Mathematics, has been awarded first place in the prestigious IMA National Celebration of the Best Maths Project 2025. The event celebrates top mathematical talent across UK universities and is organised by the Institute of Mathematics and its Applications (IMA).
Rebecca’s winning project, titled “Not Your Usual Circle: Geometry on the Integer Grid,” impressed the judges with its original investigation into integer geometry—a field focused on shapes whose vertices lie on the integer lattice. Her work introduces the concept of “integer circles,” discrete analogues of Euclidean circles that defy familiar geometric expectations. Unlike classical circles, these objects are unbounded, display rich arithmetic structure, and have positive density in the plane.
In her research, Rebecca defines these integer circles, illustrates their surprising behaviours, and explains how to compute their densities and intersection patterns. The project draws on sophisticated number-theoretic techniques, including zeta functions, Dirichlet series, and continued fractions, revealing new links between discrete geometry and analytic number theory.
The IMA National Celebration of the Best Maths Project offers a distinguished platform for emerging mathematicians to showcase their work and engage with the wider mathematical community. Rebecca’s achievement continues the University of Liverpool’s strong record of success at this event, following last year’s first-place win by Will Barnfield.
Rebecca will present her project remotely via Zoom. If you wish to attend, please register here.