Stability conditions and canonical metrics

My thesis, completed in 2023, is in the subject of geometry. Specifically it is concerned with two main problems, which make up the two parts of the thesis:

1. The first part concerns the study of D-branes in type IIB superstring theory. D-branes are regions (thought of as membranes floating in space, hence "brane") inside spacetime upon which the strings of string theory must end. These branes inside spacetime have fields over them which must satisfy a differential equation. My thesis studies this differential equation, called the "deformed Hermitian Yang–Mills equation."
   It also introduces a generalisation of this equation, called the "Z-critical equation," which goes further than the physical origins of the idea and links to the mathematical field of algebraic geometry by relating solutions of the equations to Bridgeland stability.
2. The second part studies the properties of certain kinds of geometric spaces, called fibre bundles. These are higher dimensional spaces made up out of two lower-dimensional spaces, the base and the fibre, by gluing a copy of the fibre space to every point of the base space. My thesis considers a special class of fibre bundles, called Kähler fibrations. My thesis shows that the classification of these geometric spaces is equivalent to an already existing classification of principal bundles, another kind of space which may be associated to a fibre bundle.

For a more careful introduction to the ideas of my thesis, see the blog post I have written aimed at the layperson to explain the ideas behind modern geometry and my area of research specifically here.