This page documents some of the work I did during my PhD. I studied problems in geometric analysis, an area of maths that brings together ideas from differential equations, geometry and measure theory. You can find some notes, papers and my thesis below after the descriptions of what I worked on.
For the mathematicians
My work was focussed on geometric variational problems, formulated in the languages of geometric measure theory and PDE. In particular I was interested in those problems arising from the study of the area functional of submanifolds of Euclidean space or, more generally, a Riemannian manifold. I worked on problems relating to mean curvature flow (which is like the gradient descent for area), where I was mostly interested in the process of singularity formation and the existence and regularity of weak solutions. I was also interested in the regularity of stationary varifolds (these are measure theoretic notions of minimal submanifolds, and so are critical for area), and in particular worked on trying to understand boundary behaviour, and regularity up to the boundary.
For the civilians
I worked in an area of mathematics called geometric analysis. Broadly speaking, what this means is that I worked on problems in geometry (the study of space and shapes) using techniques and ideas from analysis (this is a rather large area of mathematics that is hard to define well, but relates mostly to calculus and the ideas that arose from its development). More specifically it turns out that many problems in geometry are described by differential equations, and often one hopes that by better understanding the behaviour of the equation and its solutions, one can gain insight into the original geometric problem. This can also work the other way; often the equations that arise from these geometric problems are somehow ‘typical’ examples of a wider class of equations containing them. One hopes that perhaps insights coming from the geometric picture, could lead to development of techniques that will generalise to the wider class of equations where perhaps there is no obvious associated geometric picture or intuition.
“On short time existence of Lagrangian mean curvature flow” joint with Kim Moore. Math. Ann. (2017) 367:1473–1515. Available here.
“A regularity theorem for stationary varifolds, and an existence theorem for mean curvature flow”. Click here to download.
“Introduction to Lagrangian mean curvature flow”. Notes based on a reading seminar given by Jason Lotay at UCL.
“Mean curvature flow and related topics”. Notes based on a graduate lecture course given by Neshan Wickramasekera at Cambridge. Though the course was initially planned to focus on mean curvature flow, substantial material on regularity of minimal surfaces was covered also.