Mia Beard (Maths and Physics)

WHAT DO I RESEARCH?

I study geometry, but not the part where you memorise Euclidean axioms and circle theorems! The idea is that in classical geometry one asks 'What is the shape of this object in space?' and differential geometry asks 'What is the space itself allowed to be?'

As it turns out, this way of thinking doesn't only belong to mathematicians. It is in fact the language of some physics too. Once Einstein realised that gravity is a result of the curvature of spacetime, differential geometry becomes the method for describing this. The same geometric ideas we use to describe a curve on a surface can describe how light bends around a star.

EINSTEIN MANIFOLDS AND MINIMAL SURFACES

My first 'official' research project was on Einstein manifolds, which are special kinds of manifolds where the Ricci curvature is proportional to the metric itself. In low dimensions (one, two, and three) everything is quite polite, but by the time you reach four dimensions, the geometry becomes a bit more opinionated.

I explored some parallels which exist between Einstein four-manifolds and minimal surfaces. To my delight, I found out that certain Einstein manifolds can in fact be immersed as minimal submanifolds of higher-dimensional spheres. And in fact, even in the non-compact setting, provided that one trades the ordinary sphere for a Hilbert sphere, the same holds true.

If you're interested, see my paper here: https://www.sciencedirect.com/science/article/pii/S0393044026000513 Or https://arxiv.org/abs/2506.14972.

IN-PROGRESS CALIBRATED GEOMETRY WORK

My current research focuses on special Lagrangian submanifolds (geometric objects at the intersection of complex geometry and calibrated geometry). These are a particularly beautiful example of minimal submanifolds in complex space, and they have an important application in mirror symmetry.

In my recent paper on U(1)^{n-2}-invariant special Lagrangian n-folds, I extend Joyce's elegant framework from three dimensions to arbitrary dimension n. The key idea is to use symmetry as a tool: by imposing U(1)^{n-2}-invariance, the geometric problem reduces to a surprisingly tractable 2D system of non-linear equations. From this, you're able to observe a nice interplay between smooth geometry and singularities, with the singular behaviour controlled by polynomials that arise naturally from the symmetry structure.

See the paper here: https://arxiv.org/abs/2602.19847.