Academia

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 extended generalised (para-)complex structures and how their deformations may be used to map between equivalent equations. In particular, I wish 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 — 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, indicating where vortices might form. Techniques such as geometric numerical integrators may be used to model how such a vortex might evolve and provide justification for analytic models like geometric flows.

This work is part of a collaboration with Volodya Rubtsov and my supervisors Martin Wolf and Ian Roulstone as part of my PhD and is currently funded by an STFC studentship. This project is also supported by a 3-month visiting fellowship from the HERI department of the French Embassy in the UK to collaborate with Vladimir Salnikov in the CNRS laboratory at La Rochelle University, France.

Cellular Automata on Aperiodic Monotiles

Inspired by the discovery of the first aperiodic monotile (or Einstein tile), I have also recently begun investigating the properties of cellular automata such as Conway's Game of Life and Langton's Ants on aperiodic tilings of the Euclidean plane. It is known that on a regular grid, both of these exemplary automata are Turing complete and sustain computation, however aperiodic tilings do not admit translational symmetry, hence it remains to be seen whether the same is true for these automata on aperiodic tilings. We aim to produce simulations of these automata in code, before investigating the problem mathematically.

This work is part of a collaboration with Jamie Gabbay.

Synthetic Lorentizan Geometry and Globalisation 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 globalisation of these bounds.

This work is part of a 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.

Publications

Papers and Preprints

  • A Toponogov Globalisation Result for Lorentzian Length Spaces (2023)

    With: Tobias Beran, John Harvey, and Felix Rott

    ArXiv | PDF |

    In the synthetic geometric setting introduced by Kunzinger and Sämann, 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 to Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov–Hausdorff convergence.

  • Alexandrov's Patchwork and the Bonnet–Myers Theorem for Lorentzian (pre-)length spaces (2023)

    With: Tobias Beran and Felix Rott

    ArXiv | PDF |

    We present several key results for Lorentzian pre-length spaces with global timelike curvature bounds. Most significantly, we construct a Lorentzian analogue to Alexandrov's Patchwork, thus proving that suitably nice Lorentzian pre-length spaces with local upper timelike curvature bound also satisfy a corresponding global upper bound. Additionally, for spaces with global lower bound on their timelike curvature, we provide a Bonnet–Myers style result, constraining their finite diameter. Throughout, we make the natural comparisons to the metric case, concluding with a discussion of potential applications and ongoing work.

  • Monge–Ampère Geometry and Vortices (2023)

    With: Ian Roulstone, Volodya Rubtsov, and Martin Wolf
    Published in: Nonlinearity 2024 Vol. 37(4) Ref. 045012

    Journal | ArXiv | PDF |

    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 the (higher) Lagrangian submanifold it defines 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-Ampere 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.

Reports (unpublished)

  • Monge–Ampère Geometry and Vortices — Confirmation Report (2023)

    Supervised by Martin Wolf and Ian Roulstone

    PDF
  • The algebraic Bethe ansatz for SU(2) spin chains and beyond — Master's Thesis (2021)

    Supervised by Alessandro Torrielli and Andrea Prinsloo
    (Document To Appear)

News and Events

Browse for news from my studies, including visits to workshops and conferences (see below for events at which I have given talks), as well as my progress in obtaining my Ph.D.

Summer School in Quiberon

Quiberon, France
9-14 September 2024

Lewis Napper attended a summer school on techniques for solving inverse problems in mechanics in Quiberon, France. Such inverse problems occur when we attempt to measure physical parameters of a material using waves, or measure the temperature of the material when part of its surface is inaccessible, for example.

School Page

SSHN Visiting Fellowship

University of La Rochelle, France
31st August 2024

Lewis Napper has been awarded a Scientific High-level (SSHN) Visiting Fellowship by the Higher Education, Research, and Innovation Department of the French Embassy in the UK, to undertake a research stay at the CNRS laboratory in La Rochelle, France. Lewis will be visiting La Rochelle, from September to November 2024 inclusive, to work with Vladimir Salnikov.

Award Page | UoS Blog

UK Metric Geometry and Analysis Network Meeting

University of Cardiff, UK
12th April 2024

Lewis Napper was invited to attend the first meeting of the UK Metric Geometry and Analysis Network, held at the University of Cardiff. Topics included the geometry of optimal transport (used in financial and fluid models), persistence diagrams for imaging data, and curvature bound spaces.

Event Page

Géométrie Différentielle et Mécanique Research Network Meeting

University of Sorbonne, Paris, France
21-24 November 2023

Lewis Napper visited Paris to give an invited talk at the bi-annual meeting associated with the 'Géométrie Différentielle et Mécanique' French Research Network (CNRS GDR-GDM), held in the international conference centre at the University of Sorbonne. Lewis spoke in a session on the 23rd November (see Presentations below).

Event Page | UoS Blog

Summer School on Machine Learning in Mathematical Physics

University of Oxford, UK
17-21 July 2023

Lewis Napper visited the University of Oxford's physics department to attend an intensive summer school on Machine Learning in Mathematics and Theoretical Physics, covering topics such as transformer architecture, reinforcement learning, and quantum annealing, as well as their application in knot theory, string theory, and differential geometry.

Event Page | UoS Blog

Kings and Queens of Gravity Workshop

London Mathematical Society, UK
30-31 March 2023

Lewis Napper visited the home of the London Mathematical Society, de Morgan House in London, to attend days two and three of a three day workshop on the application of analysis, PDE theory, and differential geometry to general relativity. The event was organised by Queen Mary University London and King's College London.

Event Page

Interdisciplinary Junior Scientist Workshop on Mathmatical General Relativity

Wildberg, Germany
26 February - 10 March 2023

Lewis Napper visited Haus Saron in Wildberg, Germany to attend a two week Junior Scientist Workshop on General Relativity, organised by the Universities of Potsdam and Tübingen. He spoke about "Globalisation of timelike curvature bounds in Lorentzian length spaces" on day two of the event (see Presentations below).

Event Page | UoS Blog

Workshop on Mathematical Relativity, Scalar Curvature, and Synthetic Lorentzian Geometry

Fields Institute, Canada
3-7 October 2022

Lewis Napper visited the Fields Institute in Toronto, Canada to participate in their Workshop on Mathematical Relativity, Scalar Curvature, and Synthetic Lorentzian Geometry, as part of a thematic programme on Non-smooth Riemannian and Lorentzian Geometry. Topics included optimal transport, causal sets, and Lorentzian length spaces.

Event Page | UoS Blog

South East Mathematical Physics Seminar 16

University of Surrey, UK
28 April 2022

Lewis Napper attended the 16th session of the South East Mathematical Physics Seminar (SEMPS) at the University of Surrey. The LMS funded seminar is a collaboration between universities in the south east of the United Kingdom and covers topics including general relativity, string theory, and integrable systems.

Event Page | UoS Blog

Presentations

  • Monge–Ampère Equations and their Geometry: A Strange Relationship (26th and 27th September 2024)

    2-Part Research Group Seminar
    LaSIE CNRS, La Rochelle, France

    Slides (Part 1) | Slides (Part 2) |

    Monge–Ampère equations (MAEs) are a class of non-linear, second order partial differential equations, which underlie many models of physically interesting systems. These include: the Khokhlov–Zabolotskaya equation for three-dimensional acoustics, the Poisson equation for the pressure of an incompressible fluid, the reaction-diffusion equation, and even the Laplace and wave equations.

    The first part of this talk is intended as an introduction to the geometry of MAEs as a tool for studyiing physical systems. We begin by defining the family of MAEs in m variables and explain how they can be encoded as constraints on m-dimensional submanifolds of some 2m-dimensional phase space. Specialising to two variables, we show how the properties of these submanifolds can be used to derive properties of the solutions to a given MAE, via the Lychagin–Rubtsov theorem and metric. Finally, we provide an example, applying these tools to the Poisson equation for the pressure of an incompressible fluid flow to produce a relationship between the signature of the Lychagin–Rubtsov metric and the dominance of vorticity and strain in the fluid. While some basic knowledge of differential geometry will be useful, I will not assume much a-prior and this portion of the talk will be relatively self-contained.

    The second part of this talk will cover recent progress in the field of Monge–Ampère geometry, arising from further mathematical investigation of the Poisson equation. In particular, a covariant re-formulation allows our results to be applied to fluids on non--Euclidean backgrounds, e.g. on a sphere. Furthermore, we show that the correct choice of symplectic structure on the phase space can be used to couple the Poisson equation to the incompressibility constraint and that this construction generalises to higher dimensions if we take a multi-symplectic view. This leads to the open problem of defining a class of multi-symplectic Monge–Ampère equations.

  • Monge–Ampère Geometry of the Navier–Stokes Equations (23rd November 2023)

    GDR-GDM Meeting on Differential Geometry and Mechanics
    International Conference Centre, University of Sorbonne, Paris, France

    UoS Blog | Meeting Webpage | Slides |

    In this short primer to the paper 'Monge–Ampère Geometry and Vortices,' we recap the concepts of Monge–Ampère equations and their associated structures, before applying them to the Poisson equation for the pressure arising from the Navier–Stokes equations under the assumptions of incompressibility and inhomogeneity of density. In particular, we show that the associated Lychagin–Rubtsov metric on the phase space of the flow encodes the Laplacian of pressure, while the pull-back of this metric to a Lagrangian submanifold (which represents a solution of the origina parital differential equation) has eigenvalues and signature determined by the dominance of vorticity and strain. We conclude with comments on how this setup generalises to a framework which is both spatially covariant and multisymplectic in nature in higher dimensions, and infer a Weiss–Okubo criterion for fluid flows on an arbitrary Riemannian manifold.

  • (Higher) Monge–Ampère Geometry of the Navier–Stokes Equations (21st November 2023)

    Integrable Systems Seminar
    Beijing Institute of Mathematical Sciences and Applications, China (via Zoom).

    UoS Blog | Slides |

    The Poisson equation for the pressure of a homoheneous, incompressible Navier–Stokes flow is a key diagnostic relation for understanding the formation of vortices in turbulence. Building on the observation that, in two dimensions, the aforementioned equations is a Monge–Ampère equation for the stream function, this talk introduces a framework for studying the relation from the perspective of (multi-)symplectic geometry.

    While reviewing the geometry of Monge–Ampère equations presented by Rubtsov, D'Onofrio, and Roulstone in earlier seminars of this series, we demonstrate how an associated metric on the phase space of a two-dimensional fluid flow encodes the dominance of vorticity and strain. We then discuss how multi-symplectic geometry may be used to generalise to fluid flows on Riemannian manifolds in higher dimensions, culminating in a Weiss–Okubo-type criterion in these cases. Throughout, we make comments on how the signatures and curvatures of our structures may be interpreted in terms of the geometric and topological properties of vortices.

  • The Geometry of Monge–Ampère Equations in Fluid Dynamics (1st November 2023)

    Junior Analysis Seminar
    Imperial College, London, UK.

    UoS Blog | Seminar Webpage | Slides |

    The Poisson equation for the pressure of a homogeneous, incompressible Navier–Stokes flow is a key diagnostic relation for understanding the formation of vortices in turbulence. Building from the observation that in two dimensions, the aforementioned equation is a Monge–Ampère equation for the stream function, this talk introduces a framework for studying fluid dynamics using the geometry of such Monge–Ampère equations.

    In particular, we show that the associated Lychagin–Rubtsov metric on the phase space of the flow encodes the Laplacian of pressure, while the pull-back of this metric to a Lagrangian submanifold (which represents a solution of the origina parital differential equation) has eigenvalues and signature determined by the dominance of vorticity and strain. We highlight how this framework generalises to fluid flows on curved background and to higher dimension, by inferring a Weiss–Okubo criterion for fluid flows on an arbitrary Riemannian manifold. We conclude with some comments on generalised solutions as a model for vortex sheets and weather fronts, on the topology of vortices, and on some open questions.

    Note: This talk is intended as an introduction to Monge–Ampère geometry as a tool for studying PDEs, using Navier–Stokes flows as a case study. While some knowledge of differential geometry would be useful, I do not assume this a-priori and the talk will be relatively self contained, with the relevant concepts introduced as we proceed.

  • Monge–Ampère Geometry and the Navier–Stokes Equations (26th April 2023)

    Mathematical Physics Research Seminar
    University of Hertfordshire, UK.

    UoS Blog | Slides |

    Partial differential equations of Monge–Ampère type have been shown to correspond to specific choices of differential form on the associated phase space. Furthermore, Lagrangian submanifolds of said space may be viewed as a generalisation of solutions to a Monge–Ampère equation. Building from the observation that the Poisson equation for the pressure of an incompressible, two-dimensional, Navier–Stokes flow can be presented as an equation of Monge–Ampère type, this talk introduces a framework for studying fluid dynamics using properties of the aforementioned submanifolds. In particular, it is noted that such a submanifold may be equipped with a metric whose signature acts as a diagnostic for the dominance of vorticity and strain. We discuss how this approach may be extended to fluid flows on an arbitrary Riemannian manifold and to higher dimensions, through the use of higher (categorified) symplectic geometry. We conclude with some comments on how this facilitates symmetry reductions and some open questions.

  • Monge–Ampère Geometry and the Navier–Stokes Equations (29th March 2023)

    The first annual PGR mathematics conference.
    University of Surrey, UK.

    Conference Booklet | Slides |

    For partial differential equations of Monge-Ampère type, it has been shown that solutions correspond to Lagrangian submanifolds of the associated phase space. Building from the observation that the Poisson equation for the pressure of an incompressible, two-dimensional, Navier-Stokes flow is a Monge-Ampère equation, this talk introduces a framework for studying fluid dynamics using properties of the aforementioned submanifolds. In particular, it is noted that such a submanifold may be equipped with a metric whose signature acts as a diagnostic for the dominance of vorticity and strain. We provide an illustrative example in two dimensions and probe the motivational question: ‘What is a vortex?’ We conclude with comments on extensions to fluid flows in higher dimensions and some open questions.

  • Not all ravens are writing desks: An introduction to the topology of surfaces (24th February 2023)

    Taste of Research Undergraduate Seminar (TORUS).
    University of Surrey, UK.

    Seminar Webpage | Slides |

    Part of a series of talks by Ph.D students to introduce undergraduate students to the world of doctoral research in mathematics.
    In the tale Alice in Wonderland, Lewis Carroll asks "Why is a raven like a writing desk?" With this open question in mind, we provide a short pedagogical introduction to the topology of surfaces, touching on the notions of orientability, open and closed surfaces, and curvature. Most of our attention will be paid to the concept of genus, which is loosely described by the number of holes a closed surface has. Following some illustrative examples, we shall apply the theory to the question at hand, conclusively stating conditions under which a writing desk is topologically equivalent to a raven. Given time, we shall relate these ideas to current research into fluid dynamics, space-times, and string theory.

  • Globalisation of Curvature Bounds in Lorentzian Pre-length Spaces (27th February 2023)

    Interdisciplinary Junior Scientist Workshop in Mathematical General Relativity.
    Wildberg, Germany (Hosted by Universität Tübingen and Universität Potsdam).

    UoS Blog | Workshop Webpage | Slides |

    Inspired by results from metric geometry, we discuss the globalisation of timelike curvature bounds in Lorentzian pre-length space framework of Kunzinger and Sämann. In particular, utilising the formulation of curvature bounds in terms of triangle comparison, we present a Lorentzian analogue of the Alexandrov patchwork construction for spaces with curvature bounded above. We conclude with a summary of ongoing work on the corresponding problem for Lorentzian pre-length spaces with curvature bounded below. Should time permit, we will also discuss ideas for applications.

  • Monge–Ampère Geometry and the Navier–Stokes Equations (6th February 2023)

    Fields, Strings, and Geometry / Non-linear Waves and Geometric Fluid Dynamics group research seminar.
    University of Surrey, UK.

    Seminar Webpage | Slides |

    For partial differential equations of Monge-Ampère type, it has been shown that solutions correspond to Lagrangian submanifolds of the associated phase space. Building from the observation that the Poisson equation for the pressure of an incompressible, two-dimensional, Navier-Stokes flow is a Monge-Ampère equation, this talk introduces a framework for studying fluid dynamics using properties of the aforementioned submanifolds. In particular, it is noted that such a submanifold may be equipped with a metric whose signature acts as a diagnostic for the dominance of vorticity and strain. We provide an illustrative example in two dimensions and probe the motivational question: ‘What is a vortex?’ We conclude with comments on extensions to fluid flows in higher dimensions and some open questions.

Teaching

Unless otherwise stated all of the below activities were undertaken at and for the University of Surrey, during the course of my undergraduate and postgraduate studies.

Marking Support

Editorial Support

Computer Lab Assistant

Awards and Funding

Grants

  • STFC PhD Studentship 2021-2024
    Value: ~£56,000 |

    Science and Technology Facilities Council funding for a 3-year PhD, researching the Monge–Ampère Geometry of Vortices. The grant total includes the UKRI annual stipend and a £3000 travel grant.

  • CNRS Fellowship 2024
    Value: ~£4400 |

    A 3-month visiting fellowship granted by the French Embassy in the UK to support Franco-British research collaborations. Laureates receive financial support to visit collaborators at a CNRS laboratory in France, in this case at La Rochelle University, as well as membership to the France Alumni UK network.

  • Institut Für Mathematik Participation Grant 2023
    Value: €200 |

    Funding to attend the Interdisciplinary Junior Scientist Workshop in Mathematical General Relativity in Wildberg, Germany, provided by Universität Tübingen and Universität Potsdam. The grant also included accommodation costs not represented in the value.

  • LMS Early Career Researcher Travel Grant 2022
    Value: £500 |

    London Mathematical Society funding for travel to the Fields Institute, Toronto, Canada to attend a Workshop on Mathematical Relativity, Scalar Curvature, and Synthetic Lorentzian Geometry.

  • Fields Institute Participation Grant 2022
    Value: CAD$1000 |

    Funding from the Fields Institute to cover travel and subsistence expenses while attending their Workshop on Mathematical Relativity, Scalar Curvature, and Synthetic Lorentzian Geometry.

  • LMS Undergraduate Research Bursary 2019
    Value: £1440 |

    London Mathematical Society funded summer research project (URB-18-19-70). Includes matched funding from the University of Surrey.

Awards

  • Surrey Merit Scholarship 2017/18
    Value: £2000 |

    Scholarship awarded to undergraduate students enrolling at the University of Surrey who achieve exceptional A-level grades (A*AA or above). The scholarship also included free 1-year membership to Surrey Sports Park. See the University Webpage for more details.

  • Mathematics Department Prize for Excellence 2017-2021
    Value £400 |

    Four times winner of the Mathematics Department Prize for Excellence at the University of Surrey, awarded for the best performance in each year of an undergraduate/ master's degree.