Lewis Napper

My research interests are many-fold, though predominantly revolve around geometry, topology, and PDE theory, as well as their application to problems in mathematical physics. Feel free to contact me if you have any questions or would like to propose a project or collaboration. What follows is a summary of some of my research, past and present:

Monge--Ampère Geometry and the Topology of Vortices.

First and foremost, my studies focus on the application of Monge--Ampère geometry and higher (categorified) symplectic geometry to classifying the properties of fluid dynamical flows. On the geometric side, I am interested in the existence of almost (generalised, para-)complex structures and conditions for their integrability with respect to choices of bracket. In particular, one wishes to use such structures to classify higher dimensional Monge--Ampère equations and extensions thereof. Motivating this study is the parallel problem of providing a consistent geometric description of vortices in a fluid flow --- the Poisson equation for the pressure of an incompressible fluid flow is of Monge--Ampère type and induces structure which acts as a diagnostic tool for the dominance of vorticity and strain. This work is in collaboration with Volodya Rubtsov and my supervisors Martin Wolf and Ian Roulstone as part of my PhD.

Synthetic Lorentizan Geometry and Globalization of Curvature Bounds.

Building on my work under the LMS undergraduate research bursary (URB-18-19-70) with James Grant, I am interested in establishing results in the relatively new field of synthetic Lorentzian geometry. The fundamental objects of study are Lorentzian pre-length spaces, which are to smooth spacetimes, what metric spaces and Alexandrov geometry are to smooth Riemannian manifolds. Such low regularity structures are of special interest in general relativity. In particular, my current work involves the study of (local) curvature bounds via triangle comparison and the globalization of these bounds. This work is part of a current collaboration with Tobias Beran, Felix Rott, and John Harvey.

The Algebraic Bethe Ansatz for su(2) Spin Chains and Beyond.

Finally, the topic of my MMath final year thesis concerned the application of the Algebraic Bethe Ansatz to particles with spin, in particular to isotropic Heisenberg chains of particles with fundamental representation given by su(2) and su(3). The nested Bethe equations and their eigenvalues were re-derived in full detail and preliminary steps towards the construction of the universal R-matrix were made. This work was carried out under the supervision of Alessandro Torrielli and Andrea Prinsloo ---unfortunately the accompanying report is currently unavailable online (please contact me if you would like a copy).