IMA National Celebration of the Best Maths Project 2025
Talks by the winner and runner up of the 2025 National Best Maths Project Competition

Details

When? Wednesday 10 December 2025, 16:00 to 17:00
Where? Online via Zoom

Rebecca Sheppard (University of Liverpool) and Rebecca Maver (University of Manchester) were awarded first and second place respectively, and will deliver talks based on their respective projects.

Abstracts

Not Your Usual Circle: Geometry on the Integer Grid

Rebecca Sheppard (University of Liverpool)

Integer geometry explores objects whose vertices lie on the integer lattice $\mathbb Z^2$, with congruence defined by lattice-preserving affine transformations. In this project, I introduced remarkable geometric objects called integer circles: discrete analogues of Euclidean circle. These objects challenge our geometric intuition regarding circles. Unlike their classical counterparts, integer circles are unbounded, exhibit nontrivial arithmetic structure, and possess positive density in the plane.

In this talk, I will define integer circles, illustrate their unusual behaviour, and demonstrate how to rigorously compute their densities and intersection patterns. I use various number-theoretic tools, such as zeta functions, Dirichlet series, and continued fractions. These results reveal surprising links between discrete geometry and analytic number theory, pointing toward new directions in the study of lattice-invariant structures.

Finite Difference Methods for Solving Convection-Diffusion Equations
Rebecca Maver (University of Manchester)

Abstract: Convection-diffusion equations are a type of partial differential equation which describes the progression of important physical processes, such as the distribution of heat in a convective wind or the movement of gas particles. As with many partial differential equations, analytical solutions are often complex or unavailable. The implementation of numerical methods is, therefore, crucial. This project is an introduction to applying the technique of finite differences to convection-diffusion equations and how one analyses their performance through classical notions of consistency, stability and convergence. The second half will also derive, implement and analyse various time-stepping schemes used in conjunction with finite difference methods to investigate the behaviour of convection-diffusion equations through time. The implicit Euler, Crank-Nicolson and semi-implicit methods will be used, in particular. The scope of problems tackled begins with steady, one-dimensional equations with known exact solutions, culminating in the time-dependent, two-dimensional double glazing problem, which can only be solved by numerical means.