Dr Lewis Napper
Academic and research departments
Mathematics at the Interface Group, School of Mathematics and Physics.About
My research project
Monge--Ampère geometry and the topology of vorticesMy research currently focuses on studying Monge--Ampère type equations using higher (categorified) symplectic geometry. In particular, we wish to examine the indefinite metrics and generalised structures which arise on cotangent bundles and their k-Lagrangian submanifolds. This work is motivated by the Poisson equation for the pressure, arising from the incompressible Navier--Stokes equations, which is of Monge--Ampère type when considered in two dimensions. In this physical picture, the signatures of the metrics act as a diagnostic for the dominance of the vorticity and rate-of-strain, and can be related to the topology of the fluid flow. Our geometric framework also enables us to recast the Lundgren transformation of vortex tubes (such as Burgers' vortex) as a higher symplectic symmetry reduction.
This project is fully funded by an STFC studentship.
Supervisors
My research currently focuses on studying Monge--Ampère type equations using higher (categorified) symplectic geometry. In particular, we wish to examine the indefinite metrics and generalised structures which arise on cotangent bundles and their k-Lagrangian submanifolds. This work is motivated by the Poisson equation for the pressure, arising from the incompressible Navier--Stokes equations, which is of Monge--Ampère type when considered in two dimensions. In this physical picture, the signatures of the metrics act as a diagnostic for the dominance of the vorticity and rate-of-strain, and can be related to the topology of the fluid flow. Our geometric framework also enables us to recast the Lundgren transformation of vortex tubes (such as Burgers' vortex) as a higher symplectic symmetry reduction.
This project is fully funded by an STFC studentship.
My qualifications
News
In the media
ResearchResearch interests
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).
Research interests
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).
Teaching
Local Department Seminars
- Monge--Ampère Geometry and the Navier--Stokes Equations (6th February 2023) --- A joint seminar in the Fields, Strings, and Geometry Group, and Nonlinear Waves and Geometric Fluid Dynamics Group research seminar series.
- Not all ravens are writing desks: An introduction to the topology of surfaces (24th February 2023) --- A seminar in the Taste of Research Undergraduate Seminar Series designed by mathematics PhD students to introduce undergraduate mathematics students to a taste of doctoral research in mathematics.
- Monge--Ampère Geometry and the Navier--Stokes Equations (29th March 2023) --- A talk contributed as part of the first PGR Mathematics Conference, representing the work of the Fields, Strings, and Geometry research group.
Marking Support (2023/24)
- MAT1034 Linear Algebra
- MAT3012 Statistical Methods with Financial Applications
- MAT2003 Stochastic Processes
- MAT2009 Operations Research and Optimisation
- MAT2004 Real Analysis II
Marking Support (2022/23)
- MAT3043 Graphs and Networks
- MAT2009 Operations Research and Optimization
- MAT1032 Real Analysis I
- MAT1031 Algebra
Marking Support (2021/22)
- MAT3009 Manifolds and Topology
- MAT1033 Probability and Statistics
Computer Lab Assistant
- MAT1033 Probability and Statistics (2019/20)
- MAT2001 Numerical and Computational Methods (2019/20 and 2020/21)
Publications
In the synthetic geometric setting introduced by Kunzinger and S\"amann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet--Myers Theorem and Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov--Hausdorff convergence.
We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.
Additional publications
Available on ArXiv
- L. Napper, I. Roulstone, V. N. Rubtsov, and M. Wolf, Monge--Ampère Geometry and Vortices (2023)
- T. Beran, L. Napper, and F. Rott, Alexandrov's Patchwork and the Bonnet--Myers Theorem for Lorentzian length spaces (2023)
- T. Beran, J. Harvey, L. Napper, and F. Rott, A Toponogov globalisation result for Lorentzian length spaces (2023)